The Eigencurve

R. Coleman and B. Mazur

Let p be a prime number and C_{p} the completion of an algebraic closure of the field Q_{p} of p-adic numbers. Let N be an integer relatively prime to p. To describe our main object, we assume that p > 2, that the group of units in the ring Z/NZ is of order prime to p, and we restrict our attention (at least in this introduction) to classical modular cuspidal eigenforms f = ∑_{n=1}^{∞} a_{n}q^{n} on Γ_{0}(pN) of weight k ≥ 2, with Fourier coefficients in C_{p}, and normalized so that a_{1} = 1. By “eigenform” let us mean eigenform for the Hecke operators T_{ℓ} for primes ℓ not dividing pN, for the Atkin-Lehner operators U_{q} for primes q dividing pN and for the diamond operators ⟨d⟩ for integers d prime to Np. By the slope of such an eigenform we mean the non-negative rational number σ = ord_{p}(a_{p}), where a_{p} is (both) the p-th Fourier coefficient of f and the U_{p}-eigenvalue of f. Assume that the newform associated to f is either a newform for Γ_{0}(N) or Γ_{0}(pN) (the latter case can occur only if σ = (k − 2)/2).

By the work of Hida (cf. [H-ET] for an exposition of this theory, and for further bibliography given there) one knows that any such eigenform f of weight k ≥ 2 and slope 0 is a member of a p-adic analytic family f_{κ} of (overconvergent) p-adic modular eigenforms of slope 0 parameterized by their p-adic weights κ (and, such that f_{k} = f) for κ ranging through a small p-adic neighborhood of k in (p-adic) weight space. Let us call this result p-adic analytic variation of slope 0 eigenforms. But Hida’s results are, in fact, much more precise: If Λ_{N} := Z_{p}[[(Z/NZ)^{∗} × Z_{p}^{∗}]], Hida constructs a finite flat Λ_{N}-algebra (let us call it T_{p,N}^{o}) which is universal, in a certain sense, for slope 0 (overconvergent) eigenforms of tame level N, and such that the associated rigid analytic space to T_{p,N}^{o} (let us call it C_{p,N}^{o}) is the rigid-analytic space parameterizing p-adic analytic families of slope 0 eigenforms. If W_{N} is weight space, i.e., the rigid analytic space associated to Λ_{N}, then the p-adic families f_{κ} alluded to above are obtained from the finite flat projection C_{p,N}^{o} → W_{N}, Hida having proved that this mapping is étale at any classical modular point of weight ≥ 2 in C_{p,N}^{o}.

The natural question arising from this work of Hida for eigenforms of slope 0 is to find the appropriate generalization of that theory valid for arbitrary finite slope eigenforms. The slope 0 theory has a significant simplifying advantage over the general theory in that there is available a clean idempotent operator (call it e^{o}) projecting to the slope 0 part of the theory. This idempotent e^{o}, a p-adic idempotent reminding one a bit of the “holomorphic projector” in classical analysis, is a bounded operator on the Banach spaces involved, and in fact it has no denominators and therefore acts as an idempotent on all aspects of the theory (e.g., parabolic cohomology, as well as spaces of modular forms); moreover the image of this idempotent is of finite type over whatever is the natural base ring.

In [C-BMF] a satisfactory analogue of Hida’s p-adic analytic variation theorem for slope 0 eigenforms was established for finite slope classical eigenforms (at least for those satisfying a mild condition; cf. Cor. B5.7.1 of [C-BMF]).

The ultimate aim of the theory developed in this article is to provide a more global counterpart to the work done in [C-BMF] and construct a rigid analytic curve C_{p,N} (analogous to C_{p,N}^{o}) which parameterizes all finite slope overconvergent p-adic eigenforms of tame level N, and to study its detailed geometry: in particular, we wish to understand the nature of the projection of C_{p,N} to weight space W_{N}.

For reasons of space, and time, we do this only in the case of p > 2, and tame level N = 1 in the present article. We do, however, treat noncuspidal eigenforms as well as eigenforms on Γ_{1}(p^{m}), as opposed to the more restricted class of eigenforms delineated at the beginning of this introduction. For more precise, yet still introductory, statements concerning our main results, the reader might turn to section 1.3 (and in particular to the Theorems A,B,C,D,E,F,G formulated there). For the rest of this introduction, we suppose that p > 2, and discuss the case of tame level N = 1.

We construct a rigid analytic curve C_{p} = C_{p,N=1} over Q_{p} whose C_{p}- valued points parameterize all finite slope overconvergent p-adic eigenforms of tame level N = 1 with Fourier coefficients in C_{p}. We call C_{p} the (p-adic) eigencurve (of tame level N = 1). Hida’s rigid space C_{p}^{o} = C_{p,N=1}^{o} which parameterizes slope 0 eigenforms of tame level 1 occurs as a component part (cf. section 1.2) of our eigencurve C_{p}, but in contrast to Hida’s theory, the natural projection of C_{p} to weight space is not of finite degree. The eigencurve C_{p} has a natural embedding

c ↦ (ρ_{c}, 1/u_{c})

into the rigid analytic space X_{p} × A^{1} where X_{p} is the rigid analytic space attached to the universal deformation ring R_{p} of certain Galois (pseudo-) representations and A^{1} is the affine line. For a discussion of pseudorepresentations, see Chapter 5 below. The particular residual pseudo-repre-sentations for which R_{p} is the universal deformation ring we call p-modular residual representations (of the Galois group G_{Q,{p,∞}}). These are the residual pseudo-representations coming from the classical modular eigenforms of finite slope and level a power of p (See section 5.1 for their definition.); there are only finitely many such residual pseudo-representations, and R_{p} is a complete semi-local noetherian ring.

If c ∈ C_{p} corresponds to the overconvergent eigenform f_{c}, the first coordinate of c with respect to this embedding is the Galois pseudo-representa-tion r_{c} ∈ X_{p}(C_{p}) attached to the eigenform f_{c} and the second coordinate (1/u_{c}) is the inverse of u_{c} := the U_{p}-eigenvalue of the eigenform f_{c}. We define the eigencurve C_{p} ⊂ X_{p} × A^{1} to be the rigid analytic subspace cut out by an ideal generated by certain specific rigid analytic functions, these functions being Fredholm series over R_{p} (cf. 1.2) obtained by pullback to X_{p}×A^{1} of the characteristic series of certain completely continuous systems of operators (cf. 4.3). In general, a Fredholm series over a complete local noetherian ring R is an entire power series (with constant term 1) in one variable T over R, and we call a Fredholm variety over R (see section 1.2 below) a rigid analytic subspace of X_{p} × A^{1} which is cut out by an ideal generated by a collection of Fredholm series over R (here T is the variable parameterizing the affine line A^{1}).

Let us refer to the nilreduced rigid analytic space subjacent to the eigencurve as the reduced eigencurve C_{p}^{red}. For a brief discussion of the process of passing to the nilreduction of a rigid analytic space, see section 1.2 below. We define irreducible components there and prove under sufficiently general hypotheses that every point in such a space lies on one.

We show that each irreducible component of the reduced eigencurve C_{p}^{red} is isomorphic (at least outside of a discrete set of points) to some reduced irreducible Fredholm hypersurface over the Iwasawa ring Λ (i.e., it is a Fredholm variety defined by a single (irreducible) Fredholm series with coefficients in the Iwasawa algebra Λ) via an isomorphism that preserves projection to weight space. Each irreducible component of a Fredholm hypersurface over Λ is again a Fredholm hypersurface over Λ, and it is seen (Cor. 1.3.12 below) to either be isomorphic to the open complement of a finite set of points in the rigid analytic space attached to a finite flat Λalgebra, or else to be of infinite degree. When (and if) the first case holds, we would be in a situation very analogous to Hida’s theory for slope 0 eigenforms. But so far we have not yet been able to determine for any positive slope irreducible component of C_{p}^{red} which of these two alternatives hold!

It follows from our results that the natural projection of each irreducible component of the reduced eigencurve to weight space is componentwise almost surjective in the sense that its image avoids at most a finite number of C_{p}-valued points in the component of weight space within which it lands; in particular, each irreducible component of C_{p}^{red} has eigenforms of all (but a finite number of) weights. For a more precise statement, see Theorem B of 1.5.

It also follows from our results that any convergent eigenform of finite slope and integral weight (of tame level 1) is overconvergent if and only if its q-expansion is approximable p-adically by the q-expansions of classical eigenforms (of tame level 1). (Note that the q-expansions of Serre’s p-adic modular forms are the limits of the q-expansions of classical modular (but not necessarily eigen) forms.) For a more precise statement, see Theorem G of 1.5.

In Chapter 7, we construct a second rigid space D, by means of the Banach module theory of [C-BMF], which is evidently a curve. We prove the above assertions (in particular, that it is a curve) about C_{p}^{red} by proving them about D and showing that

D ≅ C_{p}^{red}.

From the its construction, one sees that the projection of D to weight space is locally in-the-domain finite flat, meaning that D is covered by admissible affinoid domains U such that the restriction of projection to weight space to U is a finite flat mapping of U onto its image in W.

The image of a rigid analytic morphism can be quite intricate, and, in particular, the projection C_{p}^{red} → X_{p} has an infinite number of double points (e.g., those coming from the classical modular eigenforms of level 1). The image of C_{p}^{red} in X_{p} contains the “infinite ferns” studied in the articles [GM-FM], [GM 3] and [M-IF].

We may pull back the universal pseudo-representation on X_{p} to C_{p}^{red}, via the natural projection, to obtain a rigid analytically varying pseudorepresentation on C_{p}^{red} which is realizable, at least on the complement of a certain discrete set of points on C_{p}^{red} as an O_{C_{p}^{red}}-linear continuous Gal (Q̄/Q)-representation unramified outside p with the property that the restriction of this representation to a C_{p}-valued point c ∈ C_{p} yields the p-adic Galois representation attached to the eigenform f_{c}.

Open questions. Is C_{p} reduced? (I.e., does C_{p} = D?) Is C_{p} smooth? Does C_{p}^{red} have a finite or an infinite number of components? Do any of these components have infinite genus? Is every component of C_{p}^{red} of slope > 0 of infinite degree over weight space, or are there components of positive slope that are of finite degree over weight space? Do there exist p-adic analytic families of overconvergent eigenforms of finite slope parameterized by a punctured disc, and converging, at the puncture, to an overconvergent eigenform of infinite slope? (If so, one would want to complete the eigencurve a bit by including these missing points of infinite slope.) Having constructed the (global) eigencurve allows us to ask more global questions along the lines of conjectures made by one of the authors of the present article, with Fernando Gouvêa, and bears on recent work of Daqing Wan [Wa]: Are the ramification points of the natural projection from C_{p} to weight space infinite in number? We would like to know where those ramification points are; specifically given a point c ∈ C_{p} of weight κ ∈ W let us say that an affinoid subdomain U ⊂ W containing κ is a weight-parameter space for c if there is an affinoid neighborhood V ⊂ C_{p} of c whose natural projection to U is finite étale. Can one find, given any point c corresponding to a classical eigenform of weight k of slope strictly less than k − 1, a weight-parameter space for c of radius greater than the inverse of a linear function of the slope of c? One can prove, using Wan’s results [Wa], that there are such weight-parameter spaces of radius greater than the inverse of a quadratic function of the slope if the slope is strictly less than k − 1 and not equal to (k − 1)/2.

We mentioned above that, excluding a discrete set of points (call this set Δ ⊂ C_{p}^{red}), the reduced eigencurve parametrizes a rigid-analytically varying family of Galois representations, i.e. we have a continuous representation

ρ : Gal (Q̄/Q) → GL_{2}(O_{C_{p}^{red}−Δ})

Can this family of Galois representations ρ be extended over the excluded set Δ? Is there a continuous representation of Gal (Q̄/Q) into the group of units in a rigid Azumaya algebra of rank 4 over O_{C_{p}} which, in the appropriate sense, extends ρ? Can one construct ρ (or at least ρ restricted to an appropriate C_{p}^{red} − Δ) from the cohomology of modular curves in the following sense: Let H be the polynomial algebra over Λ in the countable set of generators denoted T_{ℓ} for prime numbers ℓ ≠ p and an extra generator denoted U_{p}. For a more complete discussion of this Λ-algebra H see Chapter 6. There is a natural homomorphism of H to the ring O_{C_{p}} of rigid analytic functions on the eigencurve; in particular, in the discussion below we view O_{C_{p}} as H-algebra. Define the discrete abelian group

M := lim_{⟶} H^{1}(X_{0}(p^{n}); Q_{p}/Z_{p}),

the direct limit of étale cohomology groups of the modular curves X_{0}(p^{n}), this limit being compiled via the mappings on cohomology induced from the natural projections X_{0}(p^{n+1}) → X_{0}(p^{n}) for all n ≥ 1 (in the present brief description we omit saying which natural projections these are). The Galois group G_{Q,{p,∞}} acts on this direct limit in the natural way, as does the algebra H (each of the generators acting as the corresponding Hecke or Atkin-Lehner operator) and the actions of H and G_{Q,{p,∞}} commute, allowing us to view M as an H[[G_{Q,{p,∞}}]]-module. Let M^{∗} := Hom(M; Q_{p}/Z_{p}) be the pontrjagin dual of M, viewed as compact H[[G_{Q,{p,∞}}]]-module. There are various (possibly equivalent) ways of spreading the cohomology H-module M^{∗} over the eigencurve. The simplest way to do this is to form V := the completed tensor product of the H-module M^{∗} with the H-algebra O_{C_{p}}. Viewing V as quasi-coherent sheaf over the eigencurve, the G_{Q,{p,∞}}-action on M^{∗}, which commutes with the action of H, induces a O_{C_{p}}-linear action on V. Is it the case that, over C_{p}, or possibly just over some large portion of C_{p}, the quasi-coherent sheaf V is locally free of rank 2, and the G_{Q,{p,∞}}- representation on it is equivalent to the family of representations ρ discussed above? On the converse side of these issues, it is natural to ask whether it is the case that every irreducible 2-dimensional C_{p}-representation of G_{Q,{p,∞}} which occurs as a quotient representation space of M^{∗} ⊗_{H} H[1/U_{p}] is the Galois representation associated to an overconvergent eigenform (of slope 0, and tame level N = 1) or are there more finite-slope Galois representations in the cohomology module M^{∗} than are accounted for among the ones associated to overconvergent eigenforms? All these questions have analogues when the cohomology group

M := lim_{⟶} H^{1}(X_{0}(p^{n}); Q_{p}/Z_{p})

is replaced by (direct limits of) parabolic cohomology of higher weights, and one might ask for a theory, following Hida’s work, which deals with all weights.

Does Glenn Stevens’ construction of the p-adic L-function of a (p-adic) eigenform on Γ_{0}(pN) (cf. [St] ) work well over the eigencurve, and, in particular, does it give an L-function (with the coefficients of its Taylor expansion rigid analytic functions on C_{p}) which interpolates all the classical p-adic Lfunctions? We hope to show in a later publication, that at least a weak version of this is true. The locus of zeroes of this L-function, once constructed, would itself be a rigid-analytic curve (call it L_{p}) admitting a natural rigid analytic projection to C_{p}. This deserves study. particularly intriguing is the nature of this projection in the neighborhood of a double-zero of the L-function.

It would be interesting to construct a local version of the eigencurve using the theory of crystalline Galois representations, in the following sense. Let X now refer to the rigid-analytic universal deformation space of a fixed absolutely irreducible representation of the Galois group of Q_{p} into GL_{2}(F_{p}). It is known that X is a smooth 5-dimensional rigid-analytic ball. By a crystalline point in (the 6-dimensional) rigid-analytic space X × A^{1} − {0}, let us mean a pair (x, u) where x ∈ X classifies a crystalline representation, and u ∈ A^{1} − {0} is one of its the Frobenius eigenvalues in the Fontaine-Dieudonné module attached to the crystalline representation x. (For a general overview of crystalline, semi-stable, and potentially semistable representations, see [F].) Is it the case that the rigid-analytic closure of the set of crystalline points in X × A^{1} − {0} is a 3-dimensional rigidanalytic subvariety ? Is it a rigid analytic subspace of affine three-space?

There are also certain foundational questions which deserve much better understanding than we have, at present. For example, one still lacks a satisfactory conceptual definition of what it means for a modular form (of general p-adic weight) to be overconvergent. This notion is clear for integral weight, but at present, for general weight, we have only an ad hoc procedure based on the availability of families of Eisenstein series of general weight (See Chapter 2 below: you multiply your modular form by such an Eisenstein series to get the product to be of weight 0 and then ask that this be overconvergent as a rigid analytic function on a suitable affinoid in the modular curve). As a consequence of this awkward strategy, treatment of any serious property about overconvergent modular forms depends upon our detailed understanding of the family of Eisenstein series. It would be good to

a. have a more direct definition of overconvergent modular forms and of families of overconvergent modular forms, and at the same time

b. have a closer understanding of the properties of families of Eisenstein series (including knowledge of their zero-free regions).

An improvement in our state of knowledge of b would help in our understanding of the detailed geometry of the eigencurve.

It would be important, as well, to have some detailed computations of specific affinoid subdomains of C_{p}. In this regard, see forthcoming work of Matthew Emerton who (augmenting earlier calculations of Coleman and Teitelbaum) gives a complete description of the geometry of that part of the eigencurve for p = 2 and tame level 1 having minimal slope for their weight (cf. [Em]). See also [CTS] where some results on the low slope part of the 3adic eigencurve are proven. We might mention here that in these examples (p = 2, 3), the slope tends to zero as one approaches the “boundary” of weight space. Are there components of the p-adic eigencurve (for some p) where this phenomenon does not occur?

To be sure, a satisfactory general theory of the eigencurve must deal with all tame levels N (and as a prelude for this, one necessary task, which we carry out, is to set up the deformation theory of pseudo-representations with fixed tame level). In the present paper, although we make the construction of the eigencurve only for N = 1, in some sections (where it is easy to do so) we work with more general level N in preparation for the general theory. Our conventions concerning level will be signaled at the start of each section.

Is there an a priori deformation-theoretical approach to the eigencurve, and to the Galois representations that the eigencurve parameterizes? (We explain more precisely what we mean by this at the end of Chapter 1.) Our lack of such an approach accounts for some of the difficulty we have in analyzing local properties of the eigencurve. Is there a natural formal scheme over Z_{p} whose associated rigid analytic space is the eigencurve? One is, in any event, guaranteed (by the preprint [LvP]) that any connected onedimensional, separated, rigid analytic space over Q_{p} is the generic fiber of some formal scheme which is flat over Z_{p}. It might be interesting to study irreducible components of the closed fiber of a formal scheme whose generic fiber is (a piece of) the eigencurve. In this connection, J. Teitelbaum has a computer program that produces such irreducible components in a range of slopes (for p = 2, 3).

In a later publication we hope to present more foundational material regarding the connection between Katz modular function and convergent eigenforms (including the proof of the compatibility of the action of the diamond operators, i.e., the proof of Prop. 3.4.2 below which we omitted from this article). We are deeply indebted to Kevin Buzzard for his extremely helpful comments throughout the preparation of this article. We also wish to thank Brian Conrad for his helpful suggestions on Chapter 1 and Matthew Emerton for his close reading of an early draft.

Table of Contents.

1. Rigid analytic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1 Rigid analytic spaces attached to complete local noetherian rings. 1.2 Irreducible components and component parts. 1.3 Fredholm varieties. 1.4 Weight space. 1.5 The eigencurve as the Fredholm closure of the classical modular

locus. (Statement of the main theorems.)

2. Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1 Affinoid sub-domains of modular curves. 2.2 Eisenstein series. 2.3 Katz p-adic modular functions. 2.4 Convergent modular forms and Katz modular functions.

3. Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 Hecke eigenvectors and generalized eigenvectors. 3.2 Action on M_{k}(Np^{m}, v; K). 3.3 Action on Katz Modular Functions. 3.4 Action on M^{†}(N). 3.5 Action on weight κ forms. 3.6 Remarks about cusp forms and Eisenstein series.

4. Fredholm determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1 Completely continuous operators and Fredholm determinants. 4.2. Factoring characteristic series. 4.3. Analytic variation of the Fredholm determinant. 4.4. The Spectral Curves.

5. Galois representations and pseudo-representations . . . . . . 68 5.1. Deforming representations and pseudo-representations. 5.2. Pseudo-representations attached to Katz modular functions.

6. The eigencurve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.1 The definition of the eigencurve. 6.2. The points of the eigencurve are overconvergent eigenforms. 6.3. The projection of the eigencurve to the spectral curves. 6.4 The Eisenstein curve.

7. The eigencurve constructed as finite cover of spectral curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.1. Local pieces. 7.2. Gluing. 7.3. The relationships among the curves D_{α}. 7.4. D is reduced. 7.5. Equality of D and C^{red}. 7.6. Consequences of the relationship between D and C.

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Chapter 1. Rigid analytic varieties.

1.1 Rigid analytic spaces attached to complete local noetherian rings.

Let p be a prime number, and C_{p} the completion of a fixed algebraic closure of Q_{p}. Let v be the continuous valuation on C_{p} such that v(p) = 1. Define the absolute value | | by |a| = p^{−v(a)}, for a ∈ C_{p}. By the ring of integers O_{K} in a complete subfield K of C_{p} we mean the set of elements of absolute value at most 1. For a rigid analytic variety Y (see §9.3 of [BGR]), A(Y) will denote the ring of rigid functions on Y, for a ∈ A(Y), |a| will denote the spectral semi-norm of a (which may be infinite valued) and A^{0}(Y) will denote the sub-O_{K}-algebra of rigid functions with spectral semi-norm at most 1. For an affinoid algebra B, we let Max(B) denote the corresponding rigid variety over K.

Now let R = O_{K} be the ring of integers in a fixed finite extension K of Q_{p} in C_{p}. Let k denote the residue field of R. Let A be a complete local noetherian R-algebra with maximal ideal m_{A} and residue field A/m_{A} = k. Consider the functor A ⟶ X_{A} which attaches to each such complete noetherian local ring, its associated rigid analytic space over K. We refer to section 7 of [de J] for the construction, and for its basic properties (cf. loc. cit. Definition 7.1.3, 7.1.4a, and 7.1.5). Briefly, the construction may be given as follows: For each real number r > 0, let A_{r} be the p-adic completion of the quotient by its p-power torsion, of the ring

A[{y_{r,(a)}}]

where (a) ranges over unordered tuples of elements in m_{A} subject to the relations

p^{[k(a)r]}y_{r,(a)} = ∏(a),

where k(a) is the length of (a) and the product is over the entries of (a). (Note: A_{r} = 0 if r is sufficiently large.) If r ≥ s > 0 there is a natural Ahomomorphism from A_{s} to A_{r} determined by y_{r,(a)} ↦ p^{[k(a)r]−[k(a)s]}y_{s,(a)}. If r is rational, this ring is finitely generated over A. Indeed, if {t_{1}, …, t_{n}} generate m_{A} and r = g/h with g, h positive integers, then A_{r} is generated by y_{(a)} where the entries of (a) are in {t_{1}, …, t_{n}} and the length of (a) is at most h, because in this case, m_{A}^{h} ⊆ p^{g}A_{r}. It follows that the admissible formal schemes, in the sense of [BL-FRI&II], SpfA_{r}, where r is rational, glue together into an admissible formal scheme over R and we can make a rigid space X_{A} out of this formal scheme as in [BL-FRI&II] covered by the affinoids X_{r} := Max(A_{r} ⊗ K). In a word, X_{A} is the union of the affinoids X_{r}’s for all positive rational r. In particular, the C_{p}-valued points of X_{A} are given by (continuous) homomorphisms of the R-algebra A into C_{p} (or, equivalently, into O_{C_{p}}):

X_{A}(C_{p}) = Hom_{R}(A, C_{p}),

and we have a natural ring homomorphism A → A^{0}(X_{A}). To a complete Noetherian semi-local ring, we attach the union of the rigid spaces attached to each of its localizations at maximal ideals. An important special case for us is A = Z_{p}[[T]] (here R = Z_{p}, and K = Q_{p}). For r ≤ 1, A_{r} is the ring of rigid functions defined over Q_{p} and bounded by 1 on the disk B[0, p^{−r}] = {x ∈ C_{p} | |x| ≤ p^{−r}} in C_{p}, and X_{A} is the open unit disk, B(0, 1), viewed as rigid analytic space over Q_{p}. More generally, if one takes A to be the ring of rigid analytic functions bounded by 1 on a residue class (formal fiber) in an affinoid (see [Bo] ), X_{A} will be canonically isomorphic to that residue class. All of the rigid spaces we shall be considering in this article are separated and have a countable admissible covering by affinoid subsets. Following the terminology of [JP] one says that a rigid space X over a complete non-archimedean valued field K is paracompact if it admits an admissible covering by affinoid subsets, any one of which meets only finitely many others. If a rigid space X over a complete non-archimedean valued field K admits an admissible covering by a countable increasing sequence of affinoid subdomains X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{i} ⊂ ⋯ ⊂ X, let us call X nested. For any complete noetherian ring R, its associated rigid space X_{R} described above is nested, as is X_{R} ×A^{1}, the product of X_{R} with the affine line, which will be introduced in the next section. According to [LvP], any connected, separated, one-dimensional rigid space X over a complete nonarchimedean valued field K is paracompact (this is no longer the case in higher dimensions), and is the generic fiber of a formal scheme which is flat over Spf(O_{K}), where O_{K} is the ring of integers of K. As a general reference for the relationship between formal schemes and rigid analytic spaces, the reader might consult [L], [BL-FRI&II], [JP], [LvP], and pp. 429-436 of [Ra].

1.2. Irreducible components and component parts.

If ι: S → Z is a morphism of rigid spaces, ι is said to be a closed immersion and Z is said to be a Zariski-closed analytic subvariety (via ι) if there exists an admissible affinoid covering {U_{i}}_{i∈I} of Z such that for all i ∈ I the induced map ι: ι^{−1}(U_{i}) → U_{i} is a closed immersion of affinoid varieties [BGR §9.5.3]. It follows that for any open affinoid subvariety U of X, ι: ι^{−1}(U) → U is a closed immersion of affinoid varieties [BGR Prop. 9.5.3/2]. If S is a Zariski-closed analytic subvariety of Z, we shall refer to S, for short, as Zariski-closed and note that (since affinoids are noetherian) the category of Zariski-closed subspaces of a given rigid space Z enjoys the local descending chain condition: any infinite descending chain of Zariskiclosed subspaces of Z stabilizes finitely when restricted to any given affinoid in Z. If ι: S → X is a closed analytic subvariety of X we let the interior of S be the rigid subspace which is the union of affinoids of the form ι^{−1}(U) where U is an open affinoid subvariety of X and ι: ι^{−1}(U) → U is an isomorphism. The rigid Zariski-closure of a morphism ι: S → Z is the minimal element in the category of closed immersions W → Z through which ι factors. The fact that the rigid Zariski-closure of S exists follows from the fact that if S → Z factors through W → Z and W′ → Z it factors through W ×_{Z} W′ → Z, and from the local descending chain condition described above. If S ⊆ X_{A} ×_{K} A_{K}^{1}(C_{p}), consider the rigid space over C_{p} which is the disjoint union of the points S (we denote this space by the same letter S). We define the the rigid Zariski closure of the set of points S to be the rigid Zariski closure of the morphism i:S → X.

If X is a rigid analytic space over K, by the nilreduction of X, denoted X^{red} (cf. page 389 of [BGR]) we mean the unique rigid analytic subspace of X over K (constructed in loc. cit.) whose admissible affinoid subdomains U^{red} = Max(O_{U^{red}}) are in one:one correspondence with the admissible affinoid subdomains U = Max(O_{U}) of X, where O_{U^{red}} is the quotient of O_{U} by its nilradical. The natural closed immersion X^{red} ⊂ X induces a one:one correspondence on sets of C_{p}-valued points. One says that X is nilreduced (or reduced) if X^{red} = X.

Definition. Suppose Z is a rigid space. Then by a component part of Z we mean a (non-empty) rigid subspace of Z which is the rigid Zariski-closure in Z of an admissible open of Z. We will say that Z is irreducible if it has only one component part (i.e., itself). An irreducible component of Z is a component part which is itself irreducible.

Notes. It may be worth explicitly pointing out that the definition of irreducible above ignores nilpotent elements, and in particular (as in standard algebraic geometric usage) our definition allows the possibility that irreducible components be nonreduced; moreover, embedded components are simply not counted. As we’ll see, in Corollary 1.2.1/1, if Z is an affinoid, Z possesses only a finite number of distinct irreducible component parts, and any component part of Z may be viewed at least set-theoretically as a (finite, of course) union of irreducible component parts. Let Φ(Z) denote the set of irreducible component parts of Z. One should note that Φ(−) is not in general functorial in Z (e.g., consider the case of Z = a single point mapping to the intersection of two irreducible components of a rigid analytic space Z′). Nevertheless, rigid analytic morphisms ι : Z → Z′ which embed Z as an open subdomain in Z′ induce natural mappings Φ(ι) : Φ(Z) → Φ(Z′) and we shall make use of this in what follows. Brian Conrad and Ofer Gabber have independently proposed another definition of irreducible components of X when X is reduced. One first passes to the normalization of X (which one has to prove is a well defined rigid space), takes connected components and then defines the irreducible components of X to be the images of these components. Using this definition one can show that irreducible components have the right localization properties. Conrad has established that, in this case, these irreducible components are the same as ours.

Lemma 1.2.1. Suppose X is an affinoid. Then X is irreducible if and only if the map A(X) → A(X)_{x} is an injection for all closed points x of X. (Here A(X)_{x} denotes the localization of A(X) at x.)

Proof. Suppose 0 ≠ f ∈ A(X) such that f ↦ 0 in A(X)_{x} for some x ∈ MaxA(X). Then there exists an admissible open neighborhood U of x in X such that f ↦ 0 in A(U). It follows that f maps to 0 in the ring of functions on the Zariski-closure Z of U and hence Z ≠ X. Now suppose that A(X) injects into A(X)_{x} for all x and Z is a component part of X equal to the Zariski-closure of an admissible open U of X. Then by Proposition 9.5.3/1 of [BGR] Z is an affinoid of the form MaxA(X)/I for some ideal I of A(X). It follows that I maps to zero in A(X)_{x} for any closed point of U and so I = 0 and Z = X.

Corollary 1.2.1/1. (i) If X is an affinoid, the irreducible components of X are affinoids Z_{I} := MaxA(X)/I where I is a maximal ideal among the set S of ideals which map to zero in A(X)_{x} for some closed point x. (ii) (In particular:) If A(X) is an integral domain X is irreducible. (iii) The number of irreducible components X is equal the number of minimal prime ideals of A(X).

Proof. (i) It follows from the previous lemma that if I satisfies the above hypotheses then Z_{I} is irreducible. Conversely, suppose Z_{I} is an irreducible component part of X for some ideal I of A(X) and U is an admissible open of X contained in Z_{I}. Then I maps to zero in A(X)_{x} for all closed points x of U so I ∈ S. If I ⊂ J where J ∈ S, then Z_{J} would be a component part of Z_{I} equal to Z_{I} if and only if I = J. (ii) follows from (i). (iii) It is clear that if I ∈ S then I is contained in some minimal prime ideal of A(X). Now suppose P is a minimal prime ideal of A(X). Let I_{P} be the set of f ∈ A(X) such that f ↦ 0 in A(X)_{x} for some closed point x of Z_{P}. Then I_{P} is the ideal of all f ∈ A(X) such that there exists a t ∉ P such that tf = 0. The ideal I_{P} is contained in P and contains a power of P. It follows that I_{P} is not contained in any other minimal prime ideal. From its definition we see that I_{P} is a maximal element of S contained in P. It follows that I_{P} is a maximal element of S and the map P ↦ Z_{I_{P}} is a one to one correspondence between the minimal prime ideals of A(X) and the irreducible components of X.

We say an affinoid X is equidimensional if the Krull dimension of O_{X,x}, where x is a closed point of X is independent of x. If this happens, this number equals the Krull dimension of A(X).

Lemma 1.2.2. Suppose X is a reduced equidimensional affinoid. Then the singular locus of X has codimension at least 1 in X.

Proof. The proof of this follows the lines of the classical proof in algebraic geometry (see [Ha] Chapter 1, section 5). First, we can express A(X) as the quotient of K⟨T_{1}, …, T_{n}⟩ by an ideal generated by elements f_{1}, …, f_{m} for some m and n. If X has dimension r, the singular locus of X is the set of points where the rank of the corresponding Jacobian matrix has rank < n − r so is a Zariski closed subaffinoid. To show it has codimension at least 1 one show that the non-singular locus on each irreducible component Z of X is non-empty. By rigid Noether normalization, Corollary 6.1.3/2 of [BGR], we can find a find a finite map ϕ from Z onto B^{r}. Then the singular locus of Z is contained in the set of ramification points of ϕ. Let F be the fraction field of A := A(B^{r}). Then since A(Z) is an integral domain and Z → B^{r} is finite and surjective, A(Z) injects into A(Z) ⊗_{A} F which is a finite extension of F. It follows that Z_{U} is flat over the complement U of the zeroes of a non-zero element in A(B^{r}) and hence the discimininant ideal D in A(U) of Z_{U}/U is defined and non-zero. Moreover, the ramification points of ϕ in Z_{U} lie over the zeroes of D. Thus the non-sinular locus of Z maps onto a non-empty open of B^{r}.

Proposition 1.2.3. Suppose X is a reduced affinoid of equidimension n. If I is an ideal of A(X) and V = Max {A(X)/I}, then V is a component part of X if and only if V is reduced and the Krull dimension of O_{V,v} equals n for all closed points of V.

Proof. If V is a component part, I = ⋂_{J∈S} J where S is a set of minimal prime ideals. It follows that V is reduced and the Krull dimension of O_{V,v} equals n for all closed points v of V. Suppose V satisfies the latter hypotheses. Let Y be the maximal component part of X contained in V. Suppose Y ≠ V. Let J be the ideal corresponding to Y. Since V is reduced it follows that there exist an f ∈ J and a v_{0} ∈ V such that f doesn’t vanish at v_{0}. We can assume f(v_{0}) = 1. Let B be the subdomain of V defined by

{v ∈ V: |f(v) − 1| ≤ |p|}.

Then B ∩ Y = ∅. Now If v is point of V which is smooth in X then V contains an affinoid neighborhood of v in X and hence V contains the component part of X which is the Zariski closure of this neighborhood. Hence B is contained in the singular locus of X. Since this is a subaffinoid of codimension at least one, it follows that the Krull dimension of V at v_{0} which equals the Krull dimension of B at v_{0} is at most dimension n − 1.

Proposition 1.2.4. Suppose X is a reduced affinoid and Y is an affinoid subdomain. Then if Z is a component part of X, Z ∩ Y is a component part of Y.

Proof. This follows in the equidimensional case, from the fact that Y is reduced by Corollary 7.3.2/8 of [BGR] and Prop. 1.2.3. The general case of the lemma follows from the equidimensional case and the fact that X is the union of equidimensional component parts. Indeed, if Z is a component part of X and V is the equidimensional part of X of dimension n, the part of Z of equidimension n is a component part of V and we can apply the equidimensional part of the proposition to V, Y ∩ V and Z ∩ V.

Ofer Gabber, using the normalization definition of irreducible component has proved that arbitrary rigid spaces can be covered by irreducible components. However, the following result is sufficient for our purposes:

Proposition 1.2.5. Suppose X is a reduced rigid analytic space which has an admissible open cover by of affinoids {X_{i}: i ∈ Ω}, where Ω is some index set such that X_{ij} := X_{i} ∩ X_{j} is an affinoid for each i and j. Then every closed point of X lies on an irreducible component of X.

Proof. This proposition is true if X itself is a reduced affinoid (e.g., as can be seen from Cor. 1.2.1/1). In particular (fixing on some index 0 ∈ Ω) Proposition 1.2.5 is true for the affinoid subdomain X_{0} ⊂ X. It will then follow from

Lemma 1.2.6. Let A_{0} be an irreducible component part of X_{0}. Then A_{0} is contained in a (unique) irreducible component of X.

Proof. We will define, inductively, a collection of component parts, A_{n,i} of the affinoid subdomain X_{i} for pairs of integers i ≥ 0 and n ≥ 0 . We start the induction with A_{0,0} := A_{0} which is an irreducible component part of X_{0} and A_{0,i} = ∅ for i ≠ 0. Suppose, for a given integer n ≥ 0 we have defined a component part A_{n,i} of X_{i} for all i ∈ Z. We proceed to define the component parts A_{n+1,i} of X_{i}. Put Y_{n,j,k} := X_{j,k} ∩ A_{n,k}, which is a component part of X_{j,k} by Proposition 1.2.4. Let A_{n+1,j} be the union over k of the Zariski closures in X_{j} of Y_{n,j,k} (there are only a finite number of these, as each of them is a component part of X_{j}). For each j these A_{n,j} stabilize for large n since they are all component parts and A_{n+1,i} ⊇ A_{n,i}. Let Z be the union of the A_{n,i} over all n and i. Then (using Prop. 1.2.4) Z is a rigid subspace of X and a component part by definition. We claim that Z is irreducible. Suppose U is any non-empty affinoid subdomain contained in Z. We will show that if Ū is the rigid Zariski closure of U in X, then Ū = Z. We may suppose that U ⊂ A_{n,i} for some n and i. Let V be the rigid Zariski closure in X_{i} of U. Then V ⊆ A_{n,i}. It follows by construction that if n > 0 there exists a j so that A_{n−1,j} ∩ V contains a non-empty affinoid subdomain of X_{i}. Since this is also contained in Z we can repeat this process inductively and conclude that Ū contains a non-empty affinoid subdomain of A_{0,j} for some j ≥ 0. Therefore Ū contains A_{0,0} = A_{0}. In particular, Ū contains the union of the A_{0,j} for all j ≥ 0. Now, working up the inductive definition of A_{n,i} over all n, one sees that Y_{n,j,k} ⊂ Ū for all n, j, k and consequently, Ū = Z.

1.3. Fredholm varieties.

If A is a topological ring, recall that A⟨T ⟩ ⊂ A[[T]] is the set of power series whose coefficients tend to 0 in A. Let R = O_{K} ⊂ K where K is a complete subfield of C_{p}, as in the previous section. For any integer m, the closed disk of radius p^{m} about the origin; that is, the rigid analytic subspace of A_{K}^{1} whose C_{p}-valued points consist of {x ∈ C_{p} | |x| ≤ p^{m}} will be denoted, as usual, B[0, p^{m}]_{K} = Max(O_{K}⟨p^{m}T ⟩). We have A_{K}^{1} = ∪_{m=1}^{∞}B[0, p^{m}]_{K}. If X is a rigid space over K and F (T) is a power series with coefficients in A(X) we will say F (T) is entire over X if it is the series associated to a rigid and analytic function on X ×_{K} A_{K}^{1}. If X is a reduced affinoid, an equivalent formulation is: F (T) is entire over X if and only if the power series F (T) lies in (the intersection of) the subrings A(X)⟨p^{m}T ⟩ ⊂ A(X)[[T]] for all natural numbers m. This is so because X ×_{K} A_{K}^{1} = ∪_{m=1}^{∞} X ×_{K} B[0, p^{m}]_{K}. The set of entire power series over X, then, forms an A(X)-subalgebra of the ring of power series A(X)[[T]], denoted A(X){{T }} ⊂ A[[T]].

If A is a local ring with maximal ideal m_{A}, and P (T) = ∑_{n=1}^{∞} a_{n}T^{n} is a power series with coefficients a_{n} ∈ A, we will say that P (T) is entire over the ring A if the coefficient a_{n} is in m_{A}^{c_{n}}, where c_{n} are integers such than the sequence of real numbers {c_{n}/n} tends to ∞. The set of entire power series (in T) over a local ring A forms an A-subalgebra of the ring of power series A[[T]] which we denote A{{T }} ⊂ A[[T]]. For example, if A is a field, then A{{T }} = A[T], the ring of polynomials in T over A. If A is a complete noetherian local R-algebra with finite residue field, where R is the ring of integers in a finite extension K of Q_{p}, and if X_{A} is the rigid space over K associated to A, then we have a natural A-algebra homomorphism

A{{T }} → A(X_{A}){{T }}.

Definition. If A is a local ring, a Fredholm series over A is an entire power series over A with constant term 1. If X is a rigid space, a Fredholm series over X is an entire power series over A(X) with constant term 1. By the Fredholm function associated to a Fredholm series P (T) over A(X) we mean the rigid analytic function on X ×A^{1} determined by P (T) (i.e., the image of P (T) under the natural homomorphism A{{T }} → A(X_{A}){{T }}). If A is a complete noetherian local R-algebra with finite residue field, where R is the ring of integers in a finite extension K of Q_{p}, and if X_{A} is the rigid space over K associated to A, an A-Fredholm hypersurface in X_{A}×_{K}A_{K}^{1} is the rigid analytic subvariety cut out by a Fredholm series over A. An A-Fredholm subvariety in X_{A}×A^{1} is a rigid analytic subvariety of X_{A}×A^{1} cut out by a non-empty (possibly infinite) collection of A-Fredholm series. If S is a subset of the C_{p}-valued points of (X_{A}×A^{1})(C_{p}), by the Fredholm closure of S we mean the smallest Fredholm subvariety of X_{A} × A^{1} whose C_{p}-valued points contain S.

Note. We extend the above notions to complete semi-local noetherian rings A in the evident manner. For example, if A is a complete semi-local noetherian ring, a power series P (T) ∈ A[[T]] is a Fredholm series over A if the image of P (T) in A_{m}[[T]] is a Fredholm series, where m ranges through the maximal ideals of A, and A_{m} is the completion of A at m. Fredholm subvarieties are contained in the complement of X_{A} ×_{K} (0, 1) in X_{A} ×_{K} A_{K}^{1}. Inversion of the second coordinate in X_{A}×A^{1} therefore sends Fredholm subvarieties to closed, rigid analytic subvarieties of X_{A} ×_{K} B[0, 1]\{0}, where B[0, 1]\{0} is the punctured closed unit disk. The rigid Zariski-closure of a set of C_{p}-valued points S ⊆ X_{A} ×_{K} A_{K}^{1}(C_{p}) is a reduced rigid analytic space and therefore the rigid Zariski-closure of S factors through the nilreduction of the the Fredholm closure of S. We do not, at present, know whether it is generally true that the rigid Zariski-closure of S is equal to this nilreduction.

Lemma 1.3.2. Suppose X is a reduced affinoid.

a. The only invertible Fredholm series H(T) over X is the constant series H(T) ≡ 1.

b. For any positive integer n, put Y_{n} = X ×_{K} B[0, p^{n}] ⊂ Y = X ×_{K} A_{K}^{1}. Let G(T), F (T) be two Fredholm series over X with the property that for each n, if G_{n}(T), F_{n}(T) denote the restrictions of these Fredholm series to Y_{n}, we have that G_{n}(T) divides F_{n}(T) in the ring of rigid analytic functions on Y_{n}. Then there is a Fredholm series H(T) over X such that

F (T) = G(T)H(T).

Proof. a. We may assume X ≠ ∅. Let H(T) = 1 + a_{1}T + ⋯ + a_{n}T + ⋯ be an entire series over A and suppose there exists a k > 0 such that a_{k} ≠ 0. Then since X is reduced there exists a closed point x of X(L) for some finite extension L of K such that a_{k}(x) ≠ 0. Thus we can assume that {x} = X and so A(X) = L in which case the lemma is obvious.

b. For each n ≥ 1 let H_{n}(T) denote the rigid analytic function on Y_{n} such that F_{n}(T) = G_{n}(T)H_{n}(T), and note that H_{n}(T) is uniquely determined by this property since G_{n}(T), having constant term 1, cannot be a zero divisor. This system of functions {H_{n}}_{n} is compatible under restriction and patches together to give us a Fredholm series H(T) with the required properties.

Lemma 1.3.3. Suppose X is a reduced affinoid over K. For any positive integer n, put Y_{n} = X ×_{K} B[0, p^{n}], as before. Let Z → X ×_{K} A_{K}^{1} be a closed immersion such that for each n its intersection with Y_{n}, Z_{n} := Z ×_{K} Y_{n}, is the affinoid cut out by some rigid analytic function g_{n}(T) ∈ A(X)⟨p^{n}T ⟩ on Y_{n} with g_{n}(0) = 1. Then there is a Fredholm series G(T) over X such that for n ≥ 1,

lim_{n ≤ n′→∞} g_{n′}(T)|_{Y_{n}} = G(T)|_{Y_{n}},

where Z is the subvariety cut out by G(T).

Proof. Our hypotheses imply that we may write

g_{m+1}(T)|_{Y_{m}} = g_{m}(T)h_{m}(T)

where h_{m}(T) ∈ A(Y_{m})^{∗} and h_{m}(0) = 1. Write

h_{m}(T) = ∑_{k=0}^{∞} a_{m,k}T^{k}

where a_{m,k} ∈ A(X). The fact that h_{m}(T) is a unit on Y_{m} and h(0) = 1 implies

|a_{m,k}| < p^{−mk} for k > 0.

It follows, since X is reduced and so the spectral norm determines the topology on A(X), that the product

∏_{m≥n} h_{m}(T)

converges to a unit on Y_{n} for all n ≥ 1. This concludes the proof.

Definition. We say that an affinoid algebra A over K is relatively factorial if every F (T) ∈ A⟨T ⟩ with constant term 1 (i.e., F (0) = 1) can be factored into a (finite) product of primes (i.e., elements which generate prime ideals).

We call an affinoid X over K relatively factorial if the affinoid algebra A(X) is. It follows that such a factorization is unique up to units. (The point is that no element in A⟨T ⟩ with constant term 1 is a zero divisor.) The Tate algebra K⟨T_{1}, …, T_{n}⟩ is an example of a relatively factorial (in fact, factorial) algebra by Theorem 5.2.6/1 of [BGR].

Proposition 1.3.4. Suppose X is a relatively factorial affinoid over K such that A(X) is an integral domain and F (T) is a Fredholm series over A(X) which cuts out the (Fredholm) hypersurface Z. Then every component part W of Z is a Fredholm hypersurface whose defining equation is a divisor of F (T) in the ring A(X){{T }}.

Proof. Let V = X ×_{K} A_{K}^{1} and, as before, for positive integers n let Y_{n} = X ×_{K} B[0, p^{n}] and Z_{n} = Z ×_{V} Y_{n}. Suppose U is an admissible open in Z whose rigid Zariski-closure is W. Let C_{n} be the Zariski-closure of U ×_{V} Y_{n} in Y_{n}. Then ∪_{n}C_{n} = W and

W ×_{V} Y_{n} = ⋃_{n′≥n} C_{n′} ×_{V} Y_{n} =: W_{n} ⊂ Z_{n}.

Now because X is relatively factorial, we can write

F (T)|_{Y_{n}} = ∏_{Z∈Φ(Z_{n})} s_{Z}(T)

where the product is taken over all the irreducible component parts Z of Z_{n}, and s_{Z}(T) is a rigid analytic function on Y_{n} (with the property that s_{Z}(0) = 1) defining the irreducible component part Z. Because U ×_{V} Y_{n} is an admissible open of Z_{n} = Z ×_{V} Y_{n} and A(X) is an integral domain, there exists a collection {V_{n,m}} of irreducible reduced affinoid subdomains of Y_{n} such that U ×_{V} V_{n,m} is the affinoid cut out by F (T) restricted to V_{n,m} and the collection {U ×_{V} V_{n,m}} is an admissible open cover of U ×_{V} Y_{n} (for m ranging through an appropriate index set, call it M).

Put

h_{n,m}(T) = ∏_{Z} s_{Z}(T),

where the product is taken over those irreducible component parts Z ∈ Φ(Z_{n}) which are in the image of the natural mapping

Φ(U ×_{V} V_{n,m}) → Φ(Z_{n}),

there being such a natural mapping (as discussed in the note above) since U ×_{V} V_{n,m} is a subdomain of Z_{n}. Then U ×_{V} V_{n,m} is the affinoid in V_{n,m} cut out by the restriction of h_{n,m}(T) to V_{n,m}. An easy argument gives that the map A(Y_{n})/h_{n,m}(T)A(Y_{n}) to A(V_{n,m})/h_{n,m}(T)A(V_{n,m}) is an injection from which it follows that the Zariski-closure of U ×_{V} V_{n,m} in Y_{n} is the affinoid cut out by h_{n,m}(T). Therefore C_{n} is the affinoid in Y_{n} cut out by

k_{n}(T) := ∏_{Z} s_{Z}(T),

where the product is now taken over those irreducible component parts Z ∈ Φ(Z_{n}) which are in the union of the images of the natural mappings Φ(U ×_{V} V_{n,m}) → Φ(Z_{n}), for m ranging through the full index set M. So, for n′ ≥ n, the restriction of k_{n′}(T) to Y_{n} may be expressed as follows,

k_{n′}(T)_{|Y_{n}} = ∏_{Z} s_{Z}(T),

where Z runs through those irreducible component parts of Z_{n} which have the property that they are contained in an irreducible component of Z_{n′} which is in the image of

Φ(U ×_{V} V_{n′,m′}) → Φ(Z_{n′})

for some m′. This collection of irreducible components stabilizes for large n′. Putting g_{n}(T) := k_{n′}(T)_{|Y_{n}} for n′ large enough so that the resulting function g_{n}(T) is unchanged if we augment n′, we see that the rigid analytic function g_{n}(T) defines the subspace W_{n} ⊂ Z_{n} and (of course) divides F_{n}(T) := F (T)_{|Y_{n}}. By Lemma 1.3.3, there is a Fredholm series G(T) over X which defines W. By Lemma 1.3.2 (b) we have that G(T) divides F (T), i.e., we have a factorization F (T) = G(T)H(T) where H(T) is a (well-defined) Fredholm series over X.

Corollary 1.3.5. With notation as above, if Z is the rigid Zariski-closure in V of the affinoid B := B_{Z} cut out by s_{Z}(T) in Y_{n}, then Z is irreducible.

Proof. Suppose Z is not irreducible. Then it follows from the proposition that F (T) = G(T)H(T) for two non-trivial entire series G(T) and H(T) such that s_{Z}(T) divides the restriction of one and only one of G(T) or H(T) to Y_{n}. It follows that the rigid Zariski-closure of B is contained in the component part of Z corresponding to G(T) or H(T) which is a contradiction.

Proposition 1.3.6. Suppose X is a relatively factorial affinoid over K such that A(X) is an integral domain and F (T) is an entire series over X such that F (0) = 1 and the hypersurface Z cut out by F (T) is irreducible. Then there exists a prime element G(T) in A(X){{T }} such that G(T)^{m} = F (T) and G(0) = 1 for some positive integer m.

Proof. Set V = X × A^{1}. Let W_{n} be the induced reduced affinoid corresponding to Z ×_{V} Y_{n}. If

∏_{k=1}^{r_{n}} π_{n,k}(T)^{m_{n,k}}

is the prime factorization of F (T)|_{Y_{n}} ((π_{n,k}) ≠ (π_{n,j}) if k ≠ j) then W_{n} is the affinoid determined by

∏_{k=1}^{r_{n}} π_{n,k}(T).

Now, if ℓ ≤ n, the affinoid determined by the restriction of π_{n,k}(T) to Y_{ℓ} is a product of the π_{ℓ,j}(T) to some exponents but, by Corollary 7.3.2/10 of [BGR], it is reduced since the fiber product of the affinoid B determined by π_{n,k}(T) in Y_{n} and Y_{ℓ} is an affinoid subdomain of B. Hence all these exponents are at most 1. Since (by the dimension argument used above) every π_{ℓ,j}(T) divides the restriction of one and only one of the π_{n,k}(T) we see that the fiber product of W_{n} and Y_{ℓ} equals W_{ℓ}. It follows that W := ∪_{n}W_{n} is a Zariski closed subspace of V that satisfies the conditions of Lemma 1.3.3 and hence is the hypersurface cut out by a Fredholm series G(T). Moreover, for each n we know there exists an element H_{n}(T) ∈ A(Y_{n}) such that

G(T)H_{n}(T) = F (T).

By Lemma 1.3.2.b, these H_{n}(T) glue together into a Fredholm series H(T) such that

F (T) = G(T)H(T).

Let U be the closed subvariety of Z determined by H(T). On Y_{n} it is the affinoid determined by

∏_{k=1}^{r_{n}} π_{n,k}(T)^{m_{n,k}−1}.

Suppose there exists n and k such that m_{n,k} − 1 ≠ 0, then since the rigid Zariski-closure in U of the affinoid cut out by π_{n,k} in Y_{n} is an irreducible component of W, and since (we claim) W is irreducible, it must be equal to W and hence G(T) divides H(T). Continuing in this way we conclude F (T) = G(T)^{m} where m is equal to any of the non-zero m_{n,k}. Proof of claim. Suppose that W is not irreducible. Then we see that G(T) has a non-trivial factorization into Fredholm series.

G(T) = R(T)S(T).

It follows that there exist integers n and k, j such that 1 ≤ k, j ≤ n, π_{n,k}(T) divides R(T)|_{Y_{n}}, and π_{n,j}(T) does not. In particular, it follows that the rigid Zariski-closure of the affinoid in Y_{n} determined by π_{n,k}(T)^{m_{n,k}} is an irreducible component of the hypersurface cut out by R(T)^{m_{n,k}} but the rigid Zariski-closure of the affinoid determined by π_{n,j}(T)^{m_{n,j}} is not. Hence the rigid Zariski-closure of the first affinoid is not Z, a contradiction. Since W is both reduced and irreducible, it follows that G(T) is prime.

Theorem 1.3.7. Suppose X is a relatively factorial affinoid over K such that A(X) is an integral domain. Then, if F (T) is an entire series over X such that F (0) = 1, there exist distinct prime entire series P_{i}(T) and positive integers n_{i} such that P_{i}(0) = 1 and

F (T) = P_{1}(T)^{n_{1}} · P_{2}(T)^{n_{2}} ⋯ P_{k}(T)^{n_{k}} ⋯

Moreover, this factorization is unique up to order.

Proof. Let notation be as in Proposition 1.3.6. For each n and 1 ≤ k ≤ n, we know from Corollary 1.3.5 that the rigid Zariski-closure in Z of the affinoid B_{n,k} in Y_{n} cut out by π_{n,k}(T) is an irreducible component of Z, Z_{n,k}, which after Propositions 1.3.4 and 1.3.6 is cut out by the m_{n,k}-th power of a prime entire series P_{n,k}(T) such that P_{n,k}(0) = 1 for some positive integer m_{n,k}. To hide these unsightly double indices, let P_{γ}(T) (for γ = 1, 2, 3, …) be the list of these P_{n,k}’s, putting the indices (n, k) in lexographic order; and correspondingly, denote m_{n,k} by m_{γ}. Then the product

∏_{γ≥1} P_{γ}^{m_{γ}} (T)

converges by the argument in Lemma 1.3.3 since P_{γ}^{m_{γ}} (T) is a unit on Y_{n} for large γ and P_{γ}^{m_{γ}} (T)(0) = 1. The product converges to F (T) by Lemma 1.3.2. This yields the existence of the factorization. Now suppose Q(T) and H(T) are entire series with constant term 1 in A(X){{T }} such that Q(T) is prime and Q(T)H(T) = F (T). Then for every γ, it follows that P_{γ}(T) divides H(T) or Q(T). One of them, say P_{κ}(T), must divide Q(T) since Q(T) is not a unit and A(X){{T }} is an integral domain. Hence Q(T) = P_{κ}(T) since both series are prime and there are no non-trivial units with constant term 1 in A(X){{T }} by Lemma 1.3.2. From this, we may deduce that the factorization of F (T) is unique.

In the following, if U ⊂ X is an affinoid subdomain, and Z → X a rigid space (over X) we let Z_{U} = Z ×_{X} U denote the fiber product.

Lemma 1.3.8. Suppose X is a relatively factorial affinoid such that A(X) is an integral domain. Let F (T) = G(T)H(T) be an equation in Fredholm series over X such that G(T) ≠ 1. Suppose Z(G) ⊂ Y = X × A^{1}, the hypersurface cut out by G(T) is a non-empty component part of Z(F) ⊂ Y, the hypersurface cut out by F (T). Then if U is a non-empty affinoid subdomain of X, Z(G)_{U} is a non-empty component part of Z(F)_{U}.

Proof. The rigid space Z(G)_{U} is non-empty, for if it were empty then all the coefficients of G would have to vanish on U ⊂ X but then Lemma 1.3.1 implies G(T) = 1. The only way for Z(G)_{U} not to be a component part of Z(F)_{U} is if there were an irreducible component V ⊂ Z(G) ⊂ Y such that the following holds. If Y_{n} = X × B[0, p^{n}], V_{n} = V ×_{Y} Y_{n}. But from Lemma 1.3.1, it follows that H(T) vanishes identically when restricted to V_{n}^{red} for all n contradicting the assertion that Z(G) is a component part of Z(F).

Lemma 1.3.9. Suppose X is an affinoid over K such that A(X) is an integral domain, L is a finite Galois extension of K with Galois group Γ and P (T) is a prime entire series over X_{L} := X × L such that P (0) = 1. Then if P_{1}(T), P_{2}(T), …, P_{r}(T) is the orbit of P (T) under the action of Γ,

Q(T) := ∏_{i} P_{i}(T)

is a prime entire series over X.

Proof. Suppose there exist entire series F (T) and G(T) over X such that Q(T) divides F (T)G(T) in A(X){{T }}. Then in A(X_{L}){{T }}, P (T) divides F (T) or G(T). Suppose

P (T)H(T) = F (T).

It follows that P^{σ}(T)H^{σ}(T) = F (T) for all σ ∈ Γ. Since P^{σ}(T) doesn’t divide P (T) unless it equals P (T) by Lemma 1.3.2, we conclude that there exists an entire series R(T) over X_{L} such that

Q(T)R(T) = F (T).

Since Q and F have coefficients in A(X) and

A(X_{L}){{T }}

is an integral domain, it follows that R(T) is an entire series over X.

Lemma 1.3.10. Suppose that A is a complete local R algebra as in §1.1 and A = A^{0}(X_{A}). Suppose F (T) ∈ A{{T }} and we have a factorization

F (T) = G(T)H(T)

with G(T), H(T) ∈ A(X_{A}){{T }} and G(0) = H(0) = 1. Then G(T), H(T) ∈ A{{T }}.

Proof. Suppose G(T) = ∑_{n≥0} a_{n}T^{n} and there exists an n such that a_{n} ∉ A. Then, since A = A^{0}(X_{A}), there exists an x ∈ X_{A}(\overline{K}) such that |a_{n}(x)| > 1. Denoting G_{x}(T) := ∑_{n≥0} a_{n}(x)T^{n} we see that G_{x}(T) ∈ K̄{{T }}, has a zero in K̄ of absolute value strictly less than 1. The same must be true about F_{x}(T) which is patently false. Hence, G(T) ∈ A{{T }} and we reach the same conclusion for H(T).

The proof of the next theorem follows the general format of the proof of Theorem 1.3.7.

Theorem 1.3.11. Let A = O[[T_{1}, …, T_{m}]] be a power series ring in m variables over O, the ring of integers in a field K which is finite over Q_{p}. Denote its associated rigid space over K by X_{A}. Let F (T) be a non-trivial Fredholm series over A. Then there exist unique distinct prime Fredholm series over A, P_{i}(T), and positive integers n_{i} such that

F (T) = P_{1}(T)^{n_{1}} ⋯ P_{k}(T)^{n_{k}} · · · .

Proof. Recall the notation introduced in the construction of the rigid analytic space X_{A} in section 1.1. In particular, for each rational number, 1 ≥ r > 0, we have an affinoid subdomain of X_{r} ⊂ X_{A}. We may identify X_{A} with the open unit n dimensional polydisk v(t_{i}) > 0 and X_{r} with the affinoid subpolydisk defined over Q_{p}, v(t_{i}) ≥ r. Let B_{m} := A(X_{1/m})⊗_{Q_{p}} K where K = Q_{p}(μ_{m}, p^{1/m}). Then B_{m} ≅ K⟨X_{1}, …, X_{n}⟩ and so is relatively factorial. It follows from Theorem 1.3.7, Lemma 1.3.9 and Lemma 1.3.3 that there exists a countable set I_{n} such that

F (T)|_{X_{1/n}} = ∏_{i∈I_{n}} P_{n,i}(T)^{m_{n,i}},

where P_{n,i}(T) ∈ A(X_{1/n}){{T }} are prime entire series on X_{1/n} and the m_{n,i} are positive integers. Now if n ≥ ℓ, arguing as in the proof of Proposition 1.3.6 we see that for each i ∈ I_{ℓ}, there exists a unique j(n, ℓ, i) ∈ I_{n} such that P_{ℓ,i}(T) divides P_{n,j(n,ℓ,i)}(T)|_{X_{1/ℓ}} and it divides it exactly once. In fact, m_{ℓ,i} = m_{n,j(n,ℓ,i)}. Let S(n, ℓ, j) be the subset of I_{ℓ} such that P_{n,j}(T)|_{X_{1/ℓ}} is divisible by P_{ℓ,i}(T). Because of Lemma 1.3.8, this set is non-empty and because of Lemma 1.3.3, for j ∈ I_{n},

P_{n,j}(T)|_{X_{1/ℓ}} = ∏_{i∈S(n,ℓ,j)} P_{ℓ,i}(T).

Now let

Q_{ℓ,i}(T) = lim_{n→∞} P_{n,j(n,ℓ,i)}(T).

This equals

∏_{k∈T (ℓ,i)} P_{ℓ,k}(T),

where T (ℓ, i) = ∪_{n}S(n, ℓ, j(n, ℓ, i)). If J_{ℓ} is a subset of I_{ℓ} such that

⋃_{i∈J_{ℓ}} T (ℓ, i) = I_{ℓ} and T (ℓ, i) ∩ T (ℓ, j) = ∅

if i, j ∈ J_{ℓ}, i ≠ j, then

F (T)|_{X_{1/ℓ}} = ∏_{i∈J_{ℓ}} Q_{ℓ,i}(T)^{m_{ℓ,i}}.

Now it follows from the definitions that

Q_{n,j(n,ℓ,i)}(T)|_{X_{1/ℓ}} = Q_{ℓ,i}(T),

and hence for each i ∈ I_{ℓ} there exists a unique entire series Q_{i}(T) over X_{A} such that Q_{i}(T)|_{X_{1/ℓ}} = Q_{ℓ,i}(T) and so

F (T) = ∏_{i∈J_{ℓ}} Q_{i}(T)^{m_{ℓ,i}}.

It follows from Lemma 1.3.10 that Q_{i}(T) ∈ A{{T }} and it follows by an argument similar to that used in Theorem 1.3.7 that Q_{i}(T) is prime and that the factorization is unique.

Comments about degree and the geometry of Fredholm hypersurfaces. Suppose, for concreteness, that A = O[[T_{1}, …, T_{m}]] is a power series ring in m variables over O, the ring of integers in a field K which is finite over Q_{p}. Let X = X_{A} be the open unit ball of dimension m over K. Any Fredholm series over A (which is different from the constant function 1) is either a polynomial of (finite) degree d > 0 over A, P (T) = 1 + a_{1}T + a_{2}T^{2} + ⋯ + a_{d}T^{d}, with a_{d} nonzero, or else it is an entire power series P (T) = 1 + a_{1}T + a_{2}T^{2} + ⋯, where there are arbitrarily large integers n for which the corresponding coefficient a_{n} doesn’t vanish. Consider the Fredholm hypersurface Z ⊂ X × A^{1} whose defining equation is given by P (T) = 0. We view Z as rigid space over K. In the first case, i.e., when P (T) is a polynomial of degree d, consider the monic polynomial

P^{∗}(U) := U^{d}P (1/U) = U^{d} + a_{1}U^{d−1} + a_{2}U^{d−2} + ⋯ a_{d}.

Form the finite flat A-algebra E := A[U]/P^{∗}(U), and consider the quotient algebra E → Ē := E/(U) = A/(a_{d}). Both E and Ē are complete semilocal noetherian O-algebras. Let X_{E} and XĒ denote their respective rigid analytic spaces over K. We have a canonical immersion XĒ ↪ X_{E}, and the rigid space Z is the complement of XĒ in X_{E}. The natural projection X_{E} → X is finite flat of degree d, and therefore the projection Z → X is a mapping of generic degree d (with a decrease in the degree of fibers occurring over the zero-locus of the function a_{d} on X). If the image of a_{d} is a unit in A ⊗_{O} K, then XĒ is empty, and Z = X_{E}.

In the case where P (T) has an infinite number of nonvanishing coefficients we shall say that the Fredholm hypersurface Z is of infinite degree over X; this is a natural terminology, for, in this case, there are infinitely many points in the fiber of Z above each of the points x lying outside a proper subspace of X.

This discussion, together with Theorem 1.3.11 above, immediately gives:

Corollary 1.3.12. If Z is the Fredholm hypersurface defined by a Fredholm series over A, then each irreducible component Z_{o} of Z is either

1. a rigid space of the form X_{E} \ XĒ where X_{E} → X is the rigid space associated to a finite flat A-algebra, and where XĒ ⊂ X_{E} is the rigid space associated to a closed subscheme of Spec E which projects isomorphically onto a proper closed subscheme of Spec A.

or else is

2. of infinite degree over X.

Corollary 1.3.13. When A is isomorphic to O[[T]], any irreducible component Z_{o} of a Fredholm hypersurface defined by a Fredholm series over A has the property that its image in X omits at most a finite number of points.

Note that if A is the completion at any maximal ideal of the Iwasawa algebra Λ_{N} of the next section, where ϕ(N) is prime to p, then A satisfies the hypotheses of Cor. 1.3.13.

1.4 Weight space.

Let p be a prime number, and fix C_{p} as in 1.1. Put q = p if p > 2 and q = 4 if p = 2. Let N be a positive integer prime to p. Put Z_{p,N}^{∗} := lim_{↼} (Z/Np^{n}Z)^{∗}, and

Λ_{N} = lim_{↼} Z_{p}[(Z/Np^{n}Z)^{∗}] = Z_{p}[[(Z/NZ)^{∗} × Z_{p}^{∗}]],

the projective limits being taken as n tends to infinity. The ring Λ_{N} (sometimes referred to as the Iwasawa algebra) is the completed topological group algebra with Z_{p} coefficients of the profinite group Z_{p,N}^{∗}. The group structure of the finite group (Z/Np^{n}Z)^{∗} induces, in the standard way, a commutative (associative) Hopf algebra structure on the group ring Z_{p}[(Z/Np^{n}Z)^{∗}] (which one can view as the affine Hopf Algebra associated to the Cartier dual of the constant group scheme (Z/Np^{n}Z)^{∗} over Spec Z_{p}). Passing to the limit as n tends to ∞, we get a (completed) associative and commutative Hopf algebra structure

Λ_{N} → Λ_{N} ˆ⊗_{Z_{p}}Λ_{N}.

We have a canonical factorization,

Z_{p,N}^{∗} = (Z/NqZ)^{∗} × (1 + qZ_{p}),

the first factor being identified with the torsion subgroup of Z_{p,N}^{∗} and the second factor being the wild part of Z_{p,N}^{∗}; i.e., the multiplicative group of elements congruent to 1 mod qN. If a ∈ Z_{p,N}^{∗}, we let [a] denote the corresponding element in Λ_{N} and ⟨⟨a⟩⟩ to avoid confusion with the diamond operators the projection of a to 1 + qZ_{p} with respect to the above factorization. We have a corresponding factorization of the Iwasawa algebra,

Λ_{N} ≅ Z_{p}[(Z/NqZ)^{∗}] ⊗ Z_{p}[[(1 + qZ_{p})]],

where the second factor admits a (not very natural) isomorphism with a power series ring Z_{p}[[t]] (in one variable t over Z_{p} by sending [1 + q] ∈ Z_{p}[[(1 + qNZ_{p})]] to 1 + t ∈ Z_{p}[[t]]. If the finite group (Z/NqZ)^{∗} is of order prime to p (equivalently, if p > 2 and no prime divisor of N is congruent to 1 mod p) then Λ_{N} is a regular (noetherian) semi-local ring of Krull dimension two, and, conversely, if Λ_{N} is regular, then N has the property just stated.

Denote by W = W_{N} the rigid analytic space over Q_{p}, X_{Λ_{N}} associated to the formal Spf (Z_{p})-scheme Spf (Λ_{N}) as in §1.1. The completed Hopf algebra structure on Λ_{N} endows W_{N} with the structure of (one-parameter) commutative rigid analytic group where multiplication corresponds to the usual multiplication of characters. We refer to W_{N}, with its structure as rigid analytic group over Q_{p}, as weight space of tame level N. The C_{p}-valued points of W_{N} are the continuous C_{p}-valued characters on Z_{p,N}^{∗}. If N divides M, the natural projection Z_{p,M} → Z_{p,N} induces an injective homomorphism W_{N} ↪ W_{M} which allows us to identify the rigid analytic group W_{N} with an open and closed rigid analytic subgroup of W_{M}. In particular, for any N we identify the connected component of the identity in the rigid analytic group W_{N} with the connected component of the identity, call it B, in W_{1}, which can be identified, by restriction, with the continuous characters on 1+qZ_{p}. The group of connected components, i.e., the quotient group D_{N} := W_{N}/B may be canonically identified with the C_{p}^{∗}-dual of the finite group (Z/NqZ)^{∗} and since we have a natural restriction map from W_{N} to B we may regard W_{N} as D_{N} × B. The rigid analytic group B is isomorphic to the open disk of radius 1 about the identity element in C_{p}^{∗} with its induced multiplicative group structure, and the C_{p}-valued points of B are identified with continuous homomorphisms Z_{p}[[(1 + qNZ_{p})]] → C_{p}. Setting Λ^{(0)} = Z_{p}[[(1 + qZ_{p})]], we have that Λ^{(0)}, the ring of rigid analytic functions over Q_{p} and bounded by 1 on B, is the classical Iwasawa algebra, and the standard completed Hopf algebra structure on Λ^{(0)} induces the rigid analytic group structure on B. If κ ∈ W_{N}, we let ⟨κ⟩ ∈ B ⊂ W_{N} be the character a ↦ κ(⟨⟨a⟩⟩). It is trivial on (Z/NqZ)^{∗}.

Definition. If p > 2 and {s ∈ C_{p}: v(s) > −1+1/(p−1)}, let η_{s} : Λ_{N} → C_{p} denote the continuous character a ↦ ⟨⟨a⟩⟩^{s}, and denote by B^{∗} the space of all these characters η_{s}; i.e.,

B^{∗} := {η_{s} : a ↦ ⟨⟨a⟩⟩^{s} | s ∈ C_{p}, v(s) > −1 + 1/(p − 1)}.

If p = 2, we let

B^{∗} := {η_{s} : a ↦ ⟨⟨a⟩⟩^{s} | s ∈ C_{p}, v(s) > −1}.

By an accessible weight-character (of tame level N) we shall mean a C_{p}-valued point of W_{N} of the form χη_{k} where k ∈ Z_{p} and χ is a character of finite order. We sometimes refer to such a point κ by the coordinates (χ, k).

Let 1 be the trivial character and τ ∈ W be the the unique character which is trivial on (Z/NZ)^{∗} × (1 + qZ_{p}) and the identity on μ(Z_{p}). If k is an integer, the k-th power map (a ↦ a^{k}) on Z_{p,N}^{∗} composed with projection to C_{p}^{∗} is an accessible weight-character in the sense just defined, and has coordinates (τ^{k}, k). We refer to this homomorphism as the character of integral weight k (and trivial nebentypus). More generally, if ϵ : Z_{p,N}^{∗} → C_{p}^{∗} is any continuous character of finite order, the point ϵ·τ^{k} ·η_{k} ∈ W_{N}(C_{p}) will be said to be an arithmetic point of weight k and nebentypus character ϵ.

Most of the rigid analytic spaces we will deal with in this article will come along with a natural (rigid analytic) morphism to W_{N} (for some appropriate tame level N). We shall refer to this morphism by the phrase projection to weight space. All the rigid analytic morphisms we encounter will preserve projections to weight space.

1.5. The eigencurve as the Fredholm closure of the classical modular locus. (Statement of the main theorems)

For a prime number p and N (a tame level) an integer prime to p , let S denote the finite set of places of Q consisting of the infinite place, p, and the prime divisors of N. Let G_{Q,S} be the Galois group of the maximal algebraic extension of Q in C) which is unramified outside S. One would eventually wish to treat all primes p and all tame levels N but in the present article we prove our main theorems only for p > 2 and tame level N = 1. For the rest of this section, then, we let p be an odd prime number and N = 1.

In section 5.1 below we shall define a certain collection of residual G_{Q,S}- representations into finite fields of characteristic p which include, up to equivalence, (the semi-simplification of) all characteristic p residual Galois representations coming from classical modular forms of p-power level. These we call p-modular residual representations of G_{Q,S} of tame level 1. We also recall in detail in 5.1 the notion of pseudo-representation, following [Wi], [T], [Ny], [Ro] and [H-NO]. Briefly, if D is a topological ring, to any continuous representation ρ : G_{Q,S} → GL_{2}(D)

g ↦ ({a(g) \atop c(g)} {b(g) \atop d(g)}) ,

which sends complex conjugation in G_{Q,S} to the diagonal matrix

({1 \atop 0} {0 \atop −1}) ,

its associated 2-dimensional pseudo-representation r with values in D is taken to be the pair of (continuous) functions α, δ : G → D given by the formulas, α(g) = a(g); δ(g) = d(g), noting that α and δ, so defined, are invariants of the equivalence class of ρ. It is convenient to define, as well, the function ξ : G×G → D by the formula ξ(g, h) = b(g)c(h), and although the function ξ is determined by either function α or δ because

ξ(g, h) = α(gh) − α(g)α(h) = δ(hg) − δ(h)δ(g),

we refer to the triple of functions r := (ξ, α, δ), as the pseudo-representation associated to ρ. A (general) pseudo-representation is roughly a triple of functions satisfying the same relations as an arbitrary (ξ, α, δ) obtained from a representation in the above way. For any 2-dimensional pseudorepresentation r = (ξ, α, β) on G_{Q,S} with values in D and any g ∈ G_{Q,S} we set

trace(r(g)) = α(g) + β(g) and det(r(g)) = α(g)β(g) − ξ(g).

In 5.1 we shall consider the (complete noetherian, semi-local) ring R_{p} defined as the universal (p-complete, noetherian, semi-local) deformation ring for (degree two) modular pseudo-representations of G_{Q,S} of tame level 1. Being universal, there is a pseudo-representation r^{univ} from G_{Q,S} with values in R_{p} which specializes to any deformation of a p-modular pseudorepresentation. Here let us just note that R_{p} includes among its factor rings the universal Galois deformation rings of all absolutely irreducible modular residual representations (i.e., residual representations over finite fields of characteristic p) attached to newforms of levels dividing powers of p (see 5.1).

As we show in section 5.1, R_{p} may viewed in a natural way as a Λ-algebra (with its weight Λ-algebra structure).

Let X_{R_{p}} = X_{p} be the rigid analytic space over Q_{p} associated to the complete semi-local ring R_{p} regarded as a Z_{p}-algebra, as in §1.1, endowed with the projection to weight space, π : X_{p} → W coming from the weight Λ-algebra structure of R_{p}. Using [Wi], one can see that any two-dimensional continuous representation ρ : G_{Q,S} → GL_{2}(C_{p}) of tame level 1 whose associated residual representation is p-modular corresponds to a (unique) point x_{ρ} ∈ X_{p}(C_{p}) (given by its associated pseudo-representation).

To a normalized (classical) modular eigenform f on Γ_{1}(M) over C_{p}, where M is a positive multiple of p, of weight k and character χ with qexpansion

f(q) = c_{0} + q + c_{2}q^{2} + c_{3}q^{3} + ⋯

one can attach an odd degree 2 representation ρ_{f} over C_{p} (up to equivalence) whose associated pseudo-representation r_{f} := (ξ, α, δ) satisfies

trace(r_{f}(F rob_{l})) = c_{l} and det(r_{f}(F rob_{l})) = χ(l)l^{k−1}

for primes l not dividing M induced from the universal pseudo-representa-tion via a unique homomorphism R_{p} → C_{p}, i.e., a unique C_{p}-valued point x_{f} ∈ X_{p}(C_{p}).

If, further, M is a power of p and f is of finite slope, i.e., if its U_{p}– eigenvalue u := c_{p} is nonzero, denote by x̃_{f} the point (x_{ρ_{f}} , 1/u) ∈ (X_{p} × A^{1})(C_{p}).

Definition 1. The classical modular locus of tame level 1,

M ⊂ (X_{p} × A^{1})(C_{p})

is the set of points x̃_{f} = (x_{f}, 1/u) where f runs through all (classical) modular eigenforms of finite slope, as above, on Γ_{1}(p^{m}) for m ≥ 1.

In this article we shall give two different constructions of rigid analytic curves which parameterize the collection of all overconvergent eigenforms of tame level N = 1 and of finite slope (i.e., having non-zero U_{p}-eigenvalue).

Definition 2. The eigencurve C = C_{p} of tame level N = 1 is the Fredholm closure (see Section 1.2) in X_{p} ×A^{1} of the classical modular locus (of tame level 1).

Note: We have jumped the gun in our terminology for the above definition does not immediately allow us to see that C_{p} is a curve. In this paper we shall not define the eigencurve of general tame level, and only concern ourselves with tame level N = 1. Thus, in what follows, we will drop the tag “of tame level N = 1” and refer to the eigencurve of tame level 1 simply as the eigencurve. It follows from the above definition that C_{p} and its nilreduction C_{p}^{red} are nested in the sense defined in section 1.1, and therefore Proposition 1.2.5 implies that C_{p}^{red} is the union of its irreducible components.

Here is a statement of some of the main results of this paper. By the reduced eigencurve, we mean the nilreduction of the eigencurve C_{p}

Theorem A. Let C_{o} ⊂ X_{p} × A^{1} be an irreducible component part of the reduced eigencurve. Then there exists an element γ_{0} ∈ R_{p}^{∗} such that the mapping

X_{p} × A^{1} → W × A^{1}

induced by the Λ-algebra homomorphism Λ{{T }} → R_{p}{{T }} (T ↦ γ_{o} ·T) when restricted to C_{o} is generically an isomorphism of C_{o} onto a Fredholm hypersurface in W × A^{1}.

By “generically” we simply mean, since we are dealing with a rigid analytic curve, after the exclusion of a discrete set of points (i.e., a 0-dimensional Zariski-closed subspace).

Theorem A follows directly from Corollary 7.6.2, and its proof.

Theorem B. The natural projection of any irreducible component of the reduced eigencurve to weight space is component-wise almost surjective in the sense that given any irreducible component of the reduced eigencurve, the complement of its image in the unique irreducible component of weight space containing that image is (empty, or) consists of at most a finite number of weights.

Proof. This is Corollary 7.4.2 below. The argument for it uses Theorem A together with the Corollary 1.3.13.

Theorem C. The projection of the reduced eigencurve to weight space is locally in-the-domain finite flat in the sense that C_{p}^{red} is covered by admissible affinoid domains U such that the restriction of projection to weight space to U is a finite flat mapping of U onto its image in W.

Proof. This follows directly from the construction of D in Chapter 7 (Prop. 7.2.2) together with the fact that D ≅ C_{p}^{red} (Theorem. 7.5.1).

It follows from Theorem C that C_{p} is a curve in the sense that it is an equidimensional rigid analytic space of dimension 1.

Since the universal pseudo-representation ring R_{p} is a complete semilocal ring whose maximal ideals are in one-one correspondence with (certain) equivalence classes of semi-simple residual representations G_{Q,S} → GL_{2}(F̄_{p}) (see section 5.1 below) we have:

Theorem D. If two classical modular points of tame level N, x̃_{f}, x̃_{f′} ∈ (X_{p} × A^{1})(C_{p}) lie on the same irreducible component of the reduced eigencurve, then f ≡ f′ modulo the maximal ideal of O_{C_{p}} in the sense that their Fourier expansions are congruent modulo that maximal ideal (and their associated residual representations have equivalent semi-simplifications).

In view of Theorem D, given ρ̄ : G_{Q,S} → GL_{2}(F̄_{p}), a semi-simple representation, we will say that a component C_{j} of the eigencurve is of type ρ̄ if one (equivalently: all) of its classical modular forms have the semisimplifications of their associated residual representations equivalent to ρ̄. Define Cρ̄ to be the rigid analytic subspace of C_{p} which is the union of all components of the eigencurve of type ρ̄. We say a rigid subspace V of a rigid space U is admissibly closed-open if there exists a rigid subspace W of U disjoint from V such that {V, W } is an admissible open cover of U.

Corollary 1.5.1. The subspaces Cρ̄ are admissibly closed-open in C_{p}.

Proof. Indeed, the proof we sketched for theorem D understates the case a bit: Since the ambient rigid-analytic space X_{p} × A^{1} breaks up unto the disjoint union of rigid analytic spaces Xρ̄ × A^{1} where Xρ̄ is the universal deformation space of the pseudo-representation (of G_{Q,S}) associated to ρ̄, for ρ̄ running through the semi-simple residual‘ representations described above and since Cρ̄ = C_{p} ∩ Xρ̄ × A^{1} , Corollary 1.5.1 follows.

The classical modular forms of finite slope correspond to (certain) points on the eigencurve. What interpretation is there for the remaining points? For this we have Theorem 6.2.1 below, a paraphrase of which is given by the following theorem.

Theorem E. There is a natural one-one correspondence

c ⟷ f_{c}

between the C_{p}-valued points c on the eigencurve and overconvergent (see section 5.2) modular eigenforms f_{c} of finite slope of tame level 1. This oneone correspondence is characterized by the conditions that π(c), the natural projection of c to X_{p}, when viewed as a pseudo-representation, is the pseudorepresentation associated to the overconvergent modular eigenform f_{c} by the Gouvêa-Hida Theorem (see section 5.2), and the natural projection of c to A_{Q_{p}}^{1} is the reciprocal of the U_{p}-eigenvalue of f_{c}.

We note that the notion of an overconvergent eigenform of integral weight was introduced in [K-PMF] and its generalization to more general arithmetic weight characters was made in [C-CO] and [C-COHL] and to non-arithmetic weight in [C-BMF] (see also [G-ApM]). We might note that the compatibility of these two definitions was not formally established previously but is now, as a consequence of Corollary 2.2.6.

Theorem F. The reduced eigencurve is the rigid Zariski-closure of the classical modular locus.

Proof. By Theorem E, the points of the eigencurve correspond to overconvergent eigenforms. By Theorem B, every irreducible component of the eigencurve has points of all but a finite number of possible weights in the component of weight space to which it projects. In particular, for each irreducible component C_{o} of the eigencurve, there is a positive integer k_{o} such that C_{o} has points of all integral weights ≥ k_{o}. One immediately sees from this that every irreducible component C_{o} has points of positive integral weight k and of slope < k −1 (and different from \frac{k−1}{2}). By Corollary B5.7.1 of [C-BMF] there is an affinoid disk in C_{o} whose projection to weight space is an isomorphism C_{o} → W_{o} ⊂ W onto an affinoid W_{o} in W, and such that for a topological dense set D_{o} in the set of arithmetic weights in W_{o}(Q_{p}) the points of C_{o} mapping to D_{o} are given by classical eigenforms. This proves Theorem F.

A similar argument to that used above also gives the following result (which is Proposition 7.6.5 below).

Theorem G. If {f_{n}} is a sequence of normalized eigenforms of tame level 1 with Fourier coefficients in C_{p} such that f_{n} has weight κ_{n} ∈ W(C_{p}), v(a_{p}(f_{n})) is bounded independently of n, the sequence κ_{n} converges in (Z/(p − 1)Z) × Z_{p} to some weight κ ∈ W(C_{p}) and the sequence {f_{n}(q) ∈ C_{p}[[q]](1/q)} converges coefficientwise to a series f(q). Then, f(q) is the qexpansion of an overconvergent modular eigenform f of tame level 1, weight κ, and finite slope. Let f be a Katz modular eigenfunction (see section 2.2 below) of tame level 1, of finite slope, with (normalized) Fourier expansion

f̃ = q + a_{2}q^{2} + a_{3}q^{3} + ⋯

with coefficients a_{j} ∈ C_{p}. Suppose that f is of accessible weight-character (cf. section 1.4) (χ, s) with s ∈ Z_{p}. Then f is overconvergent if and only if f is the limit in the q-expansion topology of a sequence of classical, normalized, cuspidal-overconvergent (cf. section 3.6) modular eigenforms of tame level 1.

In section 5.2 we show that the eigencurve is universal for certain rigid analytic families of (overconvergent, finite slope) modular eigenforms. We lack, however, a completely satisfactory theory here, for we do not have an a priori concept of infinitesimal deformation of (overconvergent, finite slope) modular eigenforms. By “a priori” we mean prior to constructing the eigencurve: it would be good, for example, when given an overconvergent eigenform of finite slope f, to have a theory which produces the Zariski tangent space of the eigencurve at the point f, and produces it in a reasonably computable manner (perhaps as an appropriate cohomology group).

We will now summarize some of the key properties of the eigencurve in a purely rigid-geometric way without mentioning deformations. The p-adic eigencurve of tame level 1 is a rigid analytic curve C with the following properties: There is a rigid analytic morphism κ from C to W and rigid analytic functions t_{n}, n ≥ 1 with the following properties: Suppose

f(q) = ∑_{n≥0} a_{n}q^{n}

is a normalized eigenform on X_{1}(p^{r}), r > 0, of weight k, such that a_{p} ≠ 0 whose character on (Z/p^{r}Z)^{∗} is ϵχτ^{−k}, where χ ∈ B. Then there exists a unique point x_{f} on C such that κ(x_{f}) = ϵχη_{k} and

t_{n}(x_{f}) = a_{n}.

Call these points the classical points. On the other hand, if x ∈ C(C_{p}) such that κ(x) = ϵχη_{k} where k is an integer, χ ∈ B is a character of finite order and v(t_{p}(x)) < k − 1 then x is classical. Moreover, as we stated above, C is the rigid Zariski-closure of the classical locus. This is not, however, enough to characterize C.

Chapter 2. Modular Forms.

2.1 Affinoid sub-domains in modular curves.

If M ≥ 1 is an integer, and S is a scheme, μ_{M /S} denotes the finite flat group scheme over S given by as the kernel of multiplication by M in the multiplicative group (over S). Let p be a prime number. Fix an integer N prime to p, and m ≥ 1 and let us begin our discussion in level Np^{m}. Our running assumption in this section is that, unless otherwise explicitly mentioned, Np^{m} ≥ 5. Let Y_{1}(N · p^{m}) be the (uncompactified) modular curve classifying isomorphism classes of pairs (E, α) where E is an elliptic curve over a Q_{p}-scheme S, α : μ_{N·p^{m}} ↪ E is an injection (over the base S). This is equivalent to the [Γ_{1}[(Np^{m})] problem (over Q_{p}) as discussed in [KM]. Since Np^{m} ≥ 5, the above problem is rigid and representable in the terminology of 4.7 of [KM] (cf. pp. 327, 328 of [KM]) and we have a universal family

(u : E_{1}(Np^{m}) → Y_{1}(Np^{m}), α).

Let ω be the invertible line bundle on Y_{1}(Np^{m})_{/ Q_{p}} given by

ω = u_{∗}Ω_{E_{1}(N·p^{m})/X_{1}(N·p^{m})}^{1}.

By the discussion of (10.13) (cf. the summarizing table (10.13.9.1)) of [KM], since Np^{m} ≥ 5 we have an extension of the line bundle ω to a line bundle (which we denote by the same letter) on the compactification X_{1}(Np^{m})_{/Q_{p}}, and a canonical Kodaira-Spencer isomorphism

ω^{⊗2} ≅ Ω_{X_{1}(Np^{m})_{/ Qp}}^{1} (log cusps),

of line bundles on X_{1}(Np^{m}). Assuming now that N ≥ 5, the reader is invited to consider, as well, a noncompact model (over Z_{p}) of this moduli scheme given by the analysis in [KM] of the [Γ_{1}[(Np^{m})] problem (over Z_{p}) whose closed fiber is the (geometrically irreducible) incomplete Igusa curve of level Np^{m} (cf. Chapter 12 of [KM]).

It is sometimes useful to keep in mind the entire tower of modular curves

(∗) … → X_{1}(N · p^{n+1}) → X_{1}(N · p^{n}) → … X_{1}(N · p^{m+1}) → X_{1}(N · p^{m})

as n tends to ∞ (m ≤ n). Consider, for example, a triple (E, α_{N}, α_{∞}) where E is an elliptic curve over Z̄_{p} with ordinary reduction mod p (i.e., with either good ordinary reduction, or multiplicative type reduction), where α : μ_{N} ↪ E is an injective homomorphism (over Spec Z̄_{p}) and where α_{∞} is an embedding G_{m}[p^{∞}] ↪ E[p^{∞}] of the p-divisible group attached to G_{m} (over Spec Z̄_{p}) into the p-divisible group attached to E. Such a triple (E, α_{N}, α_{∞}) (which we will sometimes refer to as a p-trivialized elliptic curve with tame Γ_{1}(N)-structure) gives rise to a cofinal system of Spec Z̄_{p}-valued points of

(∗∗) … → Y_{1}(N · p^{n+1}) → Y_{1}(N · p^{n}) → … Y_{1}(N · p^{m+1}) → Y_{1}(N · p^{m}),

as follows. For each n restrict α_{∞} to get an embedding α_{p^{n}} : μ_{p^{n}} ↪ E, and define α_{Np^{n}} : μ_{Np^{n}} ↪ E to be the unique embedding which restricts to α_{N} and α_{p^{n}} respectively on μ_{N} and μ_{p^{n}}. We then associate to the triple (E, α_{N}, α_{∞}) the cofinal system n ↦ (E, α_{Np^{n}}). Conversely, a cofinal system of Spec Z̄_{p}-valued points of (∗∗) comes from such a triple (E, α_{N}, α_{∞}). Given such a triple, the embedding α_{∞} induces an isomorphism Ĝ_{m} ≅ Ê of formal completions at the origin. The image of the standard differential dt/t on Ĝ_{m} under this isomorphism then gives us a specific differential w generating the fiber of the line bundle ω over the point (E, α_{N}p^{n}) of Y_{1}(N · p^{n})(Q̄_{p}), for any n ≥ 1. (Although we will not use this point of view in the present paper, one way of thinking of this, is to invoke the projective limit Y_{1}(N · p^{∞}) of the projective system (*) of curves and to view the process of stipulating a generating section w ∈ ω over Y_{1}(N · p^{∞}) as giving us a canonical trivialization of the line bundle ω over Y_{1}(N · p^{∞}). See I.1 of [G] for slightly more discussion of this.)

Let us now return to the modular curve X_{1}(N · p^{m}), taken over the base Q_{p}.

First, let Z_{1}(N · p^{m}) denote the inverse image under reduction to Spec(F_{p}) of the complement of the supersingular points on the irreducible component containing the cusp ∞ in the above model of X_{1}(N · p^{m}) over Spec(Z_{p}). We have that Z_{1}(N · p^{m}) is an affinoid subdomain of the rigid analytic space X_{1}(N · p^{m}) over Q_{p} (see [C-RLC]). In §B2 of [C-BMF] a system of affinoid neighborhoods Z_{1}(N ·p^{m})(v) (called there X_{1}(N ·p^{m})(v)) of Z_{1}(N · p^{m}) in X_{1}(N · p^{m}) is described, where the parameter v (which we sometimes refer to as the radius) is a rational number allowed to vary in the range 0 ≤ v < p^{(2−m)}/(p + 1). Let I_{m} denote the set of such v. In particular, Z_{1}(N ·p^{m})(0) = Z_{1}(N ·p^{m}) and when v > 0 these neighborhoods are strict (cf. the definition of strict neighborhood below).

We shall briefly sketch the definition of Z_{1}(N · p)(v) here for p ≥ 5 (for the full story, see [C-BMF], section B2). Let A be the level 1, characteristic p, Hasse invariant modular form. For a non-negative rational number v ∈ Q, Z_{1}(Np)(v) is the affinoid subdomain of X_{1}(Np) whose points over C_{p} correspond to pairs (E, α) over S = Spec(R), where R is the ring of integers in C_{p}, as above such that v(A(\overline{E}, η̄)) ≤ v where η generates Ω_{E/S}^{1} and (\overline{E}, η̄) is the reduction of (E, η) modulo p, and α(μ_{p}) is the canonical subgroup (see Chapter 3 of [K-PMF] where the canonical subgroup is defined for all N and p ≥ 5). We will also have occasion to consider the modular curves X_{1}(Np^{m}, ℓ), where ℓ is a prime not dividing Np, which classify triples (E, α, C) where E and α are as above and C is a subgroup scheme of E of order ℓ. Then we have an affinoid subdomain Z_{1}(Np^{m}, ℓ) with affinoid neighborhoods Z_{1}(Np^{m}, ℓ)(v) for v ∈ I_{m} defined in a similar way.

In what follows let us fix a rational number v and focus attention on the affinoid sub-domain Z_{1}(N · p^{m})(v) of that width. We also reinstitute the hypothesis N · p^{m} ≥ 5.

Definition. Let V be a rigid analytic space. By an overconvergent family of rigid analytic functions on Z_{1}(N · p^{m}) parameterized by V we mean a rigid analytic function F on Z_{1}(N · p^{m}) × V with the property that there is an admissible open covering of V by affinoids V_{j} (j ∈ J) such that there are positive numbers v_{j} ∈ I_{m} (j ∈ J) for which the restriction of F to V_{j} extends to a rigid analytic function on Z_{1}(N · p^{m})(v_{j}) × V_{j}. More generally,

Definition. If X → Y is a morphism of rigid spaces over K, we say that X is affinoid over Y if for each affinoid subdomain Z in Y, X_{Z} is an affinoid. Suppose W → Y is a map of rigid spaces and X is a rigid subpace of W which is affinoid over Y, then we say that an admissible open subspace V of W is a strict neighborhood of X over Y in W if there is an admissible covering of Y by affinoids V_{j} such that for each j there exists an affinoid neighborhood U_{j} of X_{V_{j}} in V_{V_{j}} such that the morphism X̄_{V_{j}} → Ū_{V_{j}} factors through a subscheme of Ū_{V_{j}} finite over Ȳ. Finally, if X, W and Y are as above, we say that a rigid function f on X is overconvergent in W over Y if f extends to some strict neighborhood of X in W over Y. When Y is Max(K), we just say f is overconvergent on X in W.

We denote the ring of these functions on X which are overconvergent in W over Y by A^{†}(X/Y, W) and the subring of those functions bounded by 1 we denote A^{†}(X/Y, W)^{0}. We drop Y from the notation when it is a point and we drop W when it is understood from the context. In particular, the ring of overconvergent families of rigid analytic functions on Z_{1}(Nq) parameterized by V is denoted A^{†}(Z_{1}(Nq)_{V}/V). If B_{K}[0, 1] and B_{K}(0, 1) denote the affinoid and wide open unit disks over K, A^{†}(B_{Q_{p}}[0, 1]_{B_{Qp} (0,1)}/B_{Q_{p}}(0, 1))^{0} may be identified as the ring of power series in the parameter variable T over Λ^{(0)} which are of the form ∑_{n=0}^{∞} λ_{n}T^{n} such that for some positive real number a, λ_{n} ∈ I_{1}^{[an]} where I_{1} is the maximal ideal of Λ^{(0)}. (For more discussion of this notion see §A5 of [C-BMF] as well as §1 of [C-CCS].) It is less straightforward to define the notion of overconvergent modular form or family of overconvergent modular forms. Cf. section 2.4 below for the general definition but, for now, we define overconvergence only for integral weights:

Let ω^{k} for k ∈ Z denote the k-th tensor power of ω, and consider the vector space of rigid analytic sections of ω^{k} over the affinoid Z_{1}(N · p^{m})(v). More explicitly, letting K be a complete subfield of C_{p}, and Z_{1}(N ·p^{m})(v)_{/K} the affinoid over K induced from Z_{1}(N · p^{m})(v) by base change, define the K-vector space

M_{k}(Np^{m}, v; K) := ω^{k}(Z_{1}(N · p^{m})(v)_{/K}),

which has a natural K-Banach space structure, as described in section B2 of [C-BMF].

If v > 0, an element of M_{k}(Np^{m}, v; K) is called a v-overconvergent modular form (of weight k and level N · p^{m}). By definition an overconvergent modular form (of tame level N with coefficients in K) is a v-overconvergent modular form of level N · p^{m} for some v > 0, and some m > 0, i.e., it is an element in the p-adic Frechet space given by the union of these p-adic Banach spaces. (Compare this with the definition of overconvergent function, and form in A5 and B4 of [C-BMF] as well as in [C-COHL]).

The elements of M_{k}(Np^{m}, v; K) for v = 0, i.e., the space of rigid analytic sections of ω^{k} over Z_{1}(N · p^{m}) will be referred to simply as convergent; convergent modular forms on X_{1}(Np^{m}) of weight k coincide with Katz padic modular functions of that tame level and weight-character (see section 2.4 below).

The standard automorphisms and correspondences on the curve X_{1}(N · p^{m}) (which we shall list below) induce natural continuous operators on the Banach space M_{k}(Np^{m}, v; K). We explain this in detail in the next chapter. For now, we will discuss the diamond operators.

For any positive integer m, the group (Z/Np^{m}Z)^{∗} acts on the modular curve X_{1}(Np^{m}) over Q_{p} as follows. For r ∈ (Z/Np^{m}Z)^{∗}, the operator ⟨r⟩ sends the pair (E, α) to (E, r · α). This gives us a compatible action of the profinite group

Z_{p,N}^{∗} = lim_{←}(Z/Np^{m}Z)^{∗}

on the tower (∗) and more specifically, for every positive integer m, we have compatible actions of the finite quotient group (Z/Np^{m}Z)^{∗} of Z_{p,N}^{∗} on the pair

(∗∗) (X_{1}(Np^{m}) , ω_{X_{1}(Np^{m})}).

Lemma 2.1.1. The action described above induces a continuous homomorphism from the topological group Z_{p,N}^{∗} to the group Aut_{rig.an.}(Z_{1}(Np^{m})(v), ω_{Z_{1}(Np^{m})(v)}) of rigid analytic automorphisms of the pair consisting of the affinoid Z_{1}(Np^{m})(v) in X_{1}(Np^{m}) and the restriction of the line bundle ω to this affinoid. This induces a natural action of the Iwasawa algebra Λ_{N} as an algebra of continuous operators on the Banach space M_{k}(Np^{m}, v; K).

2.2 Eisenstein series.

In this section we work in tame level N = 1. We recall and continue the development of the theory of the fundamental family of modular forms, i.e., the Eisenstein family. Some of this theory was discussed and/or proved on pages 446-448 of [C-BMF] as well as in [C-CCS]. See also section 7.1 of Hida’s book [H-ET] for further proofs and discussion.

Let W = W_{N} for N = 1 and W^{+} denote that part of weight space W consisting of characters κ such that κ(−1) = 1. We have the p-adic ζ-function ζ^{∗}(κ) (notation as in section B1 of [C-BMF]) which is a rigid analytic function on W^{+} away from the point κ = 1 (at which it has a simple pole). If κ = (χ, s) is an accessible weight,

ζ^{∗}(κ) = L_{p}(χ, 1 − s),

where L_{p}(χ, s) is the Kubota-Leopoldt p-adic L-function.

For n ≥ 1 set

σ_{κ}^{∗}(n) := ∑_{d|n,(d,p)=1} κ(d)d^{−1},

which we view as an Iwasawa function on W^{+}. If κ ≠ 1 and ζ^{∗}(κ) ≠ 0, set

E_{κ}(q) = 1 + \frac{2}{ζ^{∗}(κ)} · ∑_{n=1}^{∞} σ_{κ}^{∗}(n)q^{n},

and for κ = 1, put E_{κ}(q) = 1. Compare the discussion on page 447 of loc. cit. Let us refer to this family of q-expansions E_{κ}(q), parameterized by the rigid analytic subspace

W_{eis} = {κ ∈ W^{+} and ζ^{∗}(κ) ≠ 0}

as the basic Eisenstein family. Note that B ⊂ W_{eis}. We also use the notation

E(q) := 1 + ϵ_{1}q + ϵ_{2}q^{2} + ⋯,

where the coefficients ϵ_{n} are the rigid analytic functions in A(W_{eis}) with the property that ϵ_{n}(κ) = 2σ_{κ}^{∗}(n)/ζ^{∗}(κ). The power series E(q) ∈ A(W_{eis})[[q]] has the corresponding property that

κ(E(q)) = E_{κ}(q).

One immediately checks that the power series E(q) is fixed under the action of the U-operator on A(W_{eis})[[q]] which takes power series ∑_{n=0}^{∞} a_{n}q^{n} to ∑_{n=0}^{∞} a_{pn}q^{n}. Recall that if κ = (χ, k) ∈ W^{+} is arithmetic and k > 0 and ζ^{∗}(κ) ≠ 0, then E_{κ}(q) is the q-expansion of a weight k classical (see Miyake [Mi] Chapter 7) (hence v-overconvergent, for any v) modular eigenform on X_{1}(l.c.m.(q, cond_{χ})) with character χτ^{−k}. Here cond_{χ} is the smallest power of p such that χ is trivial on 1 + cond_{χ} · Z_{p}. More generally, if κ = (χ, k) with k any integer we have that ζ^{∗}(κ)E_{κ}(q) is the q-expansion of a weight k overconvergent modular eigenform on X_{1}(l.c.m.(q, cond_{χ})) with character χτ^{−k} as we will show in Corollary 2.2.6 below when χ trivial on μ(Z_{p}). (The general case follows from this and Theorem 2.1 of [C-CCS] applied to ζ^{∗}(κ)E_{κ}(q)/E_{⟨κ⟩}(q)). (See also Proposition II.3.22 of [G-ApM].)

As discussed in [C-BMF, §B1] and elsewhere, if κ is trivial on the subgroup of roots of unity, μ(Q_{p}), in Q_{p}, then

|E_{κ}(q) − 1| < 1.

Definition. The restricted Eisenstein family E is the restriction of the basic Eisenstein family of tame level 1 to the subspace B ⊂ W (cf. notation of section 1.4) .

Remark 1. For any tame level N, as we saw in section 1.4, the rigid analytic group W_{N} is the disjoint union of a finite number of B-cosets of the form χ·B where χ is a character of finite order in D_{N}. This is important in that it will allow us to transform modular forms of arbitrary weights to functions on modular curves by division by Eisenstein series in the restricted Eisenstein family. But here we considered only tame level N = 1, and now restrict to the special case where χ = 1 is the trivial character.

Some important facts about the restricted Eisenstein family are given below.

Proposition 2.2.1. The q-expansion, E[[q]], of the restricted Eisenstein family has coefficients in Λ^{(0)}:

E(q) = 1 + ϵ_{1}q + ϵ_{2}q^{2} + ⋯ ∈ Λ^{(0)}[[q]],

and the coefficients ϵ_{j} for j ≥ 1 lie in the maximal ideal of Λ^{(0)}.

Proof. This is well known. For example, it is a consequence of Iwasawa’s theorem on p-adic L-functions [I] (cf. Theorem 16 of [S-FMZ]).

Theorem 2.2.2 (additivity). Let Z = Z_{1}(q). There exists a rigid analytic function on Z_{B×B} bounded by one and overconvergent over B × B (i.e. an element of A^{0}(Z_{B×B}/B × B))^{†}) with q-expansion

\frac{E_{α}(q)E_{β}(q)}{E_{αβ}(q)}.

Proof. Let E = E_{(τ^{0},1)}. This is the classical weight one Eisenstein series on X_{1}(q) used in Chapter B of [C-BMF]. For v(s) > 1 − 1/(p − 1) we know, by Corollary B4.5.2 of [C-BMF], that E_{(τ^{0},s)}(q)/E(q)^{s} is the q-expansion of a function on Z_{B^{∗}} overconvergent over B^{∗}. It is without zeroes on Z_{1}(q), since its q-expansion is congruent to 1, hence on Z_{1}(q)(v) for some v > 0. It follows that if v(s) and v(t) are greater that 1 − 1/(p − 1), α = (τ^{0}, s) and β = (τ^{0}, t) then

\frac{(E_{α}(q)/E(q)^{s}) · (E_{β}(q)/E(q)^{t})}{E_{αβ}(q)/E(q)^{s+t}}

is the q-expansion of a function on Z_{B^{∗}×B^{∗}} overconvergent over B^{∗} × B^{∗}. Hence, we obtain the conclusion of the theorem with B replaced by B^{∗}. We also note that the q-expansion coefficients of this function lie in Λ ˆ⊗Λ. We want to apply a generalization of Theorem 2.1 of [C-CCS] (i.e., Theorem 2.2.4 below) which we must now prepare to prove. Let I_{n} be the maximal ideal of Λ_{n} := Z_{p}[[S_{1}, …, S_{n}]]. For a ∈ Λ_{n} let v_{Λ}(a) = min{m: a ∈ I_{n}^{m}}. Suppose t = (t_{1}, …, t_{n}) is an n-tuple of real numbers < 1. We define a norm || ||_{t} on Λ_{n} by setting for

|| ∑_{M} b_{M}S^{M}||_{t} = max_{M}{|b_{M}|t^{M}}.

Here M is a multi-index. The norm || ||_{t} is the spectral norm on the disk B[t], |S_{i}| ≤ t_{i}, which is an affinoid when t_{i} ∈ p^{Q} for 1 ≤ i ≤ n. Set v_{t}(a) = − log ||a||_{t}.

Lemma 2.2.3. There exist positive constants A(t) and B(t) such that for all non-zero a ∈ Λ_{n},

A(t)v_{Λ}(a) ≤ v_{t}(a) ≤ B(t)v_{Λ}(a).

Proof. Let a = p^{m_{0}}S^{M} where M = (m_{1}, …, m_{n}) and m_{0} + ∑ M = m. Then,

||a||_{t} = |p|^{m_{0}} · t^{M},

so

v_{t}(a) = m_{0} log p + ∑_{i} m_{i}(− log t_{i}).

Thus the lemma follows if we take B(t) = max{log p, − log t_{1}, …, − log t_{n}} and A(t) = min{log p, − log t_{1}, …, − log t_{n}}.

As a corollary we get that a series over Λ_{n} which is convergent on B^{n} × B[0, 1] is overconvergent over the open unit n-polydisk B^{n} if and only if it is overconvergent over any single affinoid subpolydisk around the origin. We are ready to prove the generalization of Theorem 2.1 of [C-CCS] anticipated above. Let B_{n} be the n-dimensional unit polydisk. Then Λ_{n} may be regarded as the ring of rigid analytic functions on B_{n} defined over Q_{p} and bounded by 1.

Theorem 2.2.4 (q-expansion Principle). Suppose, t ∈ R^{n} and 0 < t_{i} < 1. Then, if G ∈ A^{†}(Z_{B[t]}/B[t]) and G(q) ∈ Λ_{n}[[q]], G uniquely analytically continues to an element of A^{†}(Z_{B_{n}}/B_{n})^{0}.

Proof. The proof goes exactly like that of Theorem 2.1 in [C-CCS] and we employ the notation of that paper. In particular if Y is an affinoid, A^{0}(Y)[X]^{†} denotes the ring of power series with coefficients in A(Y) which overconverge on B[0, 1] × Y over Y and are bounded by 1 there. In particular, if Y is the origin consider as a rigid space over Q_{p}, A^{0}(Y)[X]^{†} = Z_{p}[X]^{†}. We apply Lemma 2.2 of [C-CCS] which asserts, that there exists a finite morphism f from Z^{†} onto B[0, 1]^{†} such that f^{−1}(0) = ∞ and f̄ is separable. We base change f to a finite morphism from (Z_{B[t]}/B[t])^{†} to (B[0, 1]_{B[t]}/B[t])^{†} and let T r_{f} be the corresponding trace map. We let X be the standard parameter on A^{1} and conclude that for r ∈ A^{†}(Z)^{0}, T r_{f}(rG) is in both Λ_{n}[[X]] and A^{†}(B[0, 1]_{B[t]}/B[t]). It follows from this that if D generates the discriminant ideal in Z_{p}[X]^{†} of A^{†}(Z)^{0}/Z_{p}[X]^{†}, then DG ∈ A^{†}(Z_{B_{n}}/B_{n}). Since f̄ is separable, p |̸D. Then Lemma 2.3 of [C-CCS] generalizes immediately to

Lemma 2.2.5. Let t ∈ R^{n}, 0 < t_{i} < 1. Suppose a(X) ∈ A^{0}(B[t])[X]^{†} and suppose that there exists a D(X) ∈ Z_{p}[X]^{†} such that p |̸D(X) and D(X)a(X) ∈ Λ_{n}[X]^{†}, then a(X) ∈ Λ_{n}[X]^{†}.

However, we feel that one of the assertions made in the proof of Lemma 2.3 of [C-CCS] requires more justification. Namely, we will now justify the convergence of the sum (in the notation of [C-CCS]) S := ∑_{n} λ_{n}h_{n}(X). Recall, h_{n}(X) is an element of Z_{p}⟨X⟩ satisfying

X^{n} − r_{n}(X) = D(X)h_{n}(X),

where D(X) ∈ Z_{p}[X]^{†} and r_{n}(X) is either 0 or a polynomial of degree strictly less than d over Z_{p}. This is already enough to show that h_{n}(X) is overconvergent but we need more. Let δ^{−1} > s > 1 so that D(X) converges on B[0, s] and has no zeroes on the half open annulus A(1, s]. It follows that h_{n}(X) converges on B[0, s]. Let C = |1/D(X)|_{A[s,s]}. It follows that for large n, |h_{n}(X)| is at most Cs^{n} on the circle A[s, s] and hence (by the maximum modulus principle) on the disk B[0, s]. It follows, since |λ_{n}|_{t} ≤ δ^{n}, that the sum S converges to an element of A^{0}(B[t])[X]^{†} as claimed. We conclude the proof of the Theorem 2.2.4 the same way as that of Theorem 2.1 of [C-CCS]: Suppose b_{1}, …, b_{d} is a basis for A^{0}(Z^{†}) over Z_{p}[X]^{†}. We may write,

G = a_{1}(X)b_{1} + a_{2}(X)b_{2} + ⋯ + a_{d}(X)b_{d}

where a_{i}(X) ∈ A^{0}(B[t])[X]^{†}. Then as DG ∈ A^{†}(Z_{B_{n}}/B_{n})^{0}, it follows, since b_{1}, …, b_{n} is also a basis for A^{†}(Z_{B_{n}}/B_{n})^{0} over B_{n}, that D(X)a_{i}(X) ∈ Λ[X]^{†}. Thus we may apply the previous lemma and deduce Theorem 2.2.4.

We now give a complete proof of a claim made in §B1 of [C-BMF].

Corollary 2.2.6. If κ ∈ B is arithmetic of weight k and nebentypus ϵ, then E_{κ}(q) is the q-expansion of an overconvergent modular form in M_{k}(p^{m}, v, Q_{p}(μ_{p^{m}})) for some v > 0 where p^{m} is LCM(q, cond_{ϵ}).

Proof. This is automatically true for k > 0 since then E_{κ} is classical of the appropriate weight and level. Moreover, in this case, knowledge about the field of definition follows from the q-expansion principle. In general, we proceed by induction on −k. Suppose the corollary is true when k > −n. Let α ∈ B be the arithmetic character with coordinates (τ^{0}, 1). Suppose k = −n. Then Theorem 2.2.4 implies E_{κ}(q) equals

\frac{E_{ακ}(q)}{E_{α}(q)}F (q),

where F is an overconvergent function on Z_{1}(Nq). Since E_{α} reduces modulo p to the (p − 1)-st root of the Hasse invariant on Z_{1}(q), it has no zeroes there and hence no zeroes in a strict neighborhood of this affinoid. The corollary now follows as the induction hypothesis applies to E_{ακ}(q) which has weight −n + 1.

.

Proposition 2.2.7. Let ℓ be a prime number. The ratio of E(q) and E(q^{ℓ}),

E_{ℓ}(q) := E(q)/E(q^{ℓ}),

is the q-expansion of a rigid analytic function on Z_{1}(q, ℓ)×B if ℓ ≠ p, and on Z_{1}(q) × B if ℓ = p, overconvergent over B. The function E_{ℓ}(q) has Fourier coefficients in Λ^{(0)}, its constant term being 1, and all other coefficients lying in the maximal ideal of Λ^{(0)}.

Proof. The proof of existence of the overconvergent rigid analytic function runs along the same lines as the proof of Corollary 2.1.1 of [C-CCS] which is this proposition with ℓ = p, as did that of the additivity theorem, with Z replaced by Z_{1}(q, ℓ) if ℓ ≠ p. The statement about the coefficients follows from Proposition 2.2.1.

Remarks. For fuller control of the geometry of the eigencurve it would be useful to know explicit affinoid regions of the type X × Z_{1}(Np^{m})(v) on which some, or all of the functions E_{ℓ}(q) converge. (For a preliminary result along these lines, in the case of p = 2 see forthcoming work of M. Emerton.)

We also feel that it is generally a good idea for anyone who is just beginning the study of p-adic modular forms to spend time specifically concentrating on, and understanding the nature of, the Eisenstein family. It is at the root of much of the theory. For example:

1. (It is the ur-example of a p-adic family.) The Eisenstein family provides a very explicit example of a p-adically varying Fourier expansion (of eigenforms) parameterized by p-adic weight which interpolates to give classical modular eigenforms at positive integral weights. To our knowledge, Serre was the first person to signal the p-adic analytic variation of the Fourier coefficients of the Eisenstein family, and the fact that this could be used to provide an alternate construction of p-adic L-functions.

2. (It provides a way of understanding Hida families.) Given a classical parabolic modular eigenform f of tame level N and weight k ≥ 2 which is (p-)ordinary (i.e., which has slope zero), the restricted Eisenstein family can be used in conjunction with Hida’s projection operator to the subspace of ordinary modular forms, to enable one to see, a bit more concretely than is offered by the general theory, the analytically varying family of ordinary eigenforms of which f is a member. Briefly, just multiply f by E_{κ}(q) where κ ranges through B and then project this family to a family of ordinary modular forms using Hida’s projection operator. In the special case where the rank of the space of ordinary modular eigenforms of that tame level is 1 then things are particularly easy: the projection would then be family of ordinary modular eigenforms (for all Hecke operators) parameterized by an open subspace of weight space. But the general case is hardly more difficult.

3. (Multiplying by Eisenstein series provides us with a convenient way of passing from weight 0 to general weight. In this way we get a convenient definition of overconvergence for modular forms of arbitrary p-adic weight, and for families of modular forms by requiring the corresponding form of weight 0, or family of forms of weight 0, to be overconvergent. See, for example, the definitions given in B4 of [C-BMF], and repeated in section 2.4 below. Moreover, the multiplier function E_{p} enables one to compare the operation of the Atkin-Lehner operator U_{p} on weight 0 modular functions to the same operator on modular forms of general weight. See [C-BMF] and section 2.4. The fact that the Fourier coefficients of E_{p} are nice is one of the key ingredients for a good p-adic analytic dependence (on weight) of U_{p}-eigenvalues.

2.3. Katz p-adic Modular Functions.

For the beginning of this section fix any prime number p and an arbitrary tame level N. Denote the Iwasawa algebra Λ_{N} by Λ, for short. By a Λadic ring, we mean an algebra over Λ which is complete with respect to the Λ-adic topology, i.e., B is such a ring if and only if B is a Λ-algebra and B ≅ lim_{↼} B/rad(Λ)^{n}B where rad(Λ) is the radical of Λ.

Definition. Let B be a Λ-adic ring. By a Katz p-adic modular function over B (of tame level N) we mean a function F which assigns to any isomorphism class (E, α_{∞}, ν) of trivialized elliptic curves with level Nstructure (ν) over a Λ-adic B-algebra D an element F (E, α_{∞}, ν) ∈ D, and whose formation commutes with base change.

For a discussion of this notion and its variants (e.g., cuspidal Katz p-adic modular functions) see Chapter 4 of [K-PMF], section I.3.1 of [G-ApM], and all of [K-2], especially the appendix there. If F is a Katz modular function of tame level N over the Λ-adic ring B and a ∈ Z_{p,N}^{∗}, a = (b, c) where b ∈ (Z/NZ)^{∗} and c ∈ Z_{p}^{∗}, we set

F |⟨a⟩(E, α_{∞}, ν) = F (E, c^{−1}α_{∞}, bν),

for all isomorphism classes of triples (E, α_{∞}, ν) of trivialized elliptic curves with level N structure over a B-algebra D. If ψ : Z_{p,N}^{∗} → B^{∗} is a continuous homomorphism we will say that F has weight-character ψ if F |⟨a⟩ = ψ_{D}(a)F for all a ∈ Z_{p,N}^{∗} where ψ_{D} is the composition of ψ with the homomorphism B^{∗} → D^{∗}, induced from the structure homomorphism of the B-algebra D.

For any p-adically complete Z_{p}-algebra B, let V_{n,m}(B) be as in [G, §I.3.1]. Let Np^{m} ≥ 5. The ring V_{n,m}(B) is the ring of functions on Ẑ_{1}(Np^{m}) × B/p^{n}B where Ẑ_{1}(Np^{m}) is the underlying formal scheme over Z_{p} which is in the affinoid connected component Z_{1}(Np^{m}) of ∞ in the ordinary locus of X_{1}(Np^{m}). Then the ring of Katz holomorphic p-adic modular functions over B (of tame level N) is

lim_{↼}_{n} lim_{⇀}_{m} V_{n,m}.

For this, see ibid. and [K-2].

Proposition 2.3.1. The q-expansion E(q) is the q-expansion of a Katz p-adic modular function E over Λ^{(0)} with the identity weight-character.

Proof. Fix an integer n ≥ 0. Let R_{n} = Λ^{(0)}/([1 + q]^{p^{n}} − (1 + q)^{2p^{n}}). Then Λ = lim_{↼} R_{n}. Let E_{n}(q) be the restriction of E(q) to R_{n}. Let

χ(a) = ⟨⟨a⟩⟩^{2}ψ(a) where ψ is a character of finite order on 1 + qZ_{p}. Then since χ(E(q)) is the q-expansion of a classical weight 2 modular form with character ψτ^{−2}, and

[1 + q]^{p^{n}} − (1 + q)^{2p^{n}} = ∏_{ψ} ([1 + q] − (1 + q)^{2}ψ(1 + q)),

where the product is over all characters ψ on 1 + qZ_{p} trivial on 1 + qp^{n}Z_{p}, E_{n}(q) is the q-expansion of a Katz p-adic modular function over the normalization of R_{n}. It follows from the q-expansion principle that it is, in fact, the q-expansion of a Katz p-adic modular function over R_{n} and hence, passing to the inverse limit, E(q) is the q-expansion of a Katz p-adic modular function over Λ^{(0)}.

2.4 Convergent modular forms and Katz modular functions.

Let p be a prime number, N a tame level, and m ≥ 1, and assume, for this section, that Np^{m} ≥ 5.

Definition. We say F (q) = ∑_{n=0}^{∞} a_{n}q^{n}, a_{n} ∈ K, is the q-expansion of a convergent (resp. overconvergent) form of tame level N and weightcharacter κ ∈ W over K if F (q)/E_{⟨κ⟩}(q) is the q-expansion of a rigid (resp. overconvergent) function on Z_{1}(Nq) in X_{1}(Nq) of character κ/⟨κ⟩ for the action of (Z/NqZ)^{∗}. The K-vector space of these q-expansions with the property that the corresponding function converges on Z_{1}(Nq)(v) will be denoted M_{κ}(K, N, v).

If U is an admissible open subspace of B we also say that

F (q) = ∑_{n=0}^{∞} a_{n}q^{n},

a_{n} ∈ A(U), is the q-expansion of a family of convergent (resp. overconvergent) forms over U of tame level N if F (q)/E(q) is the q-expansion of a rigid (resp. overconvergent) function on U × Z_{1}(Nq). We say this family has type δ ∈ D_{N} if this function has character δ for the action of (Z/NqZ)^{∗}. We call the A(U) module of these forms M_{U}^{†}(N), and for v > 0, we call M_{U}^{†}(N)(v) the submodule consisting of those families over U whose respective functions converge on U × Z_{1}(Nq)(v).

Thus, in particular, it follows from the above definition that E(q^{p}) is the q-expansion of a family of overconvergent forms over B of tame level 1 and type 1. Note that when κ is arithmetic of weight k, M_{κ}(K, N, v) is contained in M_{k}(Np^{m}, v, K) for p^{m} = LCM{q, cond_{χ}} where χ is the wild part of κ for small enough v > 0 because of Corollary 2.2.6. Also, as an immediate consequence of Theorem 2.2.2 we get

Proposition 2.4.1. If F (q) and G(q) are the q-expansions of overconvergent forms of weight-characters α and β, then F (q)G(q) is the q-expansion of an overconvergent form of weight character αβ.

Theorem 2.4.2. Let F be a convergent family of modular forms over an affinoid subspace X of W, with q-expansion coefficients in A^{0}(X). Then the q-expansion of F is the q-expansion of a Katz modular function F̃ over A^{0}(X).

Proof. What we have is that F (q)/E(q) is the q-expansion of a rigid analytic function on X × Z_{1}(q). But

A^{0}(X × Z_{1}(q)) = A^{0}(X) ˆ⊗A^{0}(Z_{1}(q))

is just V_{∞,i}(A^{0}(X)), where i = 2 if p = 2 and i = 1 otherwise, so F (q)/E(q) is the q-expansion of a Katz modular function over A^{0}(X). The result now follows from 2.3.1 since A^{0}(X) is naturally a Λ-adic ring.

Chapter 3. Hecke Algebras

Fix a prime number p and a tame level N relatively prime to p. The precise conditions on these will be given at the head of each section below. Whenever we deal with a level written as Np^{m} we assume that Np^{m} ≥ 5.

3.1 Hecke eigenvectors and generalized eigenvectors.

In this section we work with arbitrary p and level N. By H = H_{N} let us mean the (commutative) polynomial algebra over the topological ring Λ_{N} in the infinitely many variables labelled T_{ℓ} for prime numbers ℓ not dividing N · p and U_{q} for primes q dividing N · p. We view H as a topological Λ_{N}-algebra given its weak topology. More explicitly, if for any finite set of monomials S in the variables T_{ℓ} and U_{q} we let H_{S} denote the free Λ_{N}- submodule of H of finite rank generated by the monomials in S, given the product topology it inherits from the topology on Λ_{N}, a subset of the topological ring H is open if and only if its intersection with every H_{S} is open. In particular, continuous ring homomorphisms from H have the property that their restriction to Λ_{N} are continuous. Purely formally, we may define the universal q-expansion

∑_{n=1}^{∞} T_{n} · q^{n} ∈ H[[q]]

by the following rules: For prime numbers ℓ which do not divide Np, we put T_{ℓ} = T_{ℓ}; for prime numbers ℓ which divide Np, we put T_{ℓ} = U_{ℓ}; and we define T_{n} for any positive integer n by the finite recursive relations summarized by the equality of formal Dirichlet series

∑_{n=1}^{∞} T_{n} · n^{s} = ∏_{ℓ}(1 − T_{ℓ} · ℓ^{−s} + [ℓ] · ℓ^{−2s})^{−1},

where the product is taken over all prime numbers ℓ, and where [ℓ] is defined to be 0 if ℓ divides pN and to be the image of ℓ ∈ lim_{↼}(Z/Np^{n}Z)^{∗} ⊂ Λ_{N} if ℓ doesn’t divide pN.

If Ψ : H → C_{p} is a C_{p}-valued character (i.e., a continuous ring homomorphism) by the weight-character of Ψ we mean the homomorphism w = w_{Ψ} : Λ_{N} → C_{p} which is the restriction of Ψ to Λ_{N}. It is sometimes useful to refer to the q-expansion

∑_{n=1}^{∞} Ψ(T_{n}) · q^{n} ∈ C_{p}[[q]]

as the Fourier expansion of the character Ψ. For any such Ψ : H → C_{p}, it is convenient to define the following ideal: First consider the ring H⊗_{Λ_{N}} C_{p} where the tensor product is made via the weight-character w : Λ_{N} → C_{p}. Define

m_{Ψ} ⊂ H ⊗_{Λ_{N}} C_{p}

to be the ideal generated by all elements of the form τ ⊗1⊗Ψ(τ) for τ ∈ H.

Definition 1. Given a continuous homomorphism w : Λ_{N} → C_{p} and a power series

f(q) := ∑_{n=1}^{∞} a_{n} · q^{n} ∈ C_{p}[[q]],

say that f(q) is a normalized Hecke eigenvector of weight w if there is a character Ψ : H → C_{p} of weight-character w whose Fourier expansion is f(q).

There are two other equivalent (and more standard) ways of formulating this notion of f(q) being a (normalized) Hecke eigenvector of weight w. To give these definitions, let us fix a weight-character w, and define the weight w action of H on C_{p}[[q]] as follows: we let the coefficient ring Λ_{N} act on C_{p}[[q]] via scalars through the homomorphism w, and define the action of T_{ℓ} by the standard rule: For primes ℓ,

f|_{w}T_{ℓ} = ∑_{n=1}^{cal1} b_{n} · q^{n} ∈ C_{p}[[q]],

where b_{n} = a_{n/ℓ}+ℓ^{−1}w([ℓ])·a_{nℓ}, where we set a_{n/l} = 0 if l |̸n and w([ℓ]) = 0 if ℓ|Np. The weight w action of H on C_{p}[[q]] extends naturally to an action of the tensor product H ⊗_{Λ_{N}} C_{p} considered above ( this tensor product being made via the weight-character w : Λ_{N} → C_{p}).

Definition 2. The power series f(q) := ∑_{n=1}^{∞} a_{n} · q^{n} ∈ C_{p}[[q]] is a normalized Hecke eigenvector of weight w if and only if the coefficient a_{1}, of q^{1} in its Fourier expansion is equal to 1, and if there is a character Ψ : H → C_{p} of weight-character w such that f(q) is an eigenvector for the weight w action of T_{ℓ} on C_{p}[[q]] with eigenvalue Ψ(T_{ℓ}), for all prime numbers ℓ. Equivalently we may ask that a_{1} = 1, and that the power series f(q) ∈ C_{p}[[q]] be annihilated by the ideal m_{Ψ} ⊂ H⊗_{Λ_{N}} C_{p} acting on C_{p}[[q]]. If this is the case, then we refer to Ψ as the Hecke eigenvalue character of f(q).

Definition 3. We will say that a nonzero power series f(q) := ∑_{n=1}^{cal1} a_{n} ·q^{n} in C_{p}[[q]], is a generalized Hecke eigenvector or just a generalized eigenvector of weight w if there is a character Ψ as above, of weightcharacter w, and some positive integer ν such that f(q) ∈ C_{p}[[q]] is annihilated by the ideal (m_{Ψ})^{ν} ⊂ H ⊗_{Λ_{N}} C_{p}. If f(q) is such a generalized eigenvector, we refer to Ψ as the Hecke eigenvalue character underlying f(q).

3.2. Action on M_{k}(Np^{m}, v; K).

In this section we work with arbitrary p and square-free tame level N, and as usual, when a level Np^{m} is invoked, we assume that Np^{m} ≥ 5. For prime numbers ℓ not dividing N · p we have the Hecke operators T_{ℓ}, and for primes q dividing N · p, we have the Atkin-Lehner operators U_{q}. These operators come from correspondences, all defined, when Np^{m} ≥ 5, on the modular curve X_{1}(Np^{m}) (cf. [M-W]) all commute with each other, and also commute with the diamond correspondences ⟨d⟩ for d ∈ (Z/Np^{m}Z)^{∗}.

Lemma 3.2.1. For q ≠ p, the Hecke, Atkin-Lehner and diamond correspondences T_{l}, U_{q} and ⟨d⟩ induce rigid analytic correspondences on the affinoid Z_{1}(Np^{m})(v) and compatible correspondences on the restriction of ω to this affinoid. These operators induce continuous Λ_{N}-linear operators on the Banach space M_{k}(Np^{m}, v; K) which commute with each other.

Proof. The main point is that if E is an elliptic curve over a scheme S of characteristic p and V er: E^{(p)} → E is Verschiebung, then if ν is a generator of Ω_{E/S}^{1},

V er^{∗}ν = A(E, ν)ν^{(p)}.

This is equivalent to the dual formula of Katz on page 97 of [K-PMF]. From this one can deduce that if h: F → E is an étale isogeny, then

A(F, h^{∗}ν) = A(E, ν).  (1)

This means one can define operators on M_{k}(Np^{m}, v; K) in the standard way. Eg., suppose m = 1 and ℓ is a prime, such that (ℓ, Np) = 1. Then if (E, α, C) is a triple corresponding to an R-valued point on Z_{1}(Np^{m}, l)(v), where R is the ring of integers in a finite extension of Q_{p} and v ∈ I_{1}, and η is a generator of Ω_{E/R}^{1}, then

v(A(\overline{E}, η̄)) = v,

where (\overline{E}, η̄) is the reduction of (E, η) modulo p. Then the image of α(μ_{p}) in E/C is the canonical subgroup and (1) implies

v = v(A(\overline{E/C}, ξ̄))

where ξ is any generator of Ω_{(E/C)/R}^{1}. Thus the two maps from X_{1}(Np, ℓ) to X_{1}(Np) take Z_{1}(Np, ℓ)(v) onto Z_{1}(Np)(v) for v ∈ I_{1}.

We also have a completely continuous operator corresponding to U_{p} acting on M_{k}(Np^{m}, v; K) for v ∈ I_{m} (see section B2 of [C-BMF]). Note: One must stipulate, specifically, whether one means to take these operators as the ones induced from direct image or the inverse image of the geometric automorphisms or correspondences listed. Our convention will be to take the direct image, i.e.,

⟨r⟩ = ⟨r⟩_{∗}, T_{ℓ} = T_{ℓ ∗}, U_{q} = U_{q ∗}.

Had we made the other choice, i.e., had we taken the inverse image, we would have had

⟨r^{−1}⟩^{∗} = ⟨r⟩_{∗}, T_{ℓ}^{∗} = ⟨ℓ⟩ · T_{ℓ ∗},

etc.; cf. the discussion and formulas in [M-W] Ch. 2, section 5.

We view H as acting on the (Banach) Λ_{N}-modules M_{k}(Np^{m}, v; K) in the evident manner. We will also have use for notation for the sub-Λ_{N}- algebra H′ ⊂ H generated by all the T_{ℓ}’s, for prime numbers ℓ not dividing N · p. Thus when N = 1, H is the polynomial algebra over H′ generated by the single symbol U_{p}, i.e., by the Atkin-Lehner operator at p. If Φ is a C_{p}-valued character on H, and if a modular form f ∈ M_{k}(Np^{m}, v; C_{p}) is an eigenvector for the action of H with Hecke eigenvalue character equal to Φ then if f(q) ∈ C_{p} is the Fourier expansion of f and if the coefficient of q in f(q) is 1, then f(q) is a normalized eigenvector in the sense of §3.1, of weight w_{Φ}. We refer to f and f(q) simply as normalized (overconvergent, modular) eigenforms; if f(q) is a generalized eigenvector in the sense §3.1, we refer to f and f(q) as generalized (overconvergent, modular) eigenforms.

3.3 Action on Katz Modular Functions.

Let N be any square-free tame level. By the Katz-Hecke Algebra T_{p}(N) we shall mean the completion of the Λ_{N}-algebra generated by the Hecke operators and diamond operators (as described in §2.3) over Z_{p} acting (faithfully) on the space V of all Katz p-adic modular functions of tame level N with respect to the compact open topology on the ring of Z_{p} endomorphisms of V (see section II.1 of [G-ApM]). As Hida shows, in Theorems 3.1 and 3.2 of [H-HA],

T_{p}(N) ≅ lim_{↼}_{ν} h_{k}(Np^{ν})

for any weight k ≥ 2 where h_{k}(Np^{ν}) is the Hecke algebra acting on weight k modular forms of level Np^{ν} over Z_{p}.

As we will show in section 5.2 (for p odd and tame level N = 1) there is a natural homomorphism R_{p} → T_{p}(1) =: T_{p}, where R_{p} is the universal deformation ring of two dimensional p-adic pseudo-representations of the Galois group of Q of tame level N.

Proposition 3.3.1. If p ≠ 2, the ring T_{p} is a complete semi-local noetherian ring with finite residue fields.

Remark. We will give a proof of Noetherian-ness motivated by the proof of Corollary III.5.7 [G-ApM], in Corollary 5.2.3 below. We will prove the rest of the proposition now (for any tame level N). Proof. . As remarked, for any fixed weight k ≥ 2, T_{p,N} is the projective limit of the Hecke algebras h_{k}(Np^{ν}) acting (faithfully) on weight k modular forms of level Np^{ν} over Z_{p}. This is enough to show it is complete, since h_{k}(Np^{ν}) is. By duality, any homomorphism of h_{k}(Np^{ν}) into a field of characteristic p comes from an eigenform of this level modulo p. Since there are only finitely many, say n, systems of eigenvalues of tame level N (cf. the Proposition 5.1.1), for large ν, there exist maximal ideals m_{1}(ν), …, m_{n}(ν) of h_{k}(Np^{ν}) such that m_{i}(ν + 1) is the inverse image of m_{i}(ν) and

h_{k}(Np^{ν}) = ⨁_{i}(h_{k}(Np^{ν}))_{m_{i}(ν)}.

Hence, m_{i} := lim_{↼}_{ν} m_{i}(ν) is a maximal ideal of T_{p,N}, (T_{p,N})_{m_{i}} is a complete

local ring and

T_{p,N} = ⨁_{i}(T_{p,N})_{m_{i}}.

3.4. Action on M^{†}(N).

In this section we work with arbitrary p, square-free tame level N, and we assume that Np^{m} ≥ 5. Let E(q) ∈ Λ^{(0)}[[q]] be the q-expansion of the restricted Eisenstein family, so that

κ(E(q)) = E_{κ}(q)

for all κ ∈ B. By Proposition 2.2.7, for any prime number ℓ the (“multiplier”) function E_{ℓ} whose q-expansion is E(q)/E(q^{ℓ}) is an overconvergent function on Z_{1}(q · ℓ) if ℓ ≠ p, on Z_{1}(p) if ℓ = p ≥ 5, and when p = 2 or p = 3 and ℓ = p we work, rather, on the affinoid Z_{1}(5q), making use of the strategy explained in section 2.5.

We set M^{†}(N) = M_{B}^{†}(N) where the latter was defined in §2.4. This is an alteration of that given in section B.4 of [C-BMF] where the definition was the same except that B was replaced by B^{∗}. For prime numbers ℓ, we will define operators T_{ℓ} copying Lemma B5.1 of [C-BMF] mutatis mutandis. If F ∈ M^{†}(N) and ℓ ∈ (Z/NZ)^{∗} × Z_{p}^{∗} define

F |⟨ℓ⟩^{∗} = ([⟨⟨ℓ⟩⟩] · E)(\frac{F}{E} |⟨ℓ⟩).

For prime numbers ℓ let ψ_{ℓ} be the operator on A(B)[[q]] given by

ψ_{ℓ}(∑_{n} a_{n}q^{n}) = ∑_{n} a_{nℓ}q^{n}.

Lemma 3.4.1. For each prime number ℓ there is a unique continuous operator T (ℓ) on M^{†}(N) such that

F |T (ℓ)(q) = ψ_{ℓ}(F (q))

when ℓ|Np, and

(F |T (ℓ))(q) = ψ_{ℓ}(F (q)) + ℓ^{−1}(F |⟨ℓ⟩^{∗})(q^{ℓ})

when ℓ does not divide Np. In fact, for each affinoid X of B, and each prime ℓ there is a v_{ℓ}(X) > 0 such that the operator T (ℓ) preserves the subspace M(N, X)(v) for all v in the range 0 < v ≤ v_{ℓ}(X).

The proof of the existence of the operators goes exactly the same as that of Lemma B5.1 of ibid., to obtain the assertion about T (ℓ) and v_{ℓ}(X), one must use Proposition 2.2.7 and the property of the Hasse invariant used in the proof of Lemma 3.2.1. The nature of the v_{ℓ}(X) depends on the analytic properties of E_{ℓ}(q).

Proposition 3.4.2. The mapping F ↦ F̃ from overconvergent modular forms to Katz modular functions respects the Hecke module structure.

Proof. We omit the proof of this proposition in this article, noting firstly that the main thing to check here is the action of the diamond operators since the other operators can be checked by looking at q-expansions, and noting secondly that this check on the action of the diamond operators is not trivial: it will be done in a subsequent article.

If we could show that the v_{ℓ}(X) could be chosen independently of ℓ we would be in a position to put a natural topology on the algebra generated by the above operators (but see Chapter 7).

3.5. Action on weight κ forms.

We use the same formulas as in the lemma of the last section. The existence of the operators follows as a corollary of that lemma by specialization. It follows from Proposition 3.4.2 that action of each of the Katz Hecke operators corresponding to T (ℓ) and ⟨d⟩ preserve M_{κ}(K, N, v) (see section 2.4) for v > 0 and sufficiently small.

3.6. Remarks about cusp forms and Eisenstein series.

For this discussion, let p > 2 and let us assume the tame level N is equal to 1. Let f be a (p-adic) overconvergent modular eigenform (of tame level 1) of finite slope. Consider its Fourier expansion f̃ = ∑_{n=0}^{∞} a_{n}q^{n} (with coefficients a_{n} = a_{n}(f) ∈ C_{p}).

Definition. We say that f is a cuspidal-overconvergent eigenform if a_{0} = 0; that is, if the constant term of its Fourier expansion vanishes.

We have included the hyphen between cuspidal and overconvergent because our condition only requires the parabolicity condition a_{0} = 0 at the cusp ∞ and therefore, for example, even certain classical Eisenstein series whose Fourier expansion at the cusp ∞ have no constant term, but whose Fourier expansion at the cusp 0 have nonzero constant term, would be counted by the above definition as cuspidal-overconvergent, even though when considered as a classical eigenform they would not be cuspidal. This is specifically the case for the eigenform on X_{0}(p) which was called in [GM-FM,CPS] the “evil twin” of the Eisenstein series of weight k and level 1.

Let us examine the properties of a non-cuspidal-overconvergent eigenform f of weight κ, of tame level 1, and of finite slope, whose Fourier expansion we may normalize to be in the form: f̃ = 1 + a_{1}q + · · ·. Let λ_{ℓ} denote the eigenvalue of the operator T_{ℓ} on f (for primes ℓ ≠ p) and λ_{ℓ} the eigenvalue of U_{p}, we have

λ_{ℓ} = λ_{ℓ} · a_{0}(f) = a_{0}(f|T_{ℓ}) = a_{0}(f)(1 + ℓ^{−1}κ([ℓ])) = (1 + ℓ^{−1}κ([ℓ])),

and similarly λ_{p} = λ_{p} · a_{0}(f) = a_{0}(f|U_{p}) = a_{0}(f) = 1. It follows that the eigenvalues of the operators T_{ℓ} and U_{p} on such an f are determined solely by the weight κ, and the eigenvalues for the operators T_{n} for n ≥ 1 are given by σ_{κ}^{∗}(n) as in section 2.2. These eigenvalues are equal to the corresponding eigenvalues of the operators in question on E_{κ} , the member of weight κ of the basic Eisenstein family, as discussed in section 2.2 (for the weights κ for which E_{κ} is defined; i.e., when ζ^{∗}(κ) ≠ 0). In particular, all these eigenforms f are of slope 0. Now make a similar computation on the coefficient a_{1}, using the above evaluation of the eigenvalues, to give a formula for all of the coefficients a_{n}(f) (n ≥ 2) as specific multiples of a_{1}(f) where the multiples depend only up n and the weight κ. For example :

a_{ℓ}(f) = a_{1}(f|T_{ℓ}) = λ_{ℓ} · a_{1}(f) = (1 + ℓ^{−1}κ([ℓ])) · a_{1}(f).

Proposition 3.6.1. The only normalized (in the sense of the previous paragraph) non-cuspidal-overconvergent eigenforms of tame level 1 are Eisenstein series. Specifically, for any weight κ, if ζ^{∗}(κ) ≠ 0 then the unique normalized non-cuspidal-overconvergent eigenform of weight κ of tame level 1 is E_{κ}, while if ζ^{∗}(κ) = 0 there are no normalized non-cuspidal-overconvergent eigenforms of weight κ of tame level 1.

Proof: Step 1. We first show that if κ ≠ 0, and if f and g are two normalized non-cuspidal-overconvergent eigenforms (of tame level 1) of weight κ, then a_{1}(f) = a_{1}(g). Suppose that we are given two such eigenforms with a_{1}(f) ≠ a_{1}(g); we may assume with no loss of generality that a_{1}(g) ≠ 0. We already know that these eigenforms have the same eigenvalues for all the T_{ℓ}’s and for U_{p}. Now consider the modular form

f − \frac{a_{1}(f)}{a_{1}(g)} · g,

which is an eigenform of weight κ and has Fourier expansion identically equal to the non-zero constant 1 − a_{1}(f)/a_{1}(g). But an overconvergent eigenform whose Fourier expansion is a constant is necessarily of weight 0 (for a discussion of this fact, see, e.g. sections 4.4, 4.5 of [K-PMF]).

Step 2. We next show that if ζ^{∗}(κ) = 0, there are no normalized noncuspidal-overconvergent eigenforms (of tame level 1) of weight κ. For in this case, despite appearances, E_{κ} is cuspidal-overconvergent, and also κ ≠ 0 (for at κ = 0, ζ^{∗} has a pole and not a zero). If there were a normalized noncuspidal-overconvergent eigenform f (of tame level 1) of weight κ, then by the discussion above, for an appropriate constant c, f −c·E_{κ} would have its Fourier expansion equal to the constant 1, which is impossible since κ ≠ 0.

Step 3. It remains to deal with the case of weight κ = 0. Here, the Eisenstein series E_{κ} has its Fourier expansion equal to the constant 1, and if there were another normalized non-cuspidal-overconvergent eigenform f (of tame level 1) of weight 0, then f − E_{κ} would have its Fourier expansion equal to a nonzero constant times ∑_{n=1}^{∞} σ_{−1}^{∗}(n)q^{n}. But this latter Fourier series is not the Fourier series of an overconvergent eigenform, by Lemma 4 of [CGJ], and therefore such an f does not exist.

Chapter 4. Fredholm determinants.

4.1. Completely continuous operators and Fredholm determinants.

Let p be any prime number, N a square-free tame level. The levels Np^{m} and qN which occur below are assumed to be ≥ 5. The results employed in this section are mainly due to Serre [S 1]. A special role is played in this theory by the Atkin-Lehner operator U_{p} ∈ H for the important reason that it acts completely continuously (cf. section A1 of [C-BMF] for the definition) on M_{k}(Np^{m}, v; K) for k ∈ Z (cf. 3.11 and 3.12 of [K-PMF], B3 of [C-BMF], and Proposition II.3.15 of [G-ApM]), and we’ll show it acts essentially completely continuously on the p-adic Frechet space of overconvergent forms of any weight κ ∈ W over C_{p}

M_{κ} := lim_{⇀}_{v→0^{+}} M_{κ}(C_{p}, v).

Since the composition of a completely continuous operator and a continuous operator is again completely continuous, it follows that any operator in the ideal U ⊂ H generated by U_{p} is completely continuous in its action on the Banach space M_{k}(Np^{m}, v; K). Let Ũ ∈ U be any element in this ideal, and denote by Ũ_{κ} the operator of M_{κ} that it induces. When k ∈ Z, the p-adic Fredholm theory applies, giving us well-defined Fredholm determinants

PŨ(Np^{m}, k, χ, T) := det(1 − T · Ũ_{k}|M_{k}(Np^{m}, v; K; χ)).

Here, for χ a character on (Z/Np^{m}Z)^{∗}, let M_{k}(Np^{m}, v; K; χ) be the subspace of M_{k}(Np^{m}, v; K) on which the diamond operators ⟨d⟩, d ∈ (Z/qNZ)^{∗} act via the character χ. These Fredholm determinants are independent of the choice of radius v (provided that v is in the range stipulated above) as one can see using proposition 4.3.2 below (Cor. II.3.18 of Gouvêa [G-ApM] and section A3 of [C-BMF]).

As we’ll see, in section 4.3, the operator Ũ_{κ} acting on M_{κ} has a Fredholm determinant PŨ(κ, T). The Fredholm determinants are power series in the variable T (indeed they are entire functions of T with coefficients in the ring of integers of K, or C_{p}, with constant term 1, and hence are Fredholm functions in the sense of section 1.2). For the basics regarding these p-adic Fredholm determinants, see Chapter A of [C-BMF]. The key fact about them is given in Theorem 4.1.1 below. For any Ũ as above, and any element ũ ∈ C_{p} define

M_{κ,{Ũ−ũ}} ⊂ M_{κ}

to be the C_{p} vector subspace consisting of vectors in the kernel of some positive power of the operator Ũ − ũ · I (where I is the identity operator) acting on M_{κ}. We shall refer to an element of M_{κ,{Ũ−ũ}} ⊂ M_{κ} as a generalized eigenform for Ũ (with eigenvalue ũ). The subspace M_{κ,{Ũ−ũ}} is stable under the action of H.

Theorem 4.1.1. Suppose κ ∈ W_{N}. Over C_{p}, the power series PŨ(κ, T) may be written as a product

PŨ(κ, T) = ∏_{j=1}^{J}(1 − ũ_{j} · T)^{e_{j}}

for J either +∞ or a positive finite integer. Here the ũ_{j}’s are distinct elements in C_{p} − {0}; the e_{j}’s are positive integers (e_{j} = the multiplicity of ũ_{j} in the product expansion above) and the ordering of the ũ_{j} is such that the rational numbers σ_{j} = ord_{p}(ũ_{j}) are increasing (but not necessarily strictly, of course). If J = +∞ then σ_{j} tends towards ∞. For any (nonzero) element ũ ∈ C_{p} − {0} the C_{p} vector subspace,

M_{κ,{Ũ−ũ}} ⊂ M_{κ}

of generalized eigenforms for Ũ with eigenvalue ũ is of C_{p}-dimension equal to the multiplicity of ũ in the product expansion of the Fredholm determinant PŨ(k, χ, T). That is, the dimension of M_{κ,{Ũ−ũ}} is zero if ũ is not equal to one of the ũ_{j}’s and is equal to e_{j} if ũ = ũ_{j}.

Proof. cf. Propositions 11 and 12 of section 7 in [S-ECC].

Definition. Let M be a K-vector space and L : M → M a K-linear operator. A nonzero vector m ∈ M will be said to have L-slope σ ∈ Q if there is a polynomial Q(T) ∈ K[T] whose Newton polygon has a single side which is slope −σ such that m is in the kernel of Q(L).

For σ a positive rational number, and Ũ as above, denote by

M_{κ,{slope(Ũ)=σ}} ⊂ M_{κ}

the C_{p}-vector subspace of vectors of slope σ. The space M_{κ,{slope(Ũ)=σ}} is generated by the vector spaces M_{κ,{Ũ−ũ_{j}}} where ũ_{j} ranges through the inverses of the zeroes of PŨ(κ, T) such that ord_{p}(ũ_{j}) = σ. The subspace M_{κ,{slope(Ũ)=σ}} is stable under the action of H.

Proposition 4.1.2. Let α ∈ H be of the form 1 + p · τ for τ ∈ H. Put Ũ = α · U_{p}. The C_{p}-vector subspace

M_{κ,{slope(Ũ)=σ}}

depends only on κ and σ. It is independent of the choice of α.

Remark. More generally, Proposition 4.1.2 holds when α is a unit in the p-completion, H_{p} := lim_{↼} H/p^{n}H of the ring H.

Proof. Let α, α′ ∈ H be two elements of the form required in our proposition, and put Ũ = α·U_{p}, and Ũ′ = α′ ·U_{p}. For any Ũ-eigenvalue ũ of slope σ we shall show that any element in M_{κ,{Ũ−ũ}} is generated by Ũ′-generalized eigenforms with Ũ′-slope equal to σ. This will establish the inclusion

M_{κ,{slope(Ũ)=σ}} ⊂ M_{κ,{slope(Ũ′)=σ}}

and, by symmetry, the opposite inclusion as well. Fix, then, a ũ as above. Note that the operators α, α′ ∈ H induce nonsingular linear transformations of the C_{p}-vector space

M := M_{κ,{Ũ−ũ}},

and more stringently they induce automorphisms of an O_{C_{p}}-lattice in M. Using the same symbols α, α′, Ũ, etc. for the endomorphisms of M that they induce, let γ = α′ · α^{−1} : M → M. The eigenvalues of γ are all units in O_{C_{p}}. Decompose the finite dimensional C_{p}-vector space M as a direct sum of (generalized) eigenspaces for the operator γ. We work on each of these (generalized) eigenspaces in turn. Let g ∈ C_{p} be an eigenvalue of γ and consider now the C_{p}-subspace M_{g} ⊂ M consisting of generalized eigenforms for γ with eigenvalue g. We have the following formula involving (commuting) elements in the algebra End_{C_{p}}(M_{g}):

γŨ − gũ · I = (γ − g · I) · Ũ + g · (Ũ − ũ · I).

Since γ − g · I and Ũ − ũ · I are nilpotent operators on M_{g} (and since all the operators in the above formula commute with each other) it follows that γŨ − gũ · I = Ũ′ − gũ · I is nilpotent as well on M_{g}, concluding the proof of our Proposition.

Remark. We denote this C_{p}-vector space M_{κ,{slope(Ũ)=σ}} simply as

M_{κ,σ}

to reflect the fact that it depends only on σ and not on the choice of α (provided that α is chosen as stipulated in the above proposition).

4.2. Factoring Characteristic Series.

Suppose A is a Banach algebra, (M, | |) is an orthonormizable Banach Amodule, and V is a completely continuous operator on M such that its norm |V | is ≤ 1. In A2 of [C-BMF] the characteristic series of V operating on M, denoted det(1 − T V) ∈ A^{o}{{T }}, was defined.

By Corollary A2.6.1 of [C-BMF], if A is semi-simple, det(1 − T V) only depends on (V and) the topology of M. Note that the definition of semisimple used here is that of [C-BMF]. Specifically, a normed ring A is semisimple if the intersection of its maximal ideals is the zero-ideal, and if, for every maximal ideal m ⊂ A, the residual norm on A/m is multiplicative. Note (cf. “Example (i)” on page 427 of [C-BMF]) that if A is a reduced affinoid algebra over a complete subfield of C_{p} and A is viewed as Banach algebra with norm equal to the supremum norm, then A is semi-simple. It follows from Corollary A2.6.1 and Lemma A1.4 of [C-BMF] that

Proposition 4.2.1. If A is semi-simple and Max(A) is a rigid space which has an admissible covering by affinoid subdomains X_{i} such that there exists an orthonormizable norm on M_{X_{i}} equivalent to the induced norm then any completely continuous operator on M over A has a characteristic series in A{T } which depends only on the topology on M.

Theorem 4.2.2. Suppose A is a semi-simple Banach algebra, (M, || ||) is an orthonormizable Banach module and N is a free submodule of M of finite rank such that there exists a continuous projector from M onto N. Then, locally, there exists a norm on M equivalent to || || such that both N and M/N with their induced norms are orthonormizable.

Proof. Suppose {e_{i}}_{i∈I}, for some index set I, is an orthonormal basis for M and {n_{i}}_{i∈S} is a basis for N where S is a finite subset of I. Write

n_{i} = ∑_{j∈I} a_{i,j}e_{j}.

It follows that we may cover Spec(A) with affine opens such that for each such open U, there exist {j_{i}}_{i∈S} such that

det ( a_{i,j_{k}} )_{i,k∈S}

is invertible on U. We may suppose U = Spec(A) and j_{i} = i. We may now change our basis for N so that a_{ij} = δ_{ij} for j ∈ S. Now {n_{i}, e_{j}}_{i∈S,j∉S} is a basis for M. We define a new norm || ||′′ on M by requiring this set to be an orthonormal basis. We claim that || ||′′ is equivalent to || ||. Suppose m ∈ M and

m = ∑_{i∈S} b_{i}n_{i} + ∑_{j∉S} b_{j}e_{j}.

Then

\begin{aligned}||m|| & ≤ max\{|b_{i}||n_{i}|, |b_{j}|: i ∈ S, j ∉ S\} \\ & ≤ R||m||′′, \end{aligned}

where R = Max{1, |n_{i}|: i ∈ S}. If we set

c_{i} = \{ \begin{matrix}b_{i} & for i ∈ S \\ b_{i} + ∑_{j∈S} b_{j}a_{ji} & for i ∉ S \end{matrix}

Then,

m = ∑_{i} c_{i}e_{i}

and

\begin{aligned}||m||′′ & ≤ max\{|c_{i}|||e_{i}||′′, |c_{j}|: i ∈ S, j ∉ S\} \\ & ≤ R||m|| \end{aligned}

because ||n_{i}|| = ||e_{i}||′′. We will henceforth suppose n_{i} = e_{i} for i ∈ S. Suppose π: M → N is a continuous projector. Then ||π|| is defined. Let F = Ker π and π′ = 1 − π. Clearly {e_{i}, π′(e_{j}): i ∈ S, j ∉ S} is a basis for M. Let || ||′ be the norm on M determined by making this an orthonormal basis. We claim that || ||′ is equivalent to || ||. Let

m = ∑_{i∈S} b_{i}e_{i} + ∑_{i∉S} b_{i}π′(e_{i}).

Then,

m = ∑_{i∈S} b_{i}e_{i} − π( ∑_{i∉S} b_{i}e_{i}) + ∑_{i∉S} b_{i}e_{i}.

It follows that

||m|| ≤ Max{1, ||π||}||m||′.

Also if m = ∑ d_{i}e_{i}, then b_{i} = d_{i} if i ∉ S and

b_{i} = d_{i} − ∑_{j∉S} d_{j}c_{ji},

if

π(e_{j}) = ∑_{k∈S} c_{jk}e_{k}.

We see that

||m||′ ≤ Max{1, ||π||}||m||.

The theorem follows.

It follows from the above results and Corollary A2.6.2 of [C-BMF] that Corollary 4.2.3. Suppose A, M and N are as in the theorem and u is a completely continuous operator on M over A such that u preserves N. Then, if u_{F} is the induced operator on F, u_{F} has a characteristic series and

det(1 − T u) = det(1 − T u|_{N}) det(1 − T u_{F}).

Remark. The proof of the last sentence of Theorem A4.5 in [C-BMF] is not complete, but that sentence follows from this corollary.

4.3. Analytic variation of the Fredholm determinant.

Fix a tame level N such that (ϕ(Nq), p) = 1. The constructions in this section will depend upon this N (but we do not systematically indicate this dependence upon N in the notation below). Under our hypothesis, the ring Λ_{N} is a regular (complete local noetherian) ring. The object of this section is to state a result (Theorem 4.3.1) which shows that the Fredholm determinants PŨ(κ, T), for κ arithmetic, vary p-adic analytically in the variable κ. This is a mild modification of work done in [C-BMF] and [C-CCS] (cf. Theorem B and Appendix I in [C-BMF] and especially Theorem 5.1 of [C-CCS]). Although by the conventions adopted in the present section the prime number p is odd, we should note that a result similar to the one we will be stating below is valid for p = 2 as well.

Now, for arbitrary κ ∈ W we let PŨ(κ; T) denote to the Fredholm determinant of the completely continuous system of operators Ũ on the system of Banach modules M_{κ}^{†}(N, 1/i) for i large enough, where M_{κ}^{N,†}(v) is the Banach module over C_{p} consisting of v-overconvergent modular forms of tame level N.

Theorem 4.3.1. For each Ũ ∈ U there is an entire power series,

PŨ(T) = ∑ a_{n}(Ũ) · T^{n} ∈ Λ_{N}{{T }}

uniquely determined by the property that for every weight κ ∈ W_{N}, the image of

PŨ(T) ∈ Λ_{N}{{T }} ⊂ Λ_{N}[[T]]

under the homomorphism Λ_{N}[[T]] → C_{p}[[T]] induced by κ : Λ_{N} → C_{p} is equal to the Fredholm determinant PŨ(κ; T) ∈ C_{p}[[T]].

Here PŨ(κ; T) refers to the Fredholm determinant of the completely continuous system of operators Ũ on the system of Banach modules M_{κ}^{†}(N, 1/i) for i large enough, where M_{κ}^{N,†}(v) is the Banach module over C_{p} consisting of v-overconvergent modular forms of tame level N.

Before we begin the proof we need to talk about completely continuous systems of operators. First suppose (M, || ||) is a Banach module over a Banach algebra (A, | |). By an orthogonal basis of M over A we mean a set {e_{i}}_{i∈I} such that every element m of M is the limit of a unique series of the form

∑_{i∈I} a_{i}e_{i},

where a_{i} ∈ A and lim_{i→∞} ||a_{i}e_{i}|| = 0 such that ||m|| = max_{i∈I}{|a_{i}| · ||m_{i}||}. If |M^{∗}| = |A^{∗}|, where M^{∗} = M − {0}, an orthogonal basis can be converted to an orthonormal basis. The proof of Proposition 3.1 of [C-CCS] yields:

Proposition 4.3.2. Suppose M and N are Banach A-modules, f: M → N is a continuous map of Banach A-modules, and U_{M}: M → M and U_{N}: N → N are completely continuous (A-linear) maps of Banach A-modules such that

\begin{matrix}M & ⟶^{U_{M}} & M \\ ↓^{f} & & ↓^{f} \\ N & ⟶^{U_{N}} & N \end{matrix}

commutes. Then if there exists an orthogonal basis B of M such that f(B) is an orthogonal basis of N,

det(1 − T U_{M}) = det(1 − T U_{N}) ∈ A{{T }}.

Now suppose given a sequence of Banach A-modules M_{n} for n ≥ 1 and continuous A-linear mappings for each n ≥ 1, f_{n} : M_{n} → M_{n+1} with the property that there is an orthogonal basis B_{n} of M_{n} such that B_{n+1} := f_{n}(B_{n}) is an orthogonal basis of M_{n+1}. Let M = lim_{⇀} M_{n}. Suppose further that |M_{n}^{∗}| = |A^{∗}| for all n ≥ 1, and that there exists an orthogonal basis of M_{0} whose image in M_{n} is orthogonal for each n. Then if U is an operator on M which restricts to a completely continuous operator on M_{n} for each n, we set

det(1 − T U) = det(1 − T U|_{M_{n}}) ∈ A{{T }},

for any n. Here we have used Proposition 4.3.2 to guarantee that det(1 − T U|_{M_{n}}) is independent of n. We say that U is a completely continuous operator on the sequence {M_{n}}_{n ≥ 1}.

If the restriction of the operator U to M_{n} is of norm ≤ 1 for all n ≥ 1, then det(1 − T U) ∈ A^{o}{{T }}.

More generally, we must allow the Banach algebra A to vary as well. So, suppose we have:

i. a sequence of Banach algebras A_{n} for integers n ≥ 1, and contractive (this means norm decreasing) ring homomorphisms A_{m} → A_{n} for m > n ≥ 1,

ii. a function n ↦ s_{n} from the set of non-negative integers to nonnegative integers, and for each n ≥ 1, a sequence of Banach A_{n}-modules {M_{n,i}}_{i≥s_{n}} such that |M_{n,i}^{∗}| = |A^{∗}| (for integers i ≥ s_{n}),

iii. continuous A_{n}-module homomorphisms f_{n,i} : M_{n,i} → M_{n,i+1} for all n ≥ 1 and i ≥ s_{n} such that there is an orthogonal basis B_{n,i} of M_{n,i} with the property that f_{n,i}(B_{n,i}) ⊂ M_{n,i+1} is an orthogonal basis of M′_{n,i+1}, and for all m ≥ n ≥ 1 and i sufficiently large, M_{m,i} is the completed tensor product of A_{m} and M_{n,i}.

Definition. We call the above structure a system of Banach modules. Given such a system of Banach modules M, to give a completely continuous system of operators V on M one must give a completely continuous operator V_{n,i} on M_{n,i} over A_{n}, for each n ≥ 1 and i ≥ s_{n}, such that all the obvious diagrams commute and such that V_{m,i} = V_{n,i} ⊗ 1 if m ≥ n ≥ 1 and i ≥ s_{m}.

Given a system of Banach modules M as above, we will say that M is a system of Banach modules over the sequence of Banach algebras {A_{n}}_{n}. Form the projective limit,

A := lim_{↼} A_{m},

of the sequence of Banach algebras and contractive mappings, A_{m} → A_{n}, given in i. above, to obtain a topological ring which we will call the limit ring A of the system M. Consider the topological ring A^{o} := lim_{↼} A_{m}^{o} ⊂ A (it is, in fact, a subring of A since projective limit over the ordered set of natural numbers is left-exact).

Proposition 4.3.3. Let V be a completely continuous system of operators on the system of Banach modules M. Let A = lim_{↼} A_{m} be the limit ring of M and A^{o} ⊂ A as above. There is a unique formal power series which we denote

det(1 − T V) ∈ A[[T]]

such that for each n ≥ 1, the natural projection of det(1 − T V) to A_{n}[[T]] is the entire power series

det(1 − T V_{n,i}|M_{n,i})

for any i ≥ s_{n}. If the operators V_{n,i} of the system V all have operator norm ≤ 1, (the norm of an operator on a Banach module was defined in section A1 of [C-BFM]) then det(1 − T V) ∈ A^{0}[[T]].

Proof. This follows from Proposition 4.3.2 together with Theorem A2.1 and Lemma A2.5 of [C-BMF].

We call det(1 − T V) ∈ A[[T]] the Fredholm determinant of the completely continuous system of operators V.

Proof of Theorem 4.3.1.

We now describe a specific system of completely continuous operators to which Proposition 4.3.2 applies.

The sequence of Banach algebras. To give the Banach algebras, let us refer to the terminology of section 1.1 where to any complete noetherian local ring R is attached a rigid space X = X_{R} given as a union of a nested sequence of affinoid subdomains, denoted X_{r} = X_{R,r} ⊂ X_{R} for positive rational numbers r, where X_{r} ⊂ X_{r′} if r ≥ r′. Let us use the same notation, when R is a complete noetherian semi-local ring, to obtain X_{R} as a union of a nested sequence of affinoid subdomains, X_{R,r} ⊂ X_{R}, by taking X_{r} = X_{R,r} to be the disjoint union of the correspondingly denoted affinoids for each of the (finite) connected components of X_{R}.

Now put R = Λ_{N} and, for every integer n ≥ 1 define the Banach algebra A_{n} to be the affinoid algebra of X_{Λ_{N} ,1/n} ⊂ X_{Λ_{N}}.

Note that for each n ≥ 1 there is a natural continuous ring homomorphism Λ_{N} → A_{n}^{o} ⊂ A_{n}, and the mappings A_{n} → A_{n+1} are Λ_{N}-algebra homomorphisms, as well as contractive homomorphism of Banach algebras.

Proposition 4.3.4. For the above sequence of Banach algebras, the natural continuous ring homomorphism

Λ_{N} → A^{0}

is an isomorphism of topological rings. Any power series in Λ_{N}[[T]] whose projection to A_{n}[[T]] is an entire power series (over A_{n}) for all n ≥ 1 is entire over Λ_{N},

Proof. . The first sentence of the proposition is (essentially) Proposition 1.1 and Corollary 1.1.1 of [C-CCS]. For the convenience of the reader we sketch the outlines of its proof. Given our assumptions on the level, the ring Λ_{N} is a product of regular local rings of the form O[[S]]; i.e., power series rings in one variable over complete discrete valuation rings O which are finite over Z_{p}. It suffices to prove the analogous statement with R = O[[S]]. Let, then, R be such a ring, let | | be the multiplicative norm on the discrete valuation ring O whose value on any uniformizer is 1/p ∈ R. Let K denote the field of fractions of O which we can take to be a (finite degree) field extension of Q_{p} contained in C_{p}. Denote the maximal ideal of R by m_{R} = (π, S) ⊂ R (here π is a uniformizer of O). As discussed (at least for the discrete valuation ring Z_{p}) in section 1.1, X = X_{R} is the open unit disk, and one sees that for n ≥ 1, A_{n} is the ring of rigid functions defined over K and bounded by 1 on the disk B[0, p^{−1/n}] = {x ∈ C_{p} | |x| ≤ p^{−1/n}} in C_{p}.

For every real number t in the range 0 < t < 1 define the t-norm | |_{t} on a power series ∑_{n=0}^{∞} b_{n}S^{n} ∈ R by the formula

| ∑_{n=0}^{∞} b_{n}S^{n}|_{t} = Max_{n}{ |b_{n}|t^{n} }

which (if t ∈ |C_{p}|) is the norm obtained by mapping an element of R into A^{o}(B[0, t]) and then taking the supremum norm of its image. One checks that for f ∈ R,

|f|_{t} ≤ |f|_{s} ≤ |f|_{t}^{log_{t}(s)}.

Now fix t = p^{−1/n} for an integer n ≥ 1, and let f ∈ (m_{R})^{ν} for some ν > 0. Then

|f|_{t} ≤ Max{ |π|^{ν}, t^{ν} } = Max{ p^{−ν}, p^{−ν/n} } = p^{−ν/n}.

Moreover, if

ν ≤ Min{ log_{t}(|f|_{t}), −log_{p}(|f|_{t}) } = Min{ −nlog_{p}(|f|_{t}), −log_{p}(|f|_{t}) },

we have

f ∈ (m_{R})^{ν}.

These two facts give us that all the norms | |_{t} for t = p^{−1/n} and n ≥ 1 are equivalent and induce the m_{R}-adic topology on R. It follows that the ring homomorphisms R → A_{n}^{o} are closed, and R ≅ lim_{↼} A_{n}^{o}, the isomorphism being in the category of topological rings. If f ∈ (m_{R})^{ν} for ν > 0 and f ∉ (m_{R})^{ν−1}, we have the inequalities (recalling that t = p^{−1/n}):

−log_{p}(|f|_{t}) ≤ ν ≤ −n · log_{p}(|f|_{t}).

The second sentence of Proposition 4.3.4 then follows. The above inequalities reduce, in the case, of n = 1 (i.e., t = 1/p) to the equality ν = −log_{p}(|f|_{t}).

The system of Banach modules.

Here we begin by defining, for integers n, i ≥ 1, the A_{n} = A(X_{Λ_{N} ,1/n})- modules M′_{n,i} as the ring of rigid analytic functions on the reduced affinoid

X_{Λ_{N} ,1/n} × Z_{1}(Nq)(v),

where the radius v = 1/i. Having stipulated this, we have already established a structure giving much of what is required in ii. and iii. to have a system of Banach modules.

Specifically, for each n ≥ 1, we take s′_{n} := 1 and consider the sequence of Banach A_{n}-modules

{M′_{n,i}}_{i≥s′_{n}},

for integers i ≥ s′_{n} = 1. We have that |M_{n,i}^{′∗}| = |A^{∗}|. Denote by

f′_{n,i} : M′_{n,i} → M′_{n,i+1}

(for all n, i ≥ 1) the continuous A_{n}-module homomorphisms induced by the natural inclusion mappings of the corresponding affinoids. Moreover, for all m ≥ n ≥ 1 and i ≥ 1 we have that M′_{m,i} is the completed tensor product of A_{m} and M′_{n,i}. Note also that each Banach A_{n}-module M′_{n,i} is orthonormizable, by Lemma A5.1 and the remark immediately following that lemma, in [C-BMF]. What remains to be done is to check that for each Banach A_{n}-module M′_{n,i}, we may find an orthogonal basis B′_{n,i} ⊂ M′_{n,i} such that f′_{n,i}(B′_{n,i}) ⊂ M′_{n,i+1} is an orthogonal basis of M′_{n,i+1}. This follows from Proposition 3.1 and Corollary 4.2.1 of [C-CCS]. Furthermore, by Proposition A5.2 of [C-BMF], combined with what has been said above, we see that the operator U (cf. section 3.4) on M′_{n,i} is a completely continuous operator (on the Banach A_{n}-module M′_{n,i}).

Remark. By the discussion in section 2.4, one has that for each n ≥ 1 there is an integer, call it σ_{n}, which is large enough so that if i ≥ σ_{n}, the mapping

F ↦ F · E(q)

defines an isomorphism of the Banach A_{n}-module, M′_{n,i} of rigid analytic functions on X_{Λ_{N} ,1/n} × Z_{1}(Nq)(v) onto the Banach A_{n}-module

M^{†}(N)_{X_{Λ_{N} ,1/n}}(1/i)

of v-overconvergent modular forms over X_{Λ_{N} ,1/n} of tame level N, where v = 1/i. If i ≥ σ_{n}, put M_{n,i} := M^{†}(N)_{X_{Λ_{N} ,1/n}}(1/i), and let us concentrate on the system of Banach modules given by the M_{n,i} (i ≥ σ_{n}). We note that the Banach A_{n}-module M_{n,i} inherits an orthogonal A_{n}-basis from M′_{n,i} which is brought (under f_{n,i}) to an orthogonal A_{n}-basis of M_{n,i+1}.

The sequence of completely continuous operators.

We now return to Ũ = α · U_{p} given in the hypothesis of Theorem 4.3.1. This operator is the product of an element α ∈ H which may be written as a polynomial in a finite number of Hecke operators T_{ℓ_{j}} (j = 1, …, μ) and the operator U_{p}, the coefficients of this polynomial being elements of Λ_{N}. The operator U_{p}, when viewed as acting on overconvergent modular forms F ∈ M_{n,i} (for i large enough) is given by the formula

F |U_{p} = EU(F/E).

Recalling Lemma 3.4.1 (and the notation v_{ℓ}(X) described in that Lemma) let us put

s_{n} := Max{ σ_{n}, v_{U_{p}}(X_{Λ_{N} ,1/n}), v_{ℓ_{j}}(X_{Λ_{N} ,1/n}); j = 1, …, μ }.

Define (for n ≥ 1 and i ≥ s_{n}) the transformations

V_{n,i} : M_{n,i} → M_{n,i}

to be the A_{n}-linear operator on (1/i)-overconvergent modular forms given by Ũ = α · U_{p}. Since U_{p} is completely continuous on M_{n,i}, so are all the operators V_{n,i}. By the formulas given for the operators T_{ℓ_{j}} and for U_{p} in section 3.4, we see that the operators V_{n,i} are all of norm ≤ 1.

We have established

Lemma 4.3.5. The above system V := {V_{n,i}}_{n,i} is a system of completely continuous operators of norm ≤ 1 on the system of Banach modules M := {M_{n,i}}_{n,i}.

It follows from Propositions 4.3.3 and 4.3.4, that the system of operators V has a Fredholm determinant,

P_{V}(T) := det(1 − T V) ∈ Λ_{N}[[T]]

such that for each n ≥ 1, the natural projection of P_{V}(T) to A_{n}[[T]] is the entire power series det(1 − T V_{n,i}|M_{n,i}) for any i ≥ s_{n}. Moreover, by comparing the definitions one sees directly that for every weight κ ∈ W_{N}, the image of PŨ(T) under the homomorphism Λ_{N}[[T]] → C_{p}[[T]] induced by κ : Λ_{N} → C_{p} is equal to the Fredholm determinant PŨ(κ; T) ∈ C_{p}[[T]]. This concludes the proof of Theorem 4.3.1.

Remark. In the special instance where the element Ũ is the Atkin-Lehner operator U_{p} itself, the coefficients of the Fredholm determinant may be given in closed form as in formulae of Appendix I of [C-BMF].

4.4 The Spectral Curves.

In this section we again suppose that (ϕ(Nq), p) = 1 and Nq ≥ 5.

Definition. For each Ũ ∈ U, define ZŨ ⊂ W_{N} × A^{1} to be the rigid space cut out by the Fredholm determinant,

ZŨ : PŨ(T) = 0,

viewed as rigid analytic (Fredholm) closed hypersurface (over Q_{p}) in the (smooth) rigid analytic surface W_{N} × A^{1} (cf. section 1.2; recall that W_{N} is weight space of tame level N and A^{1} is the affine line parameterized by the variable T).

From its definition we see that ZŨ ⊂ W_{N} ×A^{1} is a rigid analytic Zariskiclosed subvariety which is (equi-dimensional) and of dimension equal to 1 (i.e., it is a rigid analytic curve). Since ZŨ is rigid-Zariski-closed in the nested (cf. 1.1) rigid analytic space W_{N} × A^{1} it follows that ZŨ is nested. We will refer to the curve ZŨ as the Spectral curve attached to Ũ. It is easy to find examples where ZŨ is disconnected. Can ZŨ have an infinite number of connected components? As in Appendix I of [C-BMF] one sees that the natural projection mapping

π : ZŨ → W_{N}

can be of infinite degree.

Theorem 4.4.1. For any κ ∈ W_{N}(C_{p}) and any non-zero element ũ ∈ C_{p}, the C_{p}-valued point (κ, 1/ũ) ∈ W_{N} × A^{1} lies on the curve ZŨ if and only if there exists an overconvergent eigenform (eigenform for all Hecke and Atkin-Lehner operators) f with Fourier coefficients in C_{p} with the following characteristics: The overconvergent form f is of tame level N, of weightcharacter κ, and has Ũ-eigenvalue equal to ũ.

Addendum. We also have that the C_{p}-vector space of overconvergent (generalized) eigenforms with the above characteristics is of dimension equal to the ramification index of ZŨ over W_{N} at (κ, 1/ũ). Proof. . This follows immediately from Theorems 4.1.1 and 4.3.1 (compare also: Lemma A2.5 of [C-BMF] and Proposition 12 of [S-ECC] or Theorem A4.5 of [C-BMF]).

Chapter 5. Galois representations and pseudo-representations.

5.1. Deforming representations and pseudo-representations.

Let p > 2 be a prime number, and let our tame level N be an arbitrary positive integer not divisible by p. Let S be the finite set of places of Q consisting of ∞ and the prime divisors of Np. Let G_{Q,S} denote the Galois group of the maximal algebraic extension of Q in C which is unramified outside the set S. Let c ∈ G_{Q,S} denote complex conjugation.

We begin by describing the residual representations of interest to us. By the catch-phrase residual representation we shall mean a two-dimensional vector space V̄ over a finite field F of characteristic p endowed with an odd F-linear continuous G_{Q,S}-action. By odd we mean that the determinant of the operation c on V̄ is −1 ∈ F. In particular, we will be considering residual representations that come from modular eigenforms, by which we mean the following:

If m is an integer and f is a (classical, cuspidal) modular eigenform on Γ_{1}(Np^{m}) with ring of Fourier coefficients O_{f} ⊂ O_{C_{p}} where O_{f} is a (finite) complete local ring extension of Z_{p} with field of fractions of O_{f} denoted by K_{f} and with residue field isomorphic to F_{f}, let V_{f} denote the associated two dimensional (irreducible) continuous representation of G_{Q,S} over K_{f} attached to f in the usual sense (cf. Section 3 of [D-D-T]). Let Ω_{f} ⊂ V_{f} be a G_{Q,S}-stable O_{f}-lattice in the K_{f}-vector space V_{f}. Let us allow ourselves the abuse of notation

V̄_{f} = Ω_{f} ⊗_{O_{f}} F_{f},

but note that in the case where V̄_{f} is not absolutely irreducible, the isomorphism class of the F_{f}[[G_{Q,S}]]-module V̄_{f} is not uniquely determined by f; it depends, as well, on the lattice Ω_{f} ⊂ V_{f} chosen.

Definition. Let us say that a F[[G_{Q,S}]]-module V̄ comes from a modular eigenform (of tame level N) if its extension of scalars to a finite field extension field F ⊂ F′ is isomorphic as module over the ring F′[[G_{Q,S}]] to a Galois representation V̄_{f} ⊗_{F_{f}} F′, where f is a (classical) modular eigenform of tame level N, and F_{f} → F′ is a homomorphism of fields.

By a p-modular, tame level N, residual representation (a pmodular representation, for short) let us mean a two-dimensional vector space V̄ over a finite field F of characteristic p endowed with an F-linear continuous G_{Q,S}-action and satisfying the following properties.

1. The F[[G_{Q,S}]]-module V̄ descends to no proper subfield of F. That is, there is no proper subfield F′ ⊂ F and sub- F′[[G_{Q,S}]]-module V̄′ ⊂ V̄ whose change of scalars to F is isomorphic to V̄. Equivalently (since the Schur index of these representations is 1) the character of the G_{Q,S}-representation V̄ maps surjectively to F.

2. The F[[G_{Q,S}]]-module V̄ comes from some classical modular eigenform f on Γ_{1}(Np^{m}) for some positive integer m. We call the pseudorepresentation associated to the semi-simplification of a p-modular representation a p-modular pseudo-representation.

Proposition 5.1.1. For given N and p > 2 there are only a finite number of isomorphism classes of p-modular, tame level N, residual representations.

Proof. Step 1. Let us first show that there are only a finite number of such representations which are absolutely irreducible. Denote by ρ̄ an absolutely irreducible p-modular residual representation satisfying the further requirements in our proposition above. By basic representation-theoretic results over finite fields, one has that ρ̄ is defined over the field generated over F_{p} by the traces of the ρ̄(g)’s for g ∈ G_{Q,S} (for example, see the Corollary of section 6 of [M-DTR] so we have from hypothesis 1. that F is generated by traces, as described above. Moreover, the isomorphism class (over F) of the F-representation ρ̄ is determined by the character

g ↦ TraceF̄ (ρ̄(g)).

For this, cf. the Corollary of section 5 of [M3] . So we must show that there are only a finite number of character-functions occurring as characters of absolutely irreducible representations ρ̄ satisfying our hypotheses. A quick way of doing this is to make use of the work surrounding Serre’s Conjecture (see [S-RD2] and the discussion given in [Ri]). Specifically, note, that if p is odd, and ρ̄ arises from a modular form on Γ_{1}(Np^{m}) (with N prime to p) then it arises from a modular form on Γ_{1}(N). This is a well-known fact ( Remarque on p. 195 of [S-RD2]) and a proof is given in [Ri] (Theorem 2.1). Now it is also known (cf. Theorem 4.3 of [Ed]; see also the discussion in [Ri]) that any modular residual Galois representation ρ̄ with values in a finite field of characteristic p and tame level N arises from an eigenform on Γ_{1}(N) and having weight k(ρ) defined by Serre ([S-RD2] section 2). Since one easily computes from the formulas given in section 2 of [S-RD2] that k(ρ) ≤ p^{2} since it follows that if ρ̄ is a p-modular residual Galois representation of fixed tame level N, then ρ̄ is associated to one of a finite number of modular eigenforms, from which the proposition follows.

Step 2. We are left with the task of showing finiteness of the set of isomorphism classes of (absolutely) reducible p-modular, tame level N, residual representations (p > 2). Here we first show that the characters of such representations (with values in the algebraic closure F̄_{p} ) are finite in number. But any such character is the sum of two degree 1 characters ψ_{1}+ψ_{2} of G_{Q,S} where S is the collection consisting of the prime at ∞ and the prime divisors of pN. Moreover the conductors of the characters ψ which can occur are bounded. It follows (by the Kronecker-Weber theorem if nothing else) that only a finite number of ψ’s can occur. Fix two such characters ψ_{1}, ψ_{2}, let K_{{ψ_{1},ψ_{2}}}/Q denote the splitting field of the two characters (meaning the fixed field in Q̄ of the subgroup of G_{Q,S} given by the intersection of the kernels of the homomorphisms ψ_{1}, ψ_{2} : Q̄ → F̄_{p}). Now consider the set of isomorphism classes of two-dimensional F̄_{p}-vector spaces V̄ equipped with continuous representations ρ̄ : G_{Q,S} → Aut_{F}(V̄) with character equal to ψ_{1} + ψ_{2} , and admitting a line (i.e., a one-dimensional F̄_{p}-vector subspace) stabilized by the action of G_{Q,S} with character ψ_{1}. The corresponding quotient space (which is also a line) has G_{Q,S} character equal to ψ_{2}. Consider such a {V̄ , ρ̄}. The subgroup H = ker(ρ̄) ⊂ G_{Q,S} has, as fixed field, an Abelian, exponent p, extension-field of K_{{ψ_{1},ψ_{2}}} which is unramified outside the set of primes of K_{{ψ_{1},ψ_{2}}} lying above S. Since the maximal Abelian, exponent p, extension-field of K_{{ψ_{1},ψ_{2}}} which is unramified outside primes lying above S is finite, we see that the collection of all {V̄ , ρ̄}’s we are presently considering have the property that there is a fixed finite quotient group Φ of G_{Q,S} through which the ρ̄’s all factor. It follows that there are only a finite number of {V̄ , ρ̄}’s up to isomorphism.

We shall now study continuous, two-dimensional, odd G_{Q,S}-representa-tions over commutative topological rings A in which 2 is invertible; that is, continuous A-linear representations of G_{Q,S} into the topological group of automorphisms of free modules M of rank two over such rings A, for which the complex conjugation c has determinant (over A) equal to −1 ∈ A. Since 2 is invertible in A, by the symmetrizing and anti-symmetrizing with respect to c, we have a canonical decomposition of A-modules, M = M^{+} ⊕ M^{−} where

M^{±} := { m ∈ M | c(m) = ±m }.

By the c-normal matrix description for the G-action on M we mean the matrix which describes the G-action in terms of the above direct sum decomposition:

g ↦ ({a(g) \atop c(g)} {b(g) \atop d(g)}) ,

where a(g) ∈ Hom_{A}(M^{+}, M^{+}), b(g) ∈ Hom_{A}(M^{−}, M^{+}), c(g) ∈ Hom_{A}(M^{+}, M^{−}), and d(g) ∈ Hom_{A}(M^{−}, M^{−}).

By a c-normal A-basis for such an A[G_{Q,S}]-module M is meant a Abasis {x, y} such that x is an A- generator of M^{+} and y is an A-generator of M^{−}. Equivalently, it is a basis for which c(x) = x and c(y) = −y.

Remark Let A = D, a complete noetherian semi-local (commutative) ring in which 2 is invertible. Then any (odd) two-dimensional representationmodule M over D admits a c-normal D-basis . To see this, consider the quotient of D by its Jacobson radical rad(D) which may be decomposed as a product of fields D/rad(D) = ∏_{j=1}^{ν} k_{j}, and let ϵ_{j} : D → k_{j} be the projection to the j-th factor. Now form the direct sum decomposition of D-modules, M = M^{+} ⊕ M^{−} and the analogous direct sum decomposition of D/rad(D)-modules

M/rad(D) · M = (M/rad(D) · M)^{+} ⊕ (M/rad(D) · M)^{−}. By noting that the dimensions of the vector spaces (M/rad(D) · M)^{+} ⊗_{D} k_{j} over k_{j} are all 1 one sees that (M/rad(D) · M)^{±} is free of rank one over D/rad(D). Choosing generators x̄ and ȳ of the free rank one modules (M/rad(D) · M)^{+} and (M/rad(D) · M)^{−} respectively, and lifting these to elements x and y of the D- modules M^{+} and M^{−}, we see by Nakayama’s lemma that x and y are generators of M^{+} and M^{−}, respectively, giving us a surjective D-module homomorphism

h : D ⊕ D → M

defined by (a, b) ↦ ax ⊕ by. Since M and D ⊕ D are both free of rank 2 over D the surjection h is immediately seen to be an isomorphism. That is, {x, y} constitutes a c-normal D-basis of M. Now let us return to A, a commutative topological ring in which 2 is invertible and M a free rank two A-module endowed with an odd, continuous, A-linear representation of G = G_{Q,S}. Let ρ : G → Aut_{A}(M) denote this G-action. Following [Wi],

[T] (cf. also [H-NO]), and a suggestion of Buzzard, define the (continuous) functions

α = α_{ρ} : G → A; δ = δ_{ρ} : G → A

by the rules:

α(g) := \frac{trace_{A}(ρ(g)) + trace_{A}(ρ(cg))}{2},

δ(g) := \frac{trace_{A}(ρ(g)) − trace_{A}(ρ(cg))}{2}.

Define the continuous function ξ = ξ_{ρ} : G × G → A to be ξ(g, h) =:= α(gh) − α(g)α(h).

If M admits a c-normal basis for the G-action on M and if we consider the matrix representation with respect to such a basis:

g ↦ ({a(g) \atop c(g)} {b(g) \atop d(g)}) ,

we have: α(g) = a(g), δ(g) = d(g), and ξ(g, h) = b(g)c(h).

Lemma. The triple of continuous functions

r := (ξ, α, δ)

satisfy the following relations (for all g, h, k, ℓ ∈ G = G_{Q,S}):

\begin{aligned}α(g · h) & = α(g)α(h) + ξ(g, h) \\ δ(g · h) & = δ(g)δ(h) + ξ(h, g) \end{aligned}  (1)

\begin{aligned}ξ(gh, k) & = α(g)ξ(h, k) + δ(h)ξ(g, k) \\ ξ(g, hk) & = α(k)ξ(g, h) + δ(h)ξ(g, k) \end{aligned}  (2)

ξ(g, h)ξ(k, ℓ) = ξ(g, ℓ)ξ(k, h)  (3)

and α(1) = δ(1) = 1; α(c) = 1; δ(c) = −1 ξ(g, h) = 0 if either g or h is 1 or c.

(4)

Proof. We leave this as an exercise to the reader, with the suggestion that each of the formula-checks can be made to follow from a single computation in an an appropriate universal situation. For example, one can check formula (1), the second equation being the only one that needs checking, by replacing G by the group G generated by elements denoted c, g, h with the single relation c^{2} = 1, and replacing A by the Z[1/2]-algebra A which is obtained from the polynomial ring in twelve variables over Z[1/2] (four variables making up the matrix coefficients of a 2 × 2-matrix C, G, H for each of the three elements c, g, h) by inverting the determinants of G, H, and insisting that the determinant of C be 1 and its trace be 0. To check formula (1) in the context of the lemma, it suffices to check the analogue of formula (1) for the representation c ↦ C, g ↦ G, h ↦ H of G on the free module M := A ⊕ A or, since A is an integral domain, it suffices to check these formulas after extension of scalars to the field of fractions A, K. But, since K is a field, M ⊗_{A} K admits a c-normal basis, with respect to which it is easy to compute everything.

Recall that a triple r = (ξ, α, δ), where ξ : G × G → A and α, δ : G → A are continuous functions, is called a (continuous) pseudorepresentation of G = G_{Q,S} with values in A if formulas (1)-(4) are satisfied. From (1) we see that r is determined by the pair of functions (α, δ). If M is a free A-module of rank two equipped with a continuous, odd, Alinear G-action, and if ρ : G → GL_{2}(A) is the corresponding representation, the A-valued pseudo-representation r = (ξ_{ρ}, α_{ρ}, δ_{ρ}) as defined above will be referred to as the pseudo-representation attached to the G-module M, or to the representation ρ. If r = (ξ, α, δ) is a pseudo-representation with values in A, we define

Trace(r) = α + δ : G → A,

and, for g ∈ G, put

Det(r)(g) = α(g) · δ(g) − ξ(g, g).

One checks, using (1)-(4), that Det(r) : G → A^{∗} is a (continuous) homomorphism. If r is the pseudo-representation attached to ρ, we have that Trace(r) = Trace(ρ) and Det(r) = Det(ρ). It follows from a result of Carayol and Serre (cf. the Corollary of section 5 of [M-DTR] ) that if A = D is a a complete noetherian local ring, and if ρ is absolutely irreducible, then the pseudo-representation r determines ρ up to equivalence.

The theory we have discussed, requiring that M be free of rank two over A, generalizes immediately to the situation where M is locally free. But let us focus on this in the special case of continuous pseudo-representations of G = G_{Q,S} with values in the ring of rigid analytic functions A = A(Y), where Y is a rigid analytic space. If Y is a rigid analytic space, consider pairs (V, ρ) where V is now a rank two vector bundle over the rigid analytic Y and ρ is a continuous, odd, O_{Y} -linear representation of G = G_{Q,S} on V,

ρ : G → Aut_{O_{Y}} (V),

and where the action of complex conjugation, ρ(c) induces a decomposition of V into the direct sum of two line bundles, i.e. invertible O_{Y} -modules,

V = V^{+} ⊕ V^{−},

where ρ(c) fixes V^{+}, and acts as multiplication by −1 on V^{−}.

We leave to the reader the exercise of defining the pseudo-representation r with values in A(Y) associated to such a representation ρ.

One of the aspects of the usefulness of pseudo-representations is that given any pseudo-representation r = (ξ, α, δ) with values in a field K which is not of characteristic 2, a theorem of Wiles [Wi] guarantees that r is indeed the pseudo-representation attached to a representation ρ : G_{Q,S} → GL_{2}(K). Moreover, the representation ρ is irreducible if and only if there exists a pair of elements (s_{0}, t_{0}) ∈ G×G such that ξ(s_{0}, t_{0}) ≠ 0. Compare Proposition 1.1 of [H-NO]). A simple version of this theorem over rigid analytic spaces can be proved using the same method, and we now prepare for its formulation.

Definition. Let Y be a rigid analytic space over Q_{p} and r = (ξ, α, δ) a continuous pseudo-representation of G_{Q,S} into the ring A(Y) of rigid analytic functions on Y. By the ideal of reducibility of the pseudorepresentation I_{r} ⊂ O_{Y} let us mean the sheaf of O_{Y} -ideals generated by all the functions ξ(s, t) ∈ A(Y) for (all pairs) (s, t) ∈ G × G.

For each point y ∈ Y (C_{p}) evaluation of the pseudo-representation r at the point y gives a C_{p}-valued pseudo-representation r_{y} to which (by the theorem of Wiles referred to above) one can attach an equivalence class of representations ρ_{y} : G_{Q,S} → GL_{2}(C_{p}). We have that ρ_{y} is irreducible if and only y is not contained in the support of the ideal I_{r}.

Theorem 5.1.2. Let Y be a rigid analytic space over Q_{p} and r = (ξ, α, δ) a continuous pseudo-representation of G_{Q,S} with values in A(Y). Suppose that there is a pair of elements s_{0}, t_{0} ∈ G_{Q,S} such that ξ(s_{0}, t_{0}) ∈ A(Y) is a non-zero-divisor, and is a generator of the O_{Y} -module I_{r}. Then there is a continuous representation

ρ : G_{Q,S} → GL_{2}(A(Y))

whose associated pseudo-representation is r.

Proof. This is just an exercise in globalizing Wiles’ theorem; one can use the argument of Proposition 1.1 in [H-NO], with hardly any change, to prove the above theorem. Explicitly, for g ∈ G, we repeat the formulas of [H-NO] (after adjusting for the changes in powers of 2). A representation ρ is given by

g ↦ ( \begin{matrix}α(g) & \frac{ξ(g,t_{0})}{ξ(s_{0},t_{0})} \\ ξ(s_{0}, g) & δ(g) \end{matrix} ) .

Remark. In particular, if Y is a smooth rigid analytic curve, and r = (ξ, α, δ) a continuous pseudo-representation of G_{Q,S} with values in A(Y) then Y can be covered by open rigid analytic subspaces U over each of which the restriction of the pseudo-representation r satisfies the hypothesis of the above theorem, or else has the property that the function ξ is identically 0. If the open U falls into the first category, Theorem 5.1.2 applies to the restriction of r to U giving us a rigid analytic G_{Q,S}-representation over U whose corresponding pseudo-representation is r restricted to U ; if it fall in the second category, the corresponding functions α and δ restricted to U are continuous characters of G with values in O_{U}^{∗}, and r restricted to U is the pseudo-representation associated to direct sum of the two degree one G_{Q,S}-representations given by the characters α and δ. Can one get a finer, more global theorem along the above lines by considering O_{Y} -linear representations of G_{Q,S} into the groups of units in Azumaya algebras (of rank 4) over O_{Y}?

Deformations of pseudo-representations compile well forming universal deformation rings as we demonstrate below in the two dimensional case. For the general case, under the assumption that the pseudo-representation is irreducible, see [Ny] or [Ro].

Theorem 5.1.3. Let k be a finite field of characteristic p ≠ 2. Let ψ̄ denote a pseudo-representation of G = G_{Q,S} with values in k. Then there is a universal lifting of ψ̄ to a pseudo-representation ψ^{univ} of G with values in a complete noetherian local ring R^{univ}(ψ̄) with residue field k which is universal in the sense that given any pseudo-representation ψ with values in a complete noetherian local ring D with residue field k, such that ψ is a lifting of ψ̄, there is a unique continuous homomorphism of local rings

π_{ψ} : R^{univ}(ψ̄) → D

reducing to the identity on k, such that ψ is the composition of ψ^{univ} and π_{ψ}.

Proof. One can prove this easily using Schlessinger’s criterion [Sch]; compare [Ny]. Briefly, fix a pseudo-representation ψ̄ = (ᾱ, δ̄, ξ̄) of G = G_{Q,S} into k. We consider deformations of ψ̄. Namely, let Ĉ be the category of complete noetherian local rings with residue field k. For A ∈ Ĉ, let F (A) be the set of pseudo-representations of G into A lifting ψ̄. Suppose we have a Cartesian diagram of elements of Ĉ:

A_{3}
↙  ↘
A_{1}  A_{2}
↘  ↙
A_{0}

The natural map

b: F (A_{3}) → F (A_{1}) ×_{F (A_{0})} F (A_{2})

is an isomorphism and we will write down its inverse. Suppose, ψ_{i} = (α_{i}, δ_{i}, ξ_{i}) are pseudo-representations with values in A_{i} for i = 1, 2 and the images of these in A_{0} are the same. Then

e(ψ_{1}, ψ_{2}) =: ((α_{1}, α_{2}), (δ_{1}, δ_{2}), (ξ_{1}, ξ_{2}))

is a pseudo-representation with values in A_{3}. Clearly, e is the inverse of b. Thus conditions H_{1}, H_{2} and H_{4} of [Sch] are automatic. The only thing we have to check is H_{3}. That is, we must show that for A = k[ϵ] , the dual numbers over k, that F (A) = F (k[ϵ]), which is a k-vector space, is finite dimensional, or equivalently, is finite. Case 1. ψ̄ is irreducible (which, by definition, means that there is a pair of elements s and t in G such that ξ̄(s, t) ≠ 0). Then as mentioned above (and compare any of our basic references; e.g., [H-NO,§1]) we know that for any ψ ∈ F (k[ϵ]) there exists an irreducible representation ρ: G → GL_{2}(F (k[ϵ]))

such that ρ(c) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) whose associated pseudo-representation equals

ψ. Because any two representations that have the same character, take the

value ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) on c, and are irreducible, must be conjugate by a matrix of

the form ( \begin{matrix}l & 0 \\ 0 & k \end{matrix} ) where l and k are in F (k[ϵ])^{∗}, if we fix one representation,

ρ̄, into GL_{2}(k) whose associated pseudo-representation is ψ̄, then there is a unique strict equivalence class of liftings of it with pseudo-representation ψ. By [M-DGR], the vector space of strict equivalence classes of liftings of ρ̄ is isomorphic to H^{1}(G, Ad(ρ̄)) which is of finite dimension over k.

Case 2. ψ̄ is reducible. That is, ξ̄(s, t) = 0 for all s and t in G. It follows from (1) and (4) that, a =: ᾱ and d =: δ̄ are characters on G with values in k^{∗}. Let ψ = (α, β, ξ) ∈ F (k[ϵ]). Write α(s) = a(s) + ϵR(s), β(s) = b(s) + ϵS(s) and ξ(s, t) = ϵX(s, t). Then (3) is trivial and if R′ = R/a and S′ = S/d,

(1) implies that

R′(st) − (R′(s) + R′(t)) = X(s, t)/a(st)  (5)
S′(st) − (S′(s) + S′(t)) = X(s, t)/d(st).  (6)

Moreover, if y(s, t) = X(s, t)/d(st) and χ(s) = a(s)/d(s), (2) implies:

\begin{aligned}y(st, r) & = χ(s)y(t, r) + y(s, r) \\ y(s, ru) & = y(s, r)χ(u) + y(s, u). \end{aligned}  (7)

In other words, y is a continuous bi-cocycle of G with values in k where the action of G on k is via the character χ. That is, r ↦ y(·, r) is a one-cocycle h_{r} and if V is the k vector space of one-cocycles, the map r ↦ (h_{r^{−1}}) is a one-cocycle with values in V where G now acts via the character χ^{−1}. Since, G satisfies the finiteness condition Φ_{p} of [M-DGR], we see first that V is a finite dimensional and then that the space of these cocycles is finite dimensional. Finally, given y and hence ξ, we see that the difference of any two R′’s satisfying (5) or of any two S′’s satisfying (6) is a homomorphism from G into k. Since there are only finitely many such homomorphisms, the set of liftings of ψ̄ to k[ϵ] is finite.

Remark. In the case where the pseudo-representation ψ̄ comes from a continuous, odd, two-dimensional representation ρ̄ of G = G_{Q,S} with values in the finite field k with the property that ρ̄ is absolutely irreducible (or more generally, such that the only endomorphisms commuting with ρ are scalars) then there exists a universal deformation ring R = R^{univ}(ρ̄) as in [M-DGR] and a universal deformation

ρ^{univ} : G_{Q,S} → GL_{2}(R)

of the residual representation ρ̄. In this case, we have that the universal deformation ring R^{univ}(ρ̄) is canonically isomorphic to the universal deformation ring R^{univ}(ψ̄) of the pseudo-representation ψ̄ attached to ρ̄, and (making the canonical identification of these two rings) the pseudorepresentation attached to ρ^{univ} is ψ^{univ}.

From now on, in this article, we suppose that N = 1.

By Proposition 5.1.1, there are only a finite number of non-isomorphic pmodular (modular, tame level 1, residual, characteristic p) representations. Let ψ̄_{1}, ψ̄_{2}, …, ψ̄_{m} be the set of the distinct pseudo-representations corresponding to these p-modular (modular, tame level 1, residual, characteristic p) representations. So the set of ψ̄_{j}’s is in natural one:one correspondence with the set of (isomorphism classes of) semi-simplifications of p-modular (modular, tame level 1, residual, characteristic p) representations. We will refer to any one of these pseudo-representations ψ_{j} (j = 1, … m) simply as a p-modular pseudo-representation. For each p-modular ψ, let R_{ψ} denote its universal deformation ring as constructed in Theorem 5.13. Put

R_{p} = ∏_{j=1}^{j=m} R_{ψ_{j}}.

We think of R_{p} as the universal p-adic deformation ring of residual representations which are p-modular, tame level 1, (pseudo-)representations, and will refer to R_{p} as the universal deformation ring (of tame level 1) for short. Of course, to be more precise we should perhaps qualify the adjective universal to emphasize that the only residual representations we consider are the p-modular ones, defined at the beginning of this section. The ring R_{p} is a semi-local complete noetherian ring, because of the proposition proved at the beginning of this section, and it comes along with a universal pseudo-representation,

r^{univ} = (ξ^{univ}, α^{univ}, δ^{univ})

with values in R_{p}.

For each p-modular pseudo-representation ψ, let X_{ψ} denote the rigid analytic space over Q_{p} associated to R_{ψ}, and let X_{p} = ∐_{j=1}^{m} X_{ψ} be the rigid analytic space over Q_{p} associated to R_{p}. We will refer to X_{p} as the universal deformation (rigid analytic) space (of tame level 1). As hinted above, there is a well-behaved part of R_{p} and a not-so-well-behaved part. Specifically, when ψ is an irreducible p-modular pseudo-representation (in the sense that its associated ξ-function does not vanish identically), then ψ is the pseudo-representation of an irreducible (odd, degree two) Grepresentation. Any such representation is absolutely irreducible (because an irreducible, but absolutely reducible, degree two representation is seen to have Abelian image, and therefore it stabilizes the two eigenvectors of the image of c). Let V̄ denote a k-vector space of dimension two, with klinear G-action having ψ as corresponding pseudo-representation. Then V̄ , being an absolutely irreducible residual G-representation, has an associated universal deformation space (cf. [M-DGR]) which we denote RV̄ equipped with a continuous G-action on a (free rank two) RV̄ -module V (the universal deformation of the G-module V̄) which satisfies the universal lifting property for deformations of the G-module V̄ (or equivalently, the G-representation to Aut_{RV̄} V is universal for strict equivalence classes of liftings of the Grepresentation on V̄). Passage to the corresponding pseudo-representation of the G action on V induces a natural isomorphism (cf. [Ny]) R_{ψ} ≅ RV̄.

Let ψ_{j} (j = 1, … μ) be the subset of the set of p-modular pseudorepresentations which are irreducible, and ψ_{j} (j = μ + 1, … m) the pmodular pseudo-representations which are not. For each j = 1, … μ fix a degree two G-representation V̄_{j} with pseudo-representation equal to ψ_{j}. Define

R_{p}^{irr} := ∏_{j=1}^{μ} R_{ψ_{j}} ≅ ∏_{j=1}^{μ} RV̄_{j},

and

R_{p}^{red} := ∏_{j=μ+1}^{m} R_{ψ_{j}}.

Then

R_{p} = R_{p}^{irr} × R_{p}^{red},

and corresponding to this product decomposition, we have a decomposition of associated rigid analytic spaces,

X_{p} = X_{p}^{irr} ⋃ X_{p}^{red}.

The essential difference between the two factors R_{p}^{irr} and R_{p}^{red} for us is that over the first factor we have a continuous representation

ρ^{univ−irr} : G → GL_{2}(R_{p}^{irr})

satisfying a universal property vis a vis irreducible p-modular residual representations (of tame level 1) whereas over the second factor we must make do with a continuous pseudo-representation (which nevertheless does indeed satisfy a corresponding universal property for pseudo-representations). The representation ρ^{univ−irr} induces a rigid analytic, O_{X_{p}^{irr}}-linear G-represen-tation on the free O_{X_{p}^{irr}}-bundle of rank two over X_{p}^{irr}. In contrast, we do not show the existence of an analogous rigid analytic, O_{X_{p}^{red}}-linear Grepresentation over X_{p}^{red}. Nevertheless, Theorem 5.1.2 will provide partial results along these lines, applicable to open smooth neighborhoods of the modular curve.

The ring R_{p} has “a” natural Λ-algebra structure. This is defined by noting that the (determinant) mapping μ : G → R_{p} of the universal pseudorepresentation, given by the formula

μ(g) := α^{univ}(g)δ^{univ}(g) − ξ^{univ}(g)

is, in fact, a determinant: it is a continuous homomorphism to R_{p}^{∗} and factors through the Abelianization χ : G^{ab} ≅ Z_{p}^{∗}, where we have identified G^{ab} with Z_{p}^{∗} via the cyclotomic character χ; that is, if ζ is a p-power root of unity, g(ζ) = χ(g) · ζ. This mapping Z_{p}^{∗} → R_{p} extends uniquely to a continuous ring homomorphism which we will denote here μ_{det} : Λ = Z_{p}[[Z_{p}^{∗}]] → R_{p} and which gives a Λ-algebra structure to R_{p} that might be referred to as the determinant Λ-algebra structure on R_{p} (for, tensoring R_{p} via this Λ-algebra structure with a character

κ : G → Z_{p}^{∗}

gives the universal ring of pseudo-deformations of ψ̄ possessing determinant character κ; for more discussion about this see section 24 of Chapter 5 of [M-DTR]). One should note that when one works with modular forms, as we are doing, it is more conventional to twist this Λ-algebra structure μ = μ_{det} to get a Λ-algebra structure on R_{p} that gives directly the weight of the modular form, rather than the determinant of its (pseudo-)representation. The conversion formula is

μ_{wt}([γ]) = γ · μ_{det}([γ]),

where for γ ∈ Z_{p}^{∗}, we denote its image in the completed group ring Λ by [γ], and the multiplication-law in the right-hand side of the above formula is via the natural Z_{p}-algebra structure of R_{p}. We will view R_{p} from now on with its weight Λ-algebra structure, i.e., the one given by μ_{wt} structure.

The rigid analytic space X_{p} is thereby endowed with a canonical projection to weight space W_{1} = W.

5.2. Pseudo-representations attached to Katz modular functions.

Keep the notation of the previous section. Let T_{p} := T_{p}(1) denote the completion of the algebra generated by Hecke operators acting on Katz modular functions of tame level 1 as described in section 3.3. Then as proven in §3 of [H2],

Theorem 5.2.1 (Gouvêa-Hida). There exists a continuous pseudorepresentation π from G_{Q,{p}} to T_{p} such that, for primes l |̸p,

“trace”(π(F rob_{l})) = T (l) and “ det ”(π(F rob_{l})) = ⟨l⟩^{∗}/l.

In particular, as T_{p} is the inverse limit of Noetherian rings (see section 3.3), we get a natural (unique) ring homomorphism

R_{p} → T_{p}

bringing the universal pseudo-representation of tame level 1 to π.

Corollary 5.2.2. Suppose m is a maximal ideal of T_{p} such that the corresponding residual representation ρ̄ is absolutely irreducible (cf. section 5.1). Then, there exists a representationρ_{m}: G_{Q,S} → GL_{2}((T_{p})_{m}) such that, for l |̸Np,

trace(ρ(F rob_{l})) = T (l) and det(ρ(F rob_{l})) = ⟨l⟩^{∗}/l.

Corollary 5.2.3. If p > 2, the ring T_{p} of Hecke operators on Katz modular functions of tame level 1 is noetherian.

Proof. We know that T_{p} is a complete semi-local p-adic ring, so all we have to show is that the completion (T_{p})_{m} is noetherian for any maximal ideal m. Let T′_{p} denote the subring of T_{p} which is the completion of the Λ-algebra generated by the Hecke operators T (l) for (l, p) = 1. Then it follows that T′_{p} ≅ lim_{↼} h′_{k}(p^{k}) where h′_{k}(p^{k}) is the subalgebra of h_{k}(p^{k}) generated by the diamond operator and the Hecke operators T (l) for (l, p) = 1. The pseudorepresentation π above actually takes values in T′_{p}. Let m′ = m ∩ T′_{p} and let \overline{π}_{m} denote the reduction of π: G_{Q,\{p\}} → T′_{p} modulo m′.* It follows that there exists a whose image contains the image of T (l) and ⟨l⟩^{∗} for homomorphism of R(\overline{ψ}_{m}) into (T_{p})_{m} all l ≠ p. Since R(ψ̄_{m}) is compact, using the Tchebotarev density theorem, we see that this homomorphism is surjective. Hence this complete local ring (T′_{p})_{m′} is Noetherian. (When m corresponds to an absolutely irreducible representation this is the conclusion of Corollary III.5.7 of [G-ApMF].) We need the following general proposition to conclude the proof:

Proposition 5.2.4. Suppose R_{n} are complete p-adic local rings and R_{n+1} → R_{n} are surjections. Then R = lim_{↼}_{n} R_{n} is a complete local ring.

Let m be its maximal ideal and k its residue field. The ring R is Noetherian if and only if the dimensions of the Zariski tangent spaces dim_{k} mR_{n}/m^{2}R_{n} are bounded independently of n.

Proof. The Noetherianness of R implies the uniform boundedness of the dimensions of the Zariski tangent spaces. If now dim_{k} mR_{n}/m^{2}R_{n} is bounded independently of n, it follows that d =: dim_{k} m/m^{2} is finite. Suppose t_{1}, …, t_{d} ∈ m is a basis modulo m^{2}. Then there is a natural map from W (k) to R and the map

W (k)[[T_{1}, …, T_{d}]] → R,

which sends T_{i} to t_{i} is a surjection. Since W (k)[[T_{1}, …, T_{d}]] is Noetherian we may conclude the proof.

We can now complete the proof of the corollary. Let T_{n} = h_{2}(p^{n}) and T′_{n} = h′_{2}(p^{n}). Then T_{n} = T_{n}^{∗}[u_{n}], where U = lim_{↼} u_{n}. Let k = T/m and k′ = T′/m′. Since, (T′)_{m′} is Noetherian, there exists a_{1}, …, a_{d} in m′ which reduce to a basis of m′/(m′)^{2} over k′. Let g(x) ∈ k′[x] be the minimal monic polynomial for U mod m over k′, which exists since k is finite. Suppose f is its degree. Now suppose n is such that k′ ≅ T′_{n}/m′T′_{n} := m′_{n} and k ≅ T/mT := m_{n}. We identify these isomorphic fields. Let g̃ be a lifting of g to a monic polynomial in T′_{n}[x]. It follows that a_{0} =: g̃(u_{n}) ∈ m′_{n}. We can write any element α of T_{n} in the form

α = ∑_{j} b_{j}(u_{n})g̃(u_{n})^{j},

* We note that, T′_{p} is also a complete semi-local ring with finite residue fields. Moreover, the map Max(T_{p})→Max(T′_{p}) is surjective by Atkin-Lehner theory and bijective away from the ordinary points.

The Eigencurve 83 where b_{j}(x) ∈ T′_{n}[x] and deg b_{j}(x) < f. Then α ∈ m_{n} if and only if b_{0}(x) ∈ m′_{n}T′_{n}[x]. This means that, in this case, we can write

b_{0}(x) = ∑_{k=1}^{d} c_{k}(x)a_{k}

where c_{k}(x) ∈ T′_{n}[x]. We conclude,

α ≡ b_{1}(u_{n})a_{0} + c_{1}(u_{n})a_{1} + ⋯ c_{d}(u_{n})a_{d} mod mT_{n}.

In particular,

dim_{k} m_{n}/m_{n}^{2} ≤ dim_{k′} m′_{n}/(m′_{n})^{2} + 1,

which concludes the proof. Corollary. dim_{k} m/m^{2} ≤ dim_{k′} m′/(m′)^{2} + 1.

Chapter 6. The Eigencurve.

6.1. The definition of the eigencurve.

Keep the notation of Chapter 5. We assume that p > 2, and recall that we are working with tame level N = 1. Let R_{p} be the universal deformation space for (p-modular) pseudo-representations of tame level 1, as before, and X_{p} is the associated rigid analytic (universal deformation) space of tame level 1. Recall that R_{p} has a natural H′-algebra structure, and is the receiving ring for the universal pseudo-deformation r^{univ} of G_{Q,S} with the notational conventions of Chapter 5. Let Y_{p} denote the rigid analytic variety given as the product

Y = Y_{p} = X_{p} × A^{1},

where the parameter for A^{1} will be given by the variable T. We prepare to apply the constructions of section 1.2 to Y. In particular, putting R = R_{p} for short, we have a natural homomorphism of the ring of entire power series R{{T }} into the ring A(Y) of rigid analytic functions on Y.

Let α ∈ H′ be any element such that the image, ι · α ∈ R, is a unit in R (where ι : H′ → R denotes the canonical homomorphism). Let Ũ ∈ U be the product of α and U_{p}. Let PŨ(T) ∈ Λ{{T }} be the Fredholm determinant of the completely continuous system of operators associated to Ũ as discussed in Chapters 3 and 4 (and was also denoted P_{α}(T) in Chapter 5). Put

ϕŨ := PŨ(T/(ι · α))

which, by Theorem 4.3.1 and our hypothesis about α, is a Fredholm series in the ring R{{T }}, and consequently determines a Fredholm function on Y_{p}, which we will denote by the same letter. Consider the ideal I in A(Y) generated by the Fredholm functions ϕŨ for all α’s which satisfy our hypothesis.

Definition 1. The eigencurve of tame level 1 is the closed, rigid analytic Q_{p}-subvariety,

C = C_{p} ⊂ Y_{p} = X_{p} × A^{1}

whose ideal of definition is I ∈ A(Y).

Remark. We shall be proving that C is a curve (cf. below) but this fact is not evident from the definition.

Equivalently, we can think of C as follows: For every α ∈ H′ such that ι · α is a unit in R we define the mapping r_{α}: X_{p} × A^{1} → W × A^{1} by

r_{α}(x, t) = ( π(x) , \frac{t}{ι · α(x)} ),

where π : X_{p} → W is the canonical projection to weight space (see section 5.1). Then

C_{p} := ⋂_{α} r_{α}^{−1}(Z_{α·U}),

where the spectral curve Z_{α·U} is the locus of zeroes of the Fredholm function P_{α·U}(T) on W × A^{1}.

By its definition, the curve C_{p} comes along with a rigid analytic projection to the universal deformation space C_{p} → X_{p} and therefore by composition of this morphism with the canonical projection, π : X → W, we have a canonical projection, C_{p} → W, of C_{p} to weight space. There are also natural morphisms of C_{p} to ZŨ for every Ũ = α · U_{p} with ι · α ∈ R^{∗}, and these natural morphisms are compatible with projection to weight space. Explicitly, we have the ring homomorphism

ιŨ : Λ{{T̃}} → R{{T }}

which extends the natural ring homomorphism ι : Λ → R by sending the variable T̃ to T/(ι · α). The ring homomorphism ιŨ brings the Fredholm determinant PŨ(T̃) ∈ Λ{{T̃}} to ϕŨ ∈ R{{T }}. Since PŨ(T̃) generates the defining ideal of the rigid analytic curve (the Fredholm hypersurface)

ZŨ ⊂ W × A^{1}

and since ϕŨ is in the defining ideal of C_{p} ⊂ X_{p} × A^{1}, ιŨ induces a rigid analytic morphism, call it λŨ : C_{p} → ZŨ.

Given a C_{p}-valued point c ∈ C_{p} let us refer to its two coordinates c = (r, 1/u) ∈ X_{p} × A^{1} as its associated pseudo-representation r = r_{c} and inverse U_{p}-eigenvalue respectively, and we will refer to the quantity u = u_{c} as the U_{p}-eigenvalue of c. The point c induces a ring homomorphism

Ψ_{c} : H → C_{p}

as follows. The associated pseudo-representation r = r_{c} induces a homomorphism Ψ_{r} : H′ → C_{p} as in the previous discussion and Ψ_{c} is the unique (ring homomorphism) extension of Ψ_{r} to the polynomial ring H = H′[U_{p}] which sends U_{p} to u ∈ C_{p}. As in the discussion in Section 3.1 we may attach to the ring homomorphism Ψ_{c} a Fourier expansion F_{c} which we will refer to as the Fourier expansion of the point c ∈ C_{p}:

F_{c} = ∑_{n=1}^{n=∞} Ψ_{c}(T_{n})q^{n} ∈ C_{p}[[q]],

where the conventions regarding the symbols T_{n} are as in Chapter 3.

The point c ∈ C_{p} is determined by its Fourier expansion F_{c}.

6.2. The points of the eigencurve are overconvergent eigenforms.

Recall that an eigenform f for (the full Λ-algebra of operators) H, with Fourier expansion f̃ = ∑_{n=0}^{∞} a_{n}q^{n}, is called normalized if either a_{1} = 1, or else we are in the specific case of weight 0 where it is possible for a_{1} to vanish, in which case the standard recurrence relations that hold for the Fourier coefficients of a H -eigenform force all the nonconstant coefficients of f̃ to vanish; in this idiosyncratic case, let us call f normalized if f̃ is the constant power series 1. For a positive integer n this normalized eigenvector has T_{n}-eigenvalue σ_{−1}^{∗}(n).

Any eigenform for H is a multiple of a normalized eigenform.

Theorem 6.2.1. There is a one-one correspondence between the set of normalized overconvergent modular eigenforms with Fourier coefficients in C_{p}, of tame level 1, weight w ∈ W(C_{p}), having nonzero U_{p}-eigenvalue, and the set of C_{p}-valued points of weight w on the eigencurve C = C_{p}. If f is a normalized overconvergent modular eigenform (of tame level 1) with Fourier coefficients in C_{p} and with nonzero U_{p}-eigenvalue with Fourier expansion

f̃ = ∑_{n=0}^{n=∞} a_{n}q^{n} ∈ C_{p}[[q]],

with a_{1} = 1, then the point c ∈ C to which f corresponds has its Fourier expansion equal to f̃ − a_{0} (i.e., the Fourier expansion of c is that of f deprived of its constant term).

Proof. Starting with an overconvergent modular eigenform f with the characteristics formulated in the statement of the above theorem, one sees (by Theorem 4.4.1) that there is a point zŨ ∈ ZŨ for every Ũ ∈ U corresponding to f and therefore, one immediately sees from the construction of the rigid analytic space C_{p} that there is a (C_{p}-valued) point c ∈ C_{p} corresponding to f as asserted in the statement of our theorem. It is somewhat more delicate to go the other way.

For this, let us start with a C_{p}-valued point c of the eigencurve C = C_{p}. We shall fix c in the discussion below, and define a list of objects dependent upon c. Note that we are working systematically, in the proof below, over C_{p}. Denote the weight of c by w (i.e., w is the C_{p}-valued point of W which is the image of c under the canonical projection C_{p} → W).

For Ũ = α · U_{p} with α ∈ H′ such that ια is in R^{∗}, let zŨ := λŨ(c) ∈ ZŨ. The points zŨ = (w, 1/ũ) ∈ W × A^{1} are, of course, of fixed weight w. Moreover, we have that ord_{p}(ũ) is equal to σ := the slope of c, i.e. σ = ord_{p}(u) where u is the U_{p}-eigenvalue of c (Note that ord_{p}(ũ) = ord_{p}(u) since ια ∈ R^{∗}). Applying Theorem 4.1.1 to the points zŨ ∈ ZŨ, we have that, for every Ũ as above, there is at least one overconvergent modular form of weight w which is an eigenform (not just a generalized eigenform) for Ũ = αU_{p} with eigenvalue ũ. Moreover, the Ũ-eigenvalue of each such eigenform is equal to ũ = the Ũ-eigenvalue of c. The C_{p}-vector space VŨ of these overconvergent Ũ- eigenforms (of weight w, tame level 1, and Ũeigenvalue equal to ũ) is finite dimensional, and is stabilized by the action of the commutative Λ-algebra H. Therefore VŨ contains eigenforms for the action of the entire algebra H. It follows that VŨ contains at least one normalized H -eigenform. Let FŨ denote the set of all normalized eigenforms in VŨ. That is, FŨ is the set of normalized overconvergent modular forms of tame level 1 which are H -eigenforms and such that their Ũ-eigenvalue is equal to ũ. Then for every Ũ = α · U, FŨ is a finite nonempty set.

Lemma 6.2.2. The intersection of the sets FŨ (running over all Ũ’s as above) is nonempty and, in fact, consists in precisely a single eigenform f.

Remarks and Proof of Lemma 6.2.2. Assume, for a second, that there is an eigenform

f ∈ ⋂_{Ũ} FŨ.

It follows that f has the same Fourier expansion as c. To see this, first note that f has the same U_{p}-eigenvalue as c (because f ∈ F_{U_{p}}). Moreover, for τ ∈ H′ and Ũ = (1 + p · τ) · U_{p}, since f ∈ FŨ, we have that f has the same Ũ-eigenvalue as c. Since the operators U_{p} and Ũ commute, it follows that f also has the same τ-eigenvalue as c. This argument yields two things. First, we see that it suffices to show existence of f, for then its uniqueness follows. Second, we see that it suffices to show that

⋂_{Ũ} FŨ

is nonempty where the intersection is taken over a special list of Ũ’s. Specifically, let ℓ_{1}, ℓ_{2}, … be an enumeration of all the prime numbers different from p so that U_{p}, T_{ℓ_{1}}, T_{ℓ_{2}} … is an enumeration of the full list of Hecke and Atkin-Lehner operators.

Sublemma 6.2.3. If for a fixed infinite sequence of positive integers a_{1}, a_{2}, … and for the list

Ũ_{0} = U_{p}
Ũ_{1} = (1 + p^{a_{1}}T_{ℓ_{1}})Ũ_{0}
Ũ_{2} = (1 + p^{a_{2}}T_{ℓ_{2}})Ũ_{1}

…

of operators, the intersection ⋂Ũ FŨ is nonempty as Ũ runs through Ũ_{k} for k ≥ 0, then this intersection is nonempty as Ũ runs through all (1+p·τ)·U_{p}.

In the proof of Lemma 6.2.2 that we give below, we shall assume given an enumeration of the primes (different from p) as above, and we will be generating inductively a particular sequence of positive integers a_{1}, a_{2}, … as required by sublemma 6.2.3 to prove lemma 6.2.2.

Step 1. Consider all the Ũ’s of the form Ũ = α · U_{p} where α = 1 + p · τ for τ ∈ H′. For these Ũ’s, the eigenforms of fixed weight w and eigenvalue of fixed slope σ all lie in the vector space

M_{κ,σ}

defined at the end of section 4.1 (here, κ = w) which we denote simply M, for short; recall that M comes along with a natural H-module structure. By Proposition 4.1.2, this (finite dimensional) M is generated by all overconvergent generalized Ũ-eigenforms of weight w and Ũ-slope equal to σ, where Ũ is any element of U as above. In particular, it is generated by all overconvergent generalized U_{p}-eigenforms of weight w and U_{p}- slope equal to σ. Since there are only a finite number of eigenvalues for the operator U_{p} acting on the (finite dimensional) space M we may find an integer a_{1} large enough to distinguish U_{p}-eigenvalues in the sense that if λ and λ′ are distinct U_{p}- eigenvalues in its action on M, then ord_{p}(λ − λ′) < a_{1}.

Step 2. Put α_{1} := 1 + p^{a_{1}} · T_{ℓ_{1}}, and Ũ_{1} = α_{1} · U_{p}. Here, ℓ_{1} is the first prime given in our enumeration above, and a_{1} is the integer obtained in Step 1. Let

f_{1} ∈ FŨ_{1},

so that the Ũ_{1}-eigenvalue of f_{1} is the same as that of c. We claim that it then follows that both the U_{p}- and the T_{ℓ_{1}}Øeigenvalues of f_{1} are the same as those of c. To see this, let us establish notation for the U_{p}- and the T_{ℓ_{1}}- eigenvalues of f_{1} and of c by the formulas:

U_{p} · f_{1} = u_{p,(f)} · f_{1} ; U_{p} · c = u_{p,(c)} · c
T_{ℓ_{1}} · f_{1} = t_{ℓ_{1},(f)} · f_{1} ; T_{ℓ_{1}} · c = t_{ℓ_{1},(c)} · c.

Denoting by ũ_{1} the common Ũ_{1}-eigenvalue of f_{1} and c, we have two expressions for ũ_{1}; namely:

ũ_{1} = (1 + p^{a_{1}} · t_{ℓ_{1},(f)}) · u_{p,(f)} = (1 + p^{a_{1}} · t_{ℓ_{1},(c)}) · u_{p,(c)}.

Since all the elements of the above formula lie in the ring of integers of C_{p}, it follows that ord_{p}(u_{p,(f)} − u_{p,(c)}) ≥ a_{1} which implies, by Step 1, that u_{p,(f)} = u_{p,(c)}, giving also that t_{ℓ_{1},(f)} = t_{ℓ_{1},(c)}. Since there only a finite number of eigenvalues for the operator Ũ_{1} acting on the (finite dimensional) space M we may find an integer a_{2} ≥ a_{1} large enough so that if λ and λ′ are distinct Ũ_{1}- eigenvalues in the space M, then ord_{p}(λ − λ′) < a_{2} − a_{1}.

Step 3. We follow the pattern of Step 2. Put α_{2} := 1 + p^{a_{2}} · T_{ℓ_{2}}, and Ũ_{2} = α_{2} · Ũ_{1}. Let

f_{2} ∈ FŨ_{2},

so that the Ũ_{2}-eigenvalue of f_{2} is the same as that of c. With the analogous notational conventions as in Step 2 (i.e., the eigenvalue of an operator on f_{2} or on c is denoted by the symbol denoting the operator, only in lower case, and with an (f) or a (c) in the subscript) we have:

ũ_{2} = (1 + p^{a_{2}} · t_{ℓ_{2},(f)}) · ũ_{1,(f)} = (1 + p^{a_{2}} · t_{ℓ_{2},(c)}) · ũ_{1,(c)}.

By an analogous argument to that of Step 2, since a_{2} is large enough to distinguish Ũ_{1}-eigenvalues, we get from the above equation first that the Ũ_{1}-eigenvalue of f_{2} and of c are equal, and then that the T_{ℓ_{2}}- eigenvalues of f_{2} and of c are equal as well. Since the Ũ_{1}-eigenvalue of f_{2} and of c are equal, a direct application of the argument of Step 2 then gives that f_{2} and c agree on their T_{ℓ_{1}}- and U_{p}-eigenvalues as well. Let a_{3} ≥ a_{2} be an integer large enough, now, to distinguish Ũ_{2}-eigenvalues in the action of Ũ_{2} on M.

Step 4. Proceeding by induction as in Steps 2 and 3 we may obtain a sequence of integers a_{1} ≤ a_{2} ≤ … a_{k} … and corresponding operators Ũ_{k} for k = 1, 2, … and overconvergent modular forms f_{k} ∈ FŨ_{k} ⊂ M which have the property that f_{k} and c agree on their U_{p}-eigenvalues as well as their T_{ℓ_{j}}-eigenvalues for j = 1, …, k. Since the sets FŨ_{k} are finite, it follows that there is at least one element f ∈ ⋂_{k=1}^{∞} FŨ_{k} ⊂ M which, by the previous discussion, is then an eigenform for U_{p} and the Hecke and Atkin-Lehner operators T_{ℓ} for all prime numbers ℓ ≠ p. This f is then an overconvergent (normalized) modular eigenform of tame level 1 whose eigenvalues for the action of any element in H is the same as that of c. This establishes Theorem 6.2.1.

6.3. The projections of the eigencurve to the spectral curves.

But the above proof gives more, and to record what it gives, let us consider multiplicities of various spaces of generalized eigenforms. Define:

μ(c) := the dimension over C_{p} of the space of all overconvergent generalized eigenforms (for H) in the H-isotypic component corresponding to c in M = M_{k}(C_{p}; )_{σ}.

Similarly, for Ũ ∈ U of the appropriate form (i.e., Ũ = (1 + p · τ) · U_{p} with τ ∈ H) let z ∈ ZŨ be the image of c under λŨ, and put

μ(z) := the dimension over C_{p} of the space, call it MŨ ⊂ M, of all overconvergent modular forms in M = M_{k}(C_{p})_{σ} which have the same weight as c, and which are generalized Ũ-eigenforms with the same Ũ-eigenvalue as c.

We have the following geometric properties of the projections of the ZŨ’s to weight space, and the morphisms λŨ : C → ZŨ (using the Reisz Theory developed in A4 of [C-BMF]).

Proposition 6.3.1. The structure of ZŨ over weight space.

Let Ũ = (1 + p · τ) · U_{p} with τ ∈ H. Then ZŨ is a Fredholm hypersurface in W × A^{1}.

Proof. This follows from Theorems 4.3.1 and 4.4.1.

Proposition 6.3.2. The structure of λŨ : C → ZŨ.

a. The morphism λŨ : C → ZŨ is surjective.

b. If z is point in ZŨ(C_{p}) then

μ(z) = ∑ μ(c)

where the sum ranges over the points c in C(C_{p}) which map to z.

Proof. Part a follows from the fact that every point z in ZŨ(C_{p}) corresponds to a finite dimensional space of eigenforms for Ũ with fixed weight and slope by theorems 4.1.1 and 4.3.1 which is stable under the action of H. It follows that this space contains an eigenform and if c is the coresponding point of C, c maps to z.

Part b follows from the fact that the space of generalized eigenforms corresponding to a point z ∈ ZŨ(C_{p}) is of dimension μ(z) and is the direct sum isotypic components for the action of H. Moreover each such isotypic component corresponds to a point c ∈ C(C_{p}) which maps to z and has dimension μ(c).

We now state some consequences of the above Propositions and of Theorem 6.2.1 (and its proof). We will have occasion to refer below to the sequence of operators Ũ_{k} for k = 1, 2, … given in that proof.

Corollary 6.3.3. For c ∈ C as in Theorem 6.2.1 with weight w, we have: 1. If k is sufficiently large,

FŨ_{k} = {f_{c}}.

2. Let z_{k} ∈ ZŨ_{k} denote the image of c under λŨ_{k}. If k is sufficiently large, then

μ(c) = μ(z_{k}).

Proof. The first statement was proved in the course of the proof of Theorem 6.2.1. But the second comes along with it for if k is sufficiently large to guarantee that FŨ_{k} = {f_{c}}, then any Ũ_{k}-generalized eigenform in MŨ_{k} is an H-generalized eigenform in the isotypic component corresponding to c.

6.4. The Eisenstein curve.

Define the Eisenstein pseudo-representation r_{κ}^{Eis} = (ξ_{κ}^{Eis}, α_{κ}^{Eis}, δ_{κ}^{Eis}) of weight κ by the formulas ξ_{κ}^{Eis}(g, h) ≡ 0, α_{κ}^{Eis}(g) ≡ 1, and let δ_{κ}^{Eis} denote the continuous C_{p}^{∗}-valued character of G_{Q,{p,∞}} which is the determinantcharacter of representations attached to modular eigenforms of weight κ, as discussed at the end of section 5.1. Specifically, consider the composition of the homomorphisms

η_{κ} : G_{Q,{p,∞}} → Z_{p}^{∗} → Λ^{∗} → C_{p}^{∗},

where the first homomorphism is χ, the canonical homomorphism arising from action on p-power roots of 1 (so for ζ ∈ Q̄_{p} a p-power root of unity, g(ζ) = ζ^{χ(g)}) the second is the natural injection, and the third is the restriction of κ to multiplicative groups of units. Calling χ again the C_{p}^{∗}-valued character induced by χ (i.e., the composition χ : G_{Q,{p,∞}} → Z_{p}^{∗} ⊂ C_{p}^{∗}, we put

δ_{κ}^{Eis} := χ^{−1} · η_{κ}.

More systematically, one can simply define a single pseudo-representa-tion r^{Eis} = (ξ^{Eis}, α^{Eis}, δ^{Eis}) of G_{Q,{p,∞}} with values in Λ by taking ξ^{Eis} ≡ 0 and α^{Eis} ≡ 0 and δ^{Eis} the evident character G_{Q,{p,∞}} → Λ^{∗} so that κ(r^{Eis}) = r_{κ}^{Eis} for all C_{p}-valued weights κ. By universality for p-modular pseudo-representations, we then get that r^{Eis} is the pseudo-representation induced from a continuous ring-homomorphism π^{Eis} : R_{p} → Λ (which sends the radical of R_{p} to the maximal ideal in Λ). By X^{Eis} ⊂ X_{p} let us mean the rigid analytic subspace of X_{p} associated to the closed subscheme Spec (Λ) ⊂ Spec (R_{p}) and define the Eisenstein curve C_{p}^{Eis} to be the rigid subspace of X_{p} × A^{1} defined by:

C_{p}^{Eis} := X^{Eis} × 1 ⊂ X_{p} × A^{1},

that is, the U_{p}-eigenvalue ( (in the sense discussed in section 6.1) of the points of C_{p}^{Eis} is taken to be identically equal to 1. By Propositions 3.6.1, and 6.2.1, we see that the set of C_{p}-valued points of C_{p}^{Eis} is contained in the set of C_{p}-valued points of the eigencurve C_{p}, and, moreover, any point of C_{p} which is not in C_{p}^{Eis} is represented by a normalized cuspidal-overconvergent eigenform (of tame level 1 and finite slope).

Chapter 7. The eigencurve as a finite cover of a spectral curve.

In this chapter, we will construct a curve which we will call D, which we will prove is isomorphic to the reduced rigid space associated to the eigencurve. We perform this construction by patching together generalizations of the pieces made in the proof of Theorem B5.7 of [C-BMF].

7.1. Local pieces.

We will need the following generalization of Theorem B5.7 of [C-BMF] (see also Theorem 1 of [CST]): We let p be an odd prime number, and work with tame level 1. We keep the notation of the previous chapters except for the following changes: For α ∈ H let P_{α}(T) denote the Fredholm determinant of αU_{p}, Z_{α} the spectral curve of αU_{p} and λ_{α} the natural map from C_{p} to Z_{α}. (These were called P_{αU_{p}}(T), Z_{αU_{p}} and λ_{αU_{p}}, respectively, in previous sections.) We view Z_{α} as a rigid analytic Fredholm curve over Q_{p}. Let π_{α} : Z_{α} → W be the projection of Z_{α} to weight space. Now fix α ∈ H. If V be an irreducible affinoid subdomain in Z_{α}, then by Lemma A5.6 of [C-BMF], its image Y := π_{α}(V) in W is an affinoid subdomain of W. Denote by Z_{Y} ⊂ Z_{α} the inverse image of Y under the morphism π_{α}. Let C_{α} be the collection of affinoid subdomains V in Z_{α} such that V is finite over its image Y := π_{α}(V) and which are admissibly closed-open in Z_{Y}. Let C_{α}^{i} denote the set of irreducible members of C_{α}. We know using Proposition A5.8 of [C-BMF] that C_{α} is an admissible covering of Z_{α}.*

Fixing α ∈ H, for each affinoid subdomain Y ∈ W, there is a one:one correspondence

V ⟷ { P_{α}(T) = Q_{V} (T)H(T) },

where V runs through the subset of C_{α}^{i} with image Y, and where the equations P_{α}(T) = Q_{V} (T)H(T) run through all factorizations of P_{α}(T) over A := A(Y), with Q_{V} (T) a polynomial in A[T] whose constant term is one and whose leading coefficient is a unit in A, and with H(T) an entire function in A{{T }} such that (Q_{V} , H) = 1. If V and Q_{V} (T) are connected via the above one:one correspondence, we have that

* It was said in [C-BMF] that the proof of Proposition A5.8 given there only works over C_{p}, but it actually works more generally. First the phrase “x∈V ” should be replaced by “closed points x of V.” Then the affinoid W found in the proof of Lemma A5.9 can be defined, in the same way, for arbitrary K.

V = Max A(Y)⟨T ⟩/(Q_{V}^{∗} (T)),

recalling the notation Q_{V}^{∗} (T) := T^{d}Q_{V} (T^{−1}), where d = degree(Q_{V} (T)). Let M_{Y}^{†} (v) be the Banach A-module of v-overconvergent modular forms of tame level 1 over the affinoid domain Y, as in section 2.4, for sufficiently small v > 0, and M_{Y}^{†} = ∪_{v}M_{Y}^{†} (v) the A-module of overconvergent modular forms of tame level 1 over the affinoid domain Y. For each v > 0 we obtain a direct sum decomposition

M_{Y}^{†} (v) = N(V; v) ⨁ F (V; v)

of closed A(Y)-modules for which Q_{V}^{∗} (Ũ) annihilates N(V; v) and is invertible on F (V; v). Using the results of section 4.3 we see that N(V; v) is independent of v for v close to 1. The submodule N(V) ⊂ M_{Y}^{†} is then defined to be N(V; v) for v sufficiently close to 1.

Remark. In the notation of section A4 of [C-BMF], we may write

N(V) = NŨ_{Y} (Q_{V}),

where Ũ_{Y} is the restriction of Ũ = αU to M_{Y}^{†}.

By Theorem A4.5 of [C-BMF], N(V) is locally free over A = A(Y) of constant rank d_{V} := deg Q_{V} (T). Define T(V), the overconvergent finite slope Hecke algebra over V, to be the image of H ⊗ A in End_{A}(N(V)). The A-algebra T(V) is complete since End_{A}(N(V)) is finite over A = A(Y). We have that T(V) is the ring of rigid analytic functions on an affinoid D(V) over Q_{p} finite flat over Y of degree d. The argument for this can be found on pp. 437-438 of [C-BMF]. In fact, A is a PID and so N(V) is free. In summary, so far, we have associated to every element V of C_{α}^{i} with image Y ⊂ W, an affinoid D(V) which is finite and flat over Y.

V/Y ⟼ D(V)/Y .

Since any member of C_{α} is a finite disjoint union of elements in C_{α}^{i}, we may extend the above association (by taking disjoint unions) to any member V of C_{α}. Now let μ denote the projection of W × A^{1} to A^{1}. We will apply the q-expansion principle to show:

Theorem 7.1.1. Suppose V ∈ C_{α} such that Y := π_{α}(V). Let L ⊂ C_{p} be a finite extension of Q_{p}. For an L-valued point x ∈ D(V)(L), let η_{x}: T_{V} → L denote the corresponding homomorphism. Now suppose κ is a weightcharacter in Y (L), and let (D(V))_{κ} denote the fiber in D(V) over κ . For each x ∈ (D(V))_{κ}(L) the vector space W_{x} over L of weight-character κ overconvergent eigenforms F over L such that

F |T_{n} = η_{x}(T_{n})F

(recalling the notation of 3.1) is one dimensional. Moreover, the correspondence x ↦ W_{x} is a bijection between (D(V))_{κ}(L) and the one dimensional spaces of eigenforms of weight-character κ whose inverse (αU)_{κ}-eigenvalue lies in μ(V_{κ}(L)).

The proof runs along the same lines as that of Theorem B5.7 of [C-BMF]. We construct a pairing between T(V) and N(V) using q-expansions in the same way. Specifically, if τ ∈ T(V) and f ∈ N(V), define

⟨τ, f⟩ := a_{1}(τ · f),

where a_{1} denotes the coefficient of q^{1} in the Fourier expansion. The main difference between the present setting and Theorem B5.7 of [C-BMF] is that the pairing here is not perfect, in general, because of the possible existence of a non-zero constant term in the q-expansions. Our pairing leads to an exact sequence of A-modules

0 → T(V) → Hom_{A}(N(V), A)→^{ρ}M → 0,  (7.1.1)

where M is the residue field of B at the trivial character 1 if 1 ∈ N(V)_{1} and ρ(h) is then h_{1}(1) in this case, otherwise M = 0. The proof of this requires our comparison of convergent modular forms and Katz modular functions, Theorem 2.4.2, and the injectivity of the q-expansion map on Katz modular functions by the second lemma of section XI of [K-LME]. Keeping this in mind, we can (almost) reformulate this result in terms of q-expansions. In particular, we obtain,

Corollary 7.1.2. Let notation be as in the theorem and set

F_{x}(q) = ∑_{n≥1} η_{x}(T_{n})q^{n}.

Then if κ is not trivial or if κ is trivial and η_{x} is not the homomorphism which takes T_{n} to σ_{−1}^{∗}(n), F_{x}(q) is the q-expansion of a normalized overconvergent eigenform over L (minus its constant term) of weight-character κ.

Here σ_{−1}^{∗}(n) is, as in 2.2, the sum of the reciprocals of the squares of the positive divisors of n which are relatively prime to p. We have a natural finite map of D(V) to V since the image of Q_{V}^{∗} (αU), where Q_{V}^{∗} (T) = T^{d}Q_{V} (T^{−1}), in T(V) is zero, by Theorem A4.5 of [BMF] (see also Corollary 4.2.3 above). This map is not necessarily an injection but it is when V is reduced (this follows from the Cayley-Hamilton theorem). Since both V and D(V) have degree d over Y we deduce:

Proposition 7.1.3. If Q_{V} (T) is square free, then the map from D(V) to V is generically an isomorphism and moreover D(V) is reduced.

The last assertion follows from the following lemma, taking X to be D(V) and W to be π_{α}(V).

Lemma 7.1.4. Suppose X → W is a morphism of affinoids of dimension one such that W is reduced and irreducible and A(X) is finite free over A(W). Then if X is generically reduced, it is reduced.

Proof. Indeed, since X is generically reduced, of dimension one and finite over Y there exists an open Y in W such that X_{Y} is reduced (just take Y to be the complement of the image of the finitely many points where X is not reduced). Thus if e ∈ A(X) and e^{2} = 0, the image of e in A(X_{Y}) must be zero. But, the other hypotheses imply that the map A(X) → A(X_{Y}) is an injection. Thus e = 0.

7.2. Gluing.

Fix α ∈ H, and let Z = Z_{α}. C := C_{α}.

Lemma 7.2.1. If V_{1} and V_{2} belong to C so does V_{1} ∩ V_{2}. Moreover, if V_{1} ⊆ V_{2}, D(V_{1}) is naturally an affinoid subdomain of D(V_{2}).

Proof. Let M = M^{†}. Let Y_{i} be the image of V_{i} in W and Y_{1 2} = Y_{1} ∩ Y_{2}. Then, it is easy to see that V_{i} ∩ Z_{Y_{1 2}} ∈ C so we may assume V_{i} = V_{i} ∩ Z_{Y_{1 2}}. The hypotheses imply that there exists functions e_{1} and e_{2} on Z_{Y_{1 2}} such that e_{i}|_{V_{j}} = δ_{i,j}. It follows that V_{1} is the disjoint union of two subdomains U_{1} and U_{2}, disconnected from each other, where e_{2} equals 1 and 0 respectively. Hence each of these must be finite over Y_{1}. But U_{1} = V_{1} ∩ V_{2}. Now suppose V_{1} ⊆ V_{2}. First as we argued above V_{3} := V_{2} ∩ Z_{Y_{1}} ∈ C. Moreover, it is not hard to see that D(V_{3}) is naturally isomorphic to D(V_{2})_{Y_{1}} which is an affinoid subdomain of D(V_{2}). Thus we may suppose Y_{1} = Y_{2}. Let Q_{i}(T) be the factor of P_{α}(T) over Y_{1}, such that Q_{i}(0) = 1, corresponding to V_{i}. Then, again arguing as above, V_{2} = V_{1} ∪ V′_{1} where V_{1} and V′_{1} are disconnected and finite over Y_{1}. It follows that

Q_{2}(T) = Q_{1}(T)Q′_{1}(T),

where Q_{1}(T) and Q_{2}(T) are relatively prime. We may apply the theory of Riesz decomposition (as in section A4 of [C-BMF]) because αU is associated to a completely continuous system (cf. section 4.3) of operators on M_{Y_{1}} = ∪_{v}M_{Y_{1}}^{†}(v). We have

N_{Q_{2}} = N_{Q_{1}} ⊕ N_{Q_{1}}′.

From this we see that

D(V_{2}) = D(V_{1}) ∐D(V′_{1})

and so D(V_{1}) is an affinoid subdomain of D(V_{2}) as asserted.

Construction of the curve D_{α}. Now we will glue all the D(V)’s, for V ∈ C_{α}, together to make a rigid analytic space D_{α} using Proposition 9.3.2/1 of [BGR]. For U, V ∈ C_{α}, let D(U, V) denote the image of D(U ∩ V) in D(U). We know this is an affinoid subdomain. Let ϕ_{U V} be the natural map from D(U, V) to D(V, U). Clearly, ϕ_{U V} ∘ ϕ_{V U} = id, D(U, U) = D(U) and ϕ_{U U} = id. Also ϕ_{U V} induces an isomorphism

ϕ_{U V W}: D(U, V) ∩ D(U, W) → D(V, U) ∩ D(V, W)

such that

ϕ_{U V W} = ϕ_{W V U} ∘ ϕ_{U W V}

because D(U, V) ∩ D(U, W) is the image of D(U ∩ V ∩ W) in D(U). Thus the triple ({D(U)}, {D(U, V)}, {ϕ_{U,V} })_{U,V ∈C} satisfies all the hypotheses of the aforementioned proposition and the D(U)’s glue together into a rigid curve D_{α} Note that D_{0} is empty. Let z_{α} denote the projection from D_{α} onto Z_{α}. By construction,

Proposition 7.2.2. The morphism z_{α}: D_{α} → Z_{α} is finite, and the projection to weight space π : D_{α} → W has the property that it is locally in-the-domain finite flat in the sense that D_{α} is covered by affinoid subdomains D_{α}(V) ⊂ D_{α} which have the property that their images Y := π(D_{α}(V)) ⊂ W are affinoid subdomains of W and moreover D_{α}(V) is finite flat over Y. The rigid analytic spaces D_{α} are curves; that is, they are equidimensional of dimension 1.

7.3. The relationships among the curves D_{α}.

Consider the set P of all pairs S = (X, N) such that:

a. X is an affinoid in W.

b. N is a T_{X}-submodule of M_{X}^{†} (here T_{X} = H ⊗ A(X)),

c. N is locally free of finite rank over A(X) and there exists a T_{X}- module projector from M_{X}^{†} onto N which comes from a continuous projector from M_{X}(v) on N, where v > 0 is sufficiently close to 0 so that all the elements on N are the images of elements in M_{X}(v). (Note that the module N, being of finite rank, is contained in the image of the Banach space M_{V}^{†} (v) for v > 0 and sufficiently small, and the results of section 4.2 apply.)

d. The operator T (p) is invertible on N. (In this way the elements of N correspond to families of forms of finite slope).

One source of such pairs is the following: Let notation be as above. Suppose V ∈ C_{α} and Q_{V} (T) ∈ A(π_{α}(V))[T] the corresponding polynomial. Let N_{V} be the kernel of Q_{V}^{∗} (αU_{p}) in M_{X}^{†}. Then, using Theorem A4.5 of [C-BMF], we see that S_{V} := (π_{α}(V), N_{V}) ∈ P. Even though M_{X}^{†} is not naturally a Banach-module over A(X), writing M_{X}^{†} = ∪_{v}M_{X}^{†}(v) we may view M_{X}^{†} as being given by a system of Banach modules, as in the terminology of section 4.3, and using Proposition 4.3.3 an operator Ũ in the ideal U has a characteristic series det(1 − T Ũ|M_{X}^{†}) in A(X){{T }}. It follows from Corollary 4.2.3 that if α ∈ H and Ũ = αU_{p},

det(1 − T Ũ|M_{X}^{†}) = det(1 − T Ũ|N) · det(1 − T Ũ|M_{X}^{†}/N).

Suppose S = (X, N) is a pair in P. Let T(S) denote the ring generated over A(X) by the image of T_{X} in End_{X}(N). It is the algebra of rigid functions on a one dimensional affinoid D(S). If V ∈ C_{α}, D(S_{V}) = D(V) and T(S_{V}) = T(V). We also see that if S = (X, N) and S′ = (X′, N′) are elements of P such that X′ ⊂ X and N′ ⊂ N_{X′} := N ⊗_{A(X)} A(X′) then we have an induced natural ring homomorphism from the image of T(S) to T(S′) which in turn induces a natural map

D(S′) → D(S),

which we will refer to below as the “functorial map”. We also set

T_{p,OF} = lim_{↼} T(S)

(the “OF” stands for overconvergent of finite slope). Using Proposition 3.4.2 and Theorem 2.4.2 we obtain compatible maps of T_{p} into T(S) for S ∈ P and hence a map from T_{p} into T_{p,OF}.

Construction of a natural mapping from D(S) into D_{α}. Suppose α ∈ H, S = (X, N) ∈ P and α is invertible on N. Let Ũ = αU.

By Corollary A2.6.2 of [C-BMF] and Theorem 4.2.2 (since Ũ is stabilizes N and A(X) is semi-simple) we may factor the Fredholm characteristic series P_{α}(T) over X,

P_{α}(T) = Q_{S}(T)H_{S}(T)

where Q_{S}(T) = det(1 − T Ũ|_{N}) is a polynomial over A(X) whose degree is the rank of N and whose leading coefficient is a unit in A(X), and where H_{S}(T) ∈ A(X){{T }}, and in contrast to members of C_{α}, here H_{S}(T) is not necessarily relatively prime to Q_{S}(T) .

Define

Z_{S} := Z_{α,S} := Max(A(X)[T]/Q(T))

which is an affinoid and note that we have natural mappings

D(S) → Z_{S} → Z := Z_{α}.

Now because C := C_{α} is an admissible cover of Z, there exists a finite collection Z_{1}, …, Z_{n} of members of C which covers the image of Z_{S}, i.e., such that the collection {Z_{S} ×_{Z} Z_{i}}_{i} is an admissible open cover of Z_{S}. Let X_{i} = π_{α}(Z_{i}) and X′_{i} = X_{i} ∩ X. Corresponding to Z_{i} there is a Riesz theory idempotent e_{i} which acts on M_{X_{i}}^{†}. The idempotent e_{i} also may be thought of as a rigid analytic function on Z_{X_{i}} which is 1 on Z_{i} and 0 on its complement. As e_{i} ∈ T_{X_{i}} (actually in the completion of the subring generated by Ũ) e_{i} acts on N_{X_{i}}. Also, over X_{i} ∩ X_{j}, e_{i} and e_{j} commute. Let f_{i} = 1 − e_{i} and E_{i} = e_{i}N_{X′_{i}} and F_{i} = f_{i}N_{X′_{i}}. Then

N_{X′_{i}} = E_{i} ⊕ F_{i}.

Consider the pairs S_{i} := (X′_{i}, E_{i}) and Σ_{i} := (X′_{i}, F_{i}) in P. Then, we have a factorization

Q(T) = det(1 − T Ũ|_{E_{i}}) det(1 − T Ũ|_{F_{i}}),

which, given our notational conventions, may be written

Q(T) = Q_{S_{i}}(T) · Q_{Σ_{i}}(T),

the polynomial factors Q_{S_{i}} and Q_{Σ_{i}} being relatively prime because of the Riesz theory. Moreover, we have the affinoids

Z_{S_{i}} := Z_{α,S_{i}} := Max(A(X′_{i})[T]/Q_{S_{i}}(T)),

and

Z_{Σ_{i}} := Z_{α,Σ_{i}} := Max(A(X′_{i})[T]/Q_{Σ_{i}}(T)).

It is easy to see that S_{i} := (X′_{i}, E_{i}) and

S_{i j} := (X_{i j} = X′_{i} ∩ X′_{j} , E_{i j} := e_{i}e_{j}N_{X′_{i}∩X′_{j}})

all lie in P. We will have later use for the natural mapping

D(S_{i}) → D(Z_{i}).

This map is obtained as follows. First, consider the pair Z_{i} := (X_{i}, e_{i}N_{X_{i}}) ∈ P. Since X′_{i} ⊂ X_{i} and e_{i}N_{X′_{i}} ⊂ e_{i}N_{X_{i}} ⊗_{A(X_{i})} A(X′_{i}), we obtain a functorial map (see above) D(S_{i}) → D(Z_{i}). Next note that D(Z_{i}) = D(Z_{i}).

Lemma 7.3.2. The natural mapping

Z_{S_{i}} → Z_{S} ×_{Z} Z_{i}

is an isomorphism.

Proof.

Z_{S} ×_{Z} Z_{i} = (Z_{S})_{X_{i}} ×_{Z} Z_{i}

and

(Z_{S})_{X_{i}} = Z_{S_{i}} ∐ Z_{Σ_{i}}

because Q_{S_{i}}(T) and Q_{Σ_{i}}(T) are relatively prime. As Z_{Σ_{i}} ×_{Z} Z_{i} = ∅ (again using the Riesz theory)

Z_{S} ×_{Z} Z_{i} = Z_{S_{i}} ×_{Z} Z_{i}.

The lemma then follows since the natural map from Z_{S_{i}} to Z is contained in Z_{i}.

Lemma 7.3.3. The natural map

D(S_{i}) → D(S) ×_{Z} Z_{i}

is an isomorphism.

Proof. As in Lemma 7.3.2,

D(S) ×_{Z} Z_{i} = D(S_{i}) ×_{Z} Z_{i}

and because the map from D(S_{i}) to Z factors through Z_{S_{i}} we see that this equals D(S_{i}).

Lemma 7.3.4. The natural map D(S_{ij}) → D(S_{i}) ×_{D(S)} D(S_{j}) is an isomorphism. Moreover, the restriction to D(S_{ij}) of either the map from D(S_{i}) to D ×_{Z} Z_{i} or D(S_{j}) to D ×_{Z} Z_{j} is the same map to D ×_{Z} Z_{ij} (where Z_{ij} = Z_{i} ×_{Z} Z_{j}).

Proof. This comes from

\begin{aligned}D(S_{i}) ×_{D(S)} D(S_{j}) & = (D(S) ×_{Z} Z_{i}) ×_{D(S)} (D(S) ×_{Z} Z_{j}) \\ & = D(S) ×_{Z} Z_{ij} \end{aligned} .

We saw above that Z_{ij} ∈ C, but we can be more precise. Over X_{ij} := X_{i} ∩ X_{j}, we have the decomposition

M = e_{i}e_{j}M ⊕ e_{i}f_{j}M ⊕ e_{j}f_{i}M ⊕ f_{i}f_{j}M.

This corresponds to a factorization, over X_{i} ∩ X_{j},

P_{α}(T) = A(T)B(T)C(T)D(T)

where D(T) ∈ A(X_{ij}){{T }} and A(T), B(T), C(T) are polynomials in A(X_{ij})[T], and

\begin{aligned}Q_{Z_{i}}(T) & = A(T)B(T) \\ Q_{Z_{j}}(T) & = A(T)C(T). \end{aligned}

We see that Q_{Z_{ij}}(T) = A(T) and if Z_{ij} ≠ ∅, the image of Z_{ij} in B equals X_{ij} and the projector corresponding to Z_{ij} is e_{i}e_{j}. By Lemma 7.3.3, we see that D(S_{ij}) = D(S) ×_{D} D(Z_{ij}) and the map from D(S_{ij}) factors through the projection to D(Z_{ij}). From this the lemma follows.

The Lemmas 7.3.3 and 7.3.4 allow us to patch together mappings.

Proposition 7.3.5. Let S ∈ P. There is a unique map from D(S) to D_{α} characterized by the following property: Suppose V ∈ C_{α}, with image Y = π_{α}(V). Let e be the corresponding idempotent acting on M_{Y}^{†} and S′ = (Y ∩ X, eN_{Y}) ∈ P. Then the following diagram commutes:

D(S) → D_{α}
↑  ↑
D(S′) → D(V)

where the left and the bottom arrows are the natural ones coming from the inclusions S′(N) ⊆ N(S)_{Y ∩X} and S′(N) ⊆ N(V)_{Y ∩X}.

Proof. It follows from Lemma 7.3.3 that the D(S_{i}) cover D(S) and from Lemma 7.3.4 that all the morphisms D(S_{i}) → D(Z_{i}) (constructed above) glue together to give a morphism from D(S) to D_{α}. To check that one gets the same map from a different covering of the image of Z_{S} by a collection of elements of C_{α} it is enough to check this when the second covering is a refinement of the first in which case it follows easily from Lemmas 7.3.2-4. We may deduce from the above that we have a natural maps ρ_{β,α}: D_{β} → D_{α} for all β ∈ αH as follows: Suppose V_{1}, V_{2} ∈ C_{β} then, as β is invertible on N(V_{i}), by the above we have morphisms D(V_{1}) → D_{α} and D(V_{2}) → D_{α}. We must show that they agree on D(V_{1}) ×_{D_{β}} D(V_{2}) which is D(V_{1} ×_{Z_{β}} V_{2}). Hence, it is enough to prove this when V_{2} ⊆ V_{1} (i.e. V_{1} ×_{Z_{β}} V_{2} = V_{2}). Suppose Z_{1}, …, Z_{n} cover the image of Z_{S(V_{1})}. Then as N(V_{2}) is a submodule of N(V_{1})_{π_{α}(V_{2})}, Z_{1}, …, Z_{n} also cover the image of Z_{S_{V_{2}}} . We can therefore use Z_{1}, …, Z_{n} to make our maps. Let in the above notation, with S = S_{V_{j}}, S_{i}(V_{j}) = S_{i}, X_{i}(V_{j}) = X′_{i}. We must check that the following diagram commutes:

D(S_{i}(V_{1})) → D(Z_{i})
↑  ∥
D(S_{i}(V_{2})) → D(Z_{i}),

but this follows from

e_{i}(N(V_{2})_{X_{i}(V_{2})}) ⊆ e_{i}((N(V_{1})_{X(V_{2})})_{X_{i}(V_{2})}) = (e_{i}(N(V_{1})_{X_{i}(V_{1})})_{X_{i}(V_{2})}.

Lemma 7.3.6. If β ∈ αT_{p,OF} and γ ∈ βT_{p,OF}

ρ_{β,α} ∘ ρ_{γ,β} = ρ_{γ,α}.

Proof. Let U ∈ C_{γ}, V ∈ C_{β}, W ∈ C_{α}, X = π_{γ}(U), Y = π_{β}(V) and Z = π_{α}(W). Let e_{V} and e_{U} be the idempotents corresponding to V and W (We suppress mentioning in the notation where they are defined here and in the following). Then if S′ = (X ∩ Y, e_{V} N(U)) and S′′ = (Y ∩ Z, e_{W} N(V)), using Proposition 7.3.5, we see that the following diagram commutes:

D(U) ⟶^{ρ_{γ,β}} D_{β}  ⟶^{ρ_{β,α}} D_{α}
↑  ↑
D(S′) ⟶ D(V)  ↑
↑
D(S′′) ⟶ D(W).

Now if S′′′ = (X ∩ Y ∩ Z, e_{W} e_{V} N(U)), it is easy to see that

D(S′) → D(V)
↑  ↑
D(S′′′) → D(S′′)

commutes, where the right and bottom arrows are the natural ones. Now using the previous corollary and the fact that the D(S′′′)’s constructed in this way cover D(U) we see that the composition of ρ_{β,α} and ρ_{γ,β} restricted to D(U) is ρ_{γ,α}. This completes the proof of the lemma.

Since ρ_{α,α} is the identity,

Corollary 7.3.7. If β and α generate the same ideal in T_{p,OF}, then ρ_{β,α} is an isomorphism.

Let D = D_{1}. The above gives us maps from D to Z_{α} for all α ∈ T_{p,OF}^{∗}. The curve D will be shown to be isomorphic to the reduced eigencurve.

If β ∈ H such that ι · β is invertible, then the image of β in T_{p,OF} is invertible because the homomorphism from H into T_{p,OF} is the composition of the homomorphisms

H → R_{p} → T_{p} → T_{p,OF}

(see sections 3.3, 5.2 and 7.2).

Remark. If α and β are non-associate elements of T_{OF} then D_{α} and D_{β} may be very different. For example, if α = 0, D_{α} will be empty although as we’ll see D := D_{1} is quite large. If α is an idempotent, D_{α} is an admissibly closed-open subspace of D.

7.4. D is reduced.

In this section, we will prove D is nested (cf. 1.1), reduced, and each irreducible component of D maps surjectively and generic isomorphically onto a Fredholm hypersurface (see Prop. 7.4.5 below). By construction we have a map w: D → W. By the results of §7.3, D ≅ D_{α} for all α ∈ T_{p,OF}^{∗}. This means that we have finite maps z_{α}: D → Z_{α}, and again by construction the following diagrams commute:

D ⟶^{z_{α}} Z_{α}
↘ ↙
W

where the mappings to weight space are the natural ones; i.e., the mapping D → W is w and the mapping Z_{α} → W is π_{α}. Let σ_{α}: Z_{α} → A^{1} be the natural projection to the affine line. If α ∈ T_{p,OF}^{∗}, the map s: P ↦ −v(σ_{α}(z_{α}(P)) is independent of α and we call it ν. Indeed, if we think of a closed point on D as a one dimensional space of eigenforms, as we can by Theorem 7.1.1, s is just the slope of these forms. We know that every point on the nilreduction D^{red} of D lies on an irreducible component part by Proposition 1.2.5. Let Z = Z_{1}.

Lemma 7.4.1. Suppose α ∈ T_{p,OF}^{∗}. If Y is an irreducible component of D^{red} then the map Y → Z_{α}^{red} factors through a surjective map onto an irreducible component part of Z^{red}.

Proof. Let V ∈ C_{α}. Let D_{1}, …, D_{n} be the irreducible components in D(V)^{red} in Y_{V} = Y ∩ D(V). These correspond, by Corollary 1.2.1, to minimal prime ideals P_{1}, …, P_{n} of T_{V}. Since the map from D(V) to V is surjective on closed points, it follows that the pullback Q_{i} of P_{i} to V is a minimal prime ideal of A(V). and let Z_{i} = MaxA(V)/Q_{i} be the irreducible component of of V^{red} corresponding to Q_{i}. Since D(V) to V is finite, it follows from the going-up theorem, Theorem 5.11 of [A-M], that the map from D_{i} to Z_{i} is surjective. Let Z_{V} be the component part of V^{red} which is the union of the irreducible components Z_{i} for i = 1 ⋯ n. The natural map from Y_{V} to Z_{V} is finite and surjective. The Z_{V} ’s for V ∈ C_{α} glue together into a component part Z of Z_{α}^{red} and the maps glue together into a finite surjective map from Y onto Z. Since Z_{α}^{red} is Fredholm hypersurface it follows that Z is as well. Since Y is irreducible so is Z.

Write Λ = ∏_{j=1}^{p−1} Λ^{(j)} where each of the p−1 factor rings Λ^{(j)} are isomorphic to a power series ring in one variable over Z_{p}, and where the product decomposition of rings corresponds to the decomposition of the rigid space W as a (disjoint) union of its p−1 irreducible components, W = ∐_{j=1}^{p−1} W^{(j)}, where Λ^{(j)} is the algebra over Q_{p} of rigid analytic functions bounded by 1 on the rigid analytic space W^{(j)}.

Corollary 7.4.2. Let w : D → W be the projection mapping to weight space. If D is an irreducible component of D^{red} then w maps D into a unique irreducible component, W^{(j_{D})}, of W (where 1 ≤ j_{D} ≤ p − 1) and the mapping

w : D → W^{(j_{D})}

is almost surjective on C_{p}-valued points in the sense that the complement of D(C_{p}) in W^{(j_{D})}(C_{p}) is finite.

Proof. Let D be an irreducible component of D^{red}. Then D projects onto an irreducible component of Z_{α}^{red} by Lemma 7.4.1. To prove Corollary 7.4.2 it suffices, then, to prove for irreducible components of Z_{α}^{red}. By Theorem 4.3.1, Z_{α} is cut out by a Fredholm series PŨ(T) over Λ. Consider the corresponding factorization

PŨ(T) = ∏_{j=1}^{p−1} PŨ(T)^{(j)},

where PŨ(T)^{(j)} ∈ Λ^{(j)}{{T }} for j = 1 ⋯ p−1. Since Λ^{(j)} is isomorphic to a power series ring in one variable over Z_{p}, Theorem 1.3.11 applies, allowing us to identify the irreducible components of Z_{α}^{red} (and also of Z_{α}) with factors of the Fredholm series PŨ(T)^{(j)} for j = 1 ⋯ p − 1. Corollary 1.3.13 then applies, which concludes the proof of our corollary.

If W is an affinoid open in W and t ∈ Q, the conditions

w(P) ∈ W
ν(P) ≤ t

determine an affinoid open D(W, t) in D.

This follows because the conditions

π_{α}(Q) ∈ W
v(σ_{α}(Q)) ≤ t

determine an affinoid open Z_{α}(W, t) in Z_{α} for any α of which D(W, t) is the pullback so is an affinoid by the rigid Chevalley lemma Proposition 9.4.4/1 of [BGR].

Lemma 7.4.3. For W an affinoid open in W and t ∈ Q. There exists an α ∈ T_{p,OF}^{∗} such that Z_{α}(W, t) is reduced.

Proof. We know D(W, t)^{red} has finitely many irreducible components by Corollary 1.2.1/1. Using Proposition 1.2.5 we see that these are contained in a finite number of irreducible components of D^{red}, D_{1}, …, D_{n}. These components D_{ν} each map almost surjectively (in the sense described in Corollary 7.4.2) to some component W^{(j_{ν})} of W. In particular, for each ν, there is a point δ_{ν} on D_{ν} of some arithmetic weight k_{ν} and slope s_{ν} with 0 ≤ s_{ν} < (k_{ν} − 1)/2. Now choose α ∈ T_{p,OF}^{∗} as in Chapter 6 so as to distinguish all the classical forms of weight k_{ν} and slope s_{ν} for all ν = 1, ⋯, n.

Claim. Z_{α}(W, t) is reduced.

Let Z_{ν} be an irreducible component part of Z_{α} containing the image ζ_{ν} of δ_{ν} under z_{α} : D_{α} → Z_{α}. Then by the arguments in Chapter 6 (Props. 6.3.1- 6.3.3) Z_{ν} is reduced (There is only one modular form corresponding to ζ_{ν}. This implies Z_{ν} is generically reduced, but since it is an irreducible Fredholm hypersurface it must be reduced.) Now the claim and our lemma follow because Z_{α}(W, t) is the union over ν of Z_{ν} ∩ Z_{α}(W, t).

Lemma 7.4.4. D(W, t) is reduced.

Proof. If we knew Z_{α}(W, t) was an element of C_{α} we could apply Proposition 7.1.3 to conclude that D(W, t) is reduced. But it may not be so we must work harder. We go to the proof of Proposition A5.8 of [BFM]. We must generalize it slightly and replace B_{K}^{1} with W. (We also use the proof of Lemma A5.9 in loc. cit. which works even when d = 0). Translating what was proved there, we know that for u ∈ Q, u > t, and sufficiently close to t, there exists a finite covering {U_{0}, U_{1}, …, U_{d}} of W by affinoids in W such that the the affinoid V_{i} in Z_{α}, whose points in Z_{α} consist of all Q such that

π_{α}(Q) ∈ U_{i} and v(σ_{α}(Q)) ≤ u

lie in C_{α} and is finite over U_{i}. Moreover, S_{n} := ⋃_{i≥n} U_{i} contains the affinoid W_{n} in W consisting of all P ∈ W such that the fiber of π_{α} restricted to Z_{α}(W, t) at P has degree at least n. We can also assume such that U_{n} is the rigid Zariski closure of W_{n} ∩ U_{n}. Since U_{n} is the Zariski closure of W_{n} ∩ U_{n} and Z_{α}(W, t) is reduced, it follows that the Zariski closure of (V_{n})_{W_{n}∩U_{n}} is V_{n}. But (V_{n})_{W_{n}∩U_{n}} = (Z_{α}(W, t))_{U_{n}} which is reduced by Corollary 7.3.3/10 of [BGR] since Z_{α}(W, t) is reduced. Thus V_{n} is generically reduced. Now we can apply Proposition 7.1.3 to conclude that D(V_{n}) is reduced. Since the D(V_{n}) cover D(W, t) our lemma follows.

Proposition 7.4.5. The rigid analytic curve D is reduced, nested, and and each irreducible component of D maps surjectively and generic isomorphically onto a Fredholm hypersurface.

Proof. The D(W, t)’s form an admissible cover of D, so D is reduced by Lemma 7.4.4. Now suppose D is an irreducible component of D. We may suppose D is the D_{1} mentioned above. We claim it is generically isomorphic to Z_{1}. We know D(W, t) is generically isomorphic to Z_{α}(W, t) (here α is as in the previous lemmas). It follows that D_{1} ∩D(W, t) is generically isomorphic to Z_{1} ∩ Z_{α}(W, t) since these affinoids are open and Zariski dense in their respective spaces. Since Z_{1} is nested, so is D. Our proposition follows.

7.5. Equality of D and C^{red}.

Set C = C_{p}. Recall C^{red} denotes the nilreduction of C. We maintain the notation of the previous section so in particular, we have morphisms z_{α}: D → Z_{α} (which is contained in W × A^{1} − {0}) for all α ∈ T_{p,OF}^{∗}. We also have a morphism

δ: D → X_{p} × (A^{1} − {0}),

given by c ↦ (ρ(c), 1/u(c)) where ρ(c) is the pseudo-representation and u(c) is the U_{p}-eigenvalue attached to c, or more precisely: The morphism from D to X_{p} arises from the map T_{p} → T_{p,OF} described in section 7.3 and the pseudo-representation described in section 5.2. The map from D to A^{1} − {0} is the composition of the morphism from D to the spectral curve Z =: Z_{1} and the natural projection of σ_{1}: Z → A^{1} − {0}. We constructed the eigencurve C, in section 6.1, as follows: For every α ∈ H′ such that ι · α is a unit in R_{p} we have a map r_{α} from Y := X_{p} × (A^{1} − {0}) to W × (A^{1} − {0}) such that

r_{α}(x, t) = (w(x), 1/t (ι · α(x))).

Then C is the rigid analytic subspace of Y which is the intersection of the subspaces

r_{α}^{−1}(Z_{α}) := Y ×_{W×(A^{1}−{0})} Z_{α},

where α ranges over the elements in H′ as above. The key point is that

r_{α} ∘ δ = z_{α}.

Thus δ factors through C. It follows that we have a natural map ω: D → C such that the diagrams

\begin{aligned}D & ⟶^{ω} & C \\ & ↘ ↙ \\ & Z_{α} \end{aligned}

commute for all α ∈ T_{OC}^{∗}.

Theorem 7.5.1. The map ω factors through a map to C^{red} and induces an isomorphism (which we continue to denote by the same letter)

ω : D ≅ C^{red}.

Proof. That ω factors through C^{red} follows from the fact that D is reduced. That ω is a bijection on closed points follows from Theorems 6.2.1 and 7.1.1. To prove it is an isomorphism of rigid spaces, we will first prove that it is a generic isomorphism and then prove it is a local isomorphism. For the first assertion we need the following lemma. For the second assertion we will show below that the morphism from D to X_{p} × A^{1} is locally a closed immersion.

Lemma 7.5.2. Suppose E_{0} and E_{1} are reduced rigid spaces (over K a complete subfield of C_{p}), and h_{0} : E_{0} → E_{1} and h_{1} : E_{1} → E_{0} are rigid analytic mappings which induce mappings on K̄-valued points which are two-sided inverses of one another; i.e., the compositions of h_{0} : E_{0}(K̄) → E_{1}(K̄) and h_{1} : E_{1}(K̄) → E_{0}(K̄) in either order yield the identity mapping on the relevant set of K̄-valued points. Then h_{0} : E_{0} ≅ E_{1} and h_{1} : E_{1} ≅ E_{0} are rigid analytic isomorphisms between E_{0} and E_{1} and are two-sided inverses of one another as rigid analytic mappings.

Proof. It will suffice to show that if E is a reduced rigid space over K, and h: E → E is a morphism of rigid spaces over K such that h is the identity on \overline{K}-valued points, then h is the identity. Let S be an admissible cover of E by affinoid subdomains. Let V ∈ S. Then there exists an admissible cover U_{i} of V by affinoid opens and V_{i} ∈ S such that h|_{U_{i}} factors through the inclusion V_{i} → E. Now since, h(V (\overline{K})) = V (\overline{K}), h(U_{i}(\overline{K})) ⊆ (V ∩ V_{i})(\overline{K}) and since V is an affinoid subdomain of E, V ∩ V_{i} is an affine subdomain of V_{i}. It follows from Proposition 7.2.2/1 of [BGR] that h|_{U_{i}} actually factors through V ∩ V_{i} and so h|_{V} factors through the inclusion V → E. Thus we may suppose E is an affinoid. Let A = A(E). We need to show h^{∗}: A → A is the identity. Suppose a ∈ A. Let m be a maximal ideal of A and (extending scalars, if necessary) suppose c ∈ \overline{K} such that a − c ∈ m. Then h^{∗}a − c ∈ m and so h^{∗}a − a ∈ m for all maximal ideals m. Since A is reduced (and semi-simple), it follows that h^{∗}a = a.

Now we apply this to our eigencurve. Let α ∈ H whose image in T_{p,OF} is invertible and suppose V is a reduced component part of Z_{α}. Let D_{V} be the fiber above V in D and C_{V} the fiber above V in C^{red}. (Thus, by definition, C_{V} is reduced.) Then we have a commutative diagram

D_{V} ⟶^{ω} C_{V}
↘ ↙
V

and we know ω is one:one on closed points and the restriction of z_{α} to D_{V} is generically an isomorphism onto V using the fact that D is reduced and Lemma 7.4.1. Let S be the set of points of V where z_{α} is not an isomorphism, V′ = V − S, D′ = D_{V} − z_{α}^{−1}(S). C′ = C_{V} − λ_{α}^{−1}(S), ω′ the restriction of ω to D′ etc. Let μ: V′ → D′ be the inverse of z_{α}: D′ → V′. Then

(μ ∘ λ′) ∘ ω′ = id.

Applying Lemma 7.5.2 with E_{0} = D′ , E_{1} = C′ and h_{0} = ω′, h_{1} = μ ∘ λ′ that ω|_{D_{V}}: D_{V} → C_{V} is generically an isomorphism. Since every component part of D composed of finitely many irreducible components is generically isomorphic to a reduced component part of Z_{α} for some α ∈ T_{p,OF}^{∗} we may conclude that ω is a generic isomorphism. Now we will show ω is a local isomorphism. Let V be a open in C := C_{1} and Y its image in W. Let X be an affinoid open in X_{p} and U a affinoid in A^{1} such that the image of X in W is Y and such that the image of D(V) in X_{p} × A^{1} is contained in X × U (more precisely, such that the morphism D(V) → X_{p} × A^{1} factors through X × U for after we get this second property, we can get the first by taking fiber products). Since A(D(V)) is generated over A(Y) by the image of the Hecke operators which can be identified as functions on X_{p} × A^{1} and A(Y) maps into A(X × U) the map from D(V) to X × U is a closed immersion.

Claim. For any point on D there exists a V ∈ C such that P ∈ D(V) and an X and U as above such that the image of δ(D(V)) equals δ(D)∩(X ×U).

Suppose for the moment that we have this. Let E be the intersection of C with X ×U. It is a subaffinoid and its points are the same as those of D(V). It follows that D(V) maps via a closed immersion onto the nilreduction of E. Since it is a bijection on closed points and D(V) is reduced we conclude that it is an isomorphism. Thus the claim implies that ω is a local isomorphism. To prove the claim, suppose ν(P) = t. Then using the analysis of Lemma A5.9 of [BGR], we see there exists an affinoid open neighborhood W of w(P) in W such that the affinoid Z(W, t) ∈ C (this is the affinoid called Z_{1}(W, t) in the last section). The choices V = Z(W, t), X = W and U = B[0, p^{t}] will fulfill the requirements of the claim.

We now prove the theorem. Suppose X is an affinoid open in X_{p} whose image in W is an affinoid open W. The map from D(W, t) to C^{red} factors through C(X , t) := C^{red} ∩ (X × B[0, p^{t}]). Moreover, both D(W, t) and C(X , t) are affinoids and the map f: D(W, t) → C(X , t) is a bijection on closed points. It follows from the above that f is an isomorphism after the removal of finitely many points S in D(W, t) and is a local isomorphism. Let A = D(W, t) − S, B = C(X , t) − ω(S) and let ψ: B → A be the inverse of ω|_{A}. Now choose affinoid neighborhoods U of S in D(W, t) and V of ω(S) in C(X , t) such that ω restricts to an isomorphism U → V. We can do this because S is finite and ω is a injection on closed points. Let ϕ: V → U be the inverse of ω|_{U}. It is clear that ψ and ϕ glue together into a morphism ρ(X , t): C(X , t) → D(W, t) inverse to ω|_{D(W,t)}. As X and t vary the affinoids C(X , t) for an admissible open covering of C^{red} and the ρ(X , t) glue together into an inverse of ω.

7.6. Consequences of the relationship between D and C.

The first consequence of section 7.5 is that we now know that the eigencurve is a curve since D is by construction. Moreover, Theorems A and B of Chapter 1, section 1.5, follow from the results of sections 7.4 and 7.5. We also established

Corollary 7.6.1. Every irreducible component of the reduced eigencurve C contains a classical point.

Corollary 7.6.2. Each irreducible component of C^{red} is generically isomorphic to a Fredholm hypersurface over W.

For the record, let us extract a slightly more precise statement that is useful in understanding the geometry of the rigid-analytic curve C. In the special case of arithmetic weight κ = (χ, k) where the slope σ of c is less than k − 1 and different from (k − 2)/2 it follows from [C-BMF] that the C_{p}-vector space M_{k}(C_{p}; χ)_{σ} consists entirely of classical modular forms, and therefore it follows from standard facts that H acts semi-simply (with multiplicity one) on M = M_{κ,σ}. In particular, μ(c) = 1. We therefore get:

Corollary 7.6.3. Let c be a C_{p}-valued point of the eigencurve C = C_{p} of weight-character w = χ⊗η_{k} ∈ W and of slope σ < k −1, and σ ≠ (k −2)/2. Then c is a smooth point of the curve C^{red} and the projection C^{red} → W is locally an isomorphism in a neighborhood of c.

One should note that if σ ≥ k − 1 the point c may indeed be ramified in the projection mapping of C to weight space. We do not have to go far to see this phenomenon. Say that a point c ∈ C of weight w = χ ⊗ η_{k} is of critical slope if the slope of c is equal to k − 1. There are examples (cf. [C-OC]) of points c of critical slope ramified under the projection mapping of C to weight space.

From the previous discussion, we deduce:

Corollary 7.6.4. If β ∈ T_{p,OF}^{∗}, and d_{β} ∈ Z_{β}, then the map D → Z_{β} is finite flat of degree at most μ(d_{β}) in a neighborhood of d_{β}. Moreover, if any of the points d ∈ D above d_{β} satisfy the conditions of the previous corollary, they all do and this degree equals μ(d_{β}).

We may now prove Theorem G stated in section 1.5:

Proposition 7.6.5. If {f_{n}} is a sequence of normalized eigenforms of tame level 1 with Fourier coefficients in C_{p} such that f_{n} has weight κ_{n} ∈ W(C_{p}), v(a_{p}(f_{n})) is bounded independently of n and the sequence {f_{n}(q) ∈ C_{p}[[q]](1/q)} converges coefficientwise to a series f(q). Then the sequence κ_{n} converges in (Z/(p − 1)Z)×Z_{p} to some weight κ ∈ W(C_{p}) and f(q) is the q-expansion of an overconvergent modular eigenform f of tame level 1, weight κ, and finite slope. Let f be a Katz modular eigenfunction (see section 2.2) of tame level 1, of finite slope, with (normalized) Fourier expansion

f̃ = q + a_{2}q^{2} + a_{3}q^{3} + ⋯

with coefficients a_{j} ∈ C_{p}. Suppose that f is of accessible weight-character (cf. section 1.4) (χ, s) with s ∈ Z_{p}. Then f is overconvergent if and only if f is the limit in the q-expansion topology of a sequence of classical, normalized, cuspidal-overconvergent (cf. section 3.6) modular eigenforms of tame level 1.

Proof. That the κ_{n} converge to a κ ∈ W(C_{p}) follows from Theorem 2 of [S 3]. We will give two proofs of the rest of the first part of this proposition.

For the first proof note that because the slopes are bounded and the weights converge, eventually all the f_{n} correspond to points P_{n} on an affinoid of the form D(V) for some V ∈ C_{1}. Moreover in the notation of Theorem 7.1.1, η_{P_{n}}(T )_{n} is the n-coefficient of f_{n}(q). Since the images of Hecke operators generate the ring of functions on D(V) over the weight space, it follows that the points P_{n} converge to a point P on D(V) such that η_{P} (T_{n}) is the n-coefficient of f(q). By Theorem 7.1.1, P correspond to an eigenform of weight κ with q-expansion f(q) and finite slope.

For the second proof and the proof of the rest of this theorem, let us first consider the various natural topologies on sets of eigenforms. For every prime number ℓ ∈ P we have a natural continuous mapping

T_{ℓ} : (X_{p} × (A^{1} \ {0}))(C_{p}) → C_{p}

by putting T_{ℓ}(x, 1/u) := T_{ℓ}(x) if ℓ ≠ p, where we view the Hecke operator T_{ℓ} as rigid analytic function on X_{p}, and putting T_{p}(x, 1/u) := u. On defines the q-expansion topology on the set X_{p} × (A^{1} \ {0})(C_{p}) by means of these functions. Explicitly, let P be the set of all prime numbers, and C_{p}^{P} the product of C_{p} with itself countably many times, indexed by P and given the natural product topology. Consider the mapping

T : X_{p} × (A^{1} \ {0})(C_{p}) → C_{p}^{P},

where for any ℓ ∈ P, the ℓ-th entry of T (c) is T_{ℓ}(c). By definition, the q-expansion topology on the set X_{p} × (A^{1} \ {0})(C_{p}) is the topology this set inherits via the mapping T to the topological space C_{p}^{P}.

Lemma 7.6.6. The natural topology that the set X_{p} × (A^{1} \ {0})(C_{p}) inherits as the C_{p}-valued points of the rigid analytic space X_{p} × (A^{1} \ {0}) (which we will call the rigid analytic topology) is the same as its qexpansion topology.

Proof. This follows from the fact that if H′ is the Λ-algebra generated by the T_{ℓ}’s (for ℓ ≠ p) (cf. Chapter 2), the natural ring homomorphism of H′ to R_{p} is topologically dense. This fact can be regarded as given by a ”standard argument” at least for those components of R_{p} which actually correspond to representations rather than pseudo-representations (for if you take such a component , e.g., a completion of R_{p} with respect to a maximal ideal m (call it R_{m}) and if you consider R′_{m} ⊂ R_{m} which is the closure of the image of H′ in R_{m}, one can see that (by continuity and Cebotarev) the image of the entire Galois group under the trace mapping is contained in R′_{m}. We may assume this is a local ring with the same residue field as R_{m} and then by standard Schur-type arguments you see that the universal representation over R_{m} descends to a representation r′ over R′_{m}. Universality of R_{m} allows us to conclude that R′_{m} = R_{m} (for if not, there would be two distinct ringhomomorphisms R_{m} → R_{m} inducing the same representation: the identity and the projection to R′).

Now this argument works word-for-word (only is easier) for pseudorepresentations, because we don’t need ”standard Schur-type arguments.” The fact that the trace descends immediately implies (from the definition of pseudo-representation) that the entire pseudo-representation descends.

The first part of our proposition is a consequence of this lemma. For Proposition 6.2.1 allows us to identify our sequence {f_{n}} with a sequence of C_{p}-valued points of (the nilreduction of) the eigencurve. But since the reduced eigencurve is a Zariski-closed rigid analytic subspace, its set of C_{p}- valued points is a closed subset of X_{p} × (A^{1} \ {0})(C_{p}) in the rigid analytic topology. By the lemma, this set is also closed in the q-expansion topology.

As for the second part, let K be the set of all Katz modular eigenfunctions of tame level 1, of finite slope, of arbitrary weight, but with Fourier expansion as hypothesized in our proposition. We view K as a topological space, given the q-expansion topology. Let c : K → X_{p} × A^{1}(C_{p}) be the mapping which assigns to a Katz modular eigenform f ∈ K the point c(f) ∈ X_{p} × A^{1}(C_{p}) whose coordinate in X_{p}(C_{p}) corresponds to the associated pseudo-representation of f, as given by the Gouvêa-Hida Theorem (see section 5.2) and whose coordinate in A^{1}(C_{p}) is the reciprocal of its U_{p}-eigenvalue. The mapping c is a homeomorphism of the topological space K onto its image in X_{p} × (A^{1} \ {0})(C_{p}). This may be seen from the above lemma by noting that the T_{ℓ}-eigenvalues (for all ℓ ≠ p) together with the U_{p}-eigenvalue of f determine the q-expansion of f (see the discussion about this in section 3.1 ).

Now take a Katz modular eigenform f ∈ K satisfying all the hypotheses of our proposition, and in particular which is of accessible weight-character. Suppose that f is overconvergent. The point c(f) is then, by Theorem 6.2.1, a C_{p}-valued point of the reduced eigencurve, and since f is of finite slope and accessible weight, using the argument in the proof of Corollary 7.6.3 one sees that f is the limit of classical eigenforms of tame level 1 (with normalized q-expansions).

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Department of Mathematics, University of California at Berkeley Department of Mathematics, Harvard University