Frank Calegari Detailed Research Plan

Detailed Research Plan

1 Introduction

My research centers on a relatively new and exciting area of number theory: the interplay between Galois representations and their deformations. Inspired by the work of H. Hida [22], B. Mazur [25] defined the notion of a deformation of a Galois representation. This work was instrumental both technically and conceptually for the subsequent work of A. Wiles [40] and R. Taylor–Wiles [38] on Fermat’s Last Theorem.

In another direction, drawing inspiration from Hida but also from pioneering work of B. Dwork, J.-P. Serre, and N. Katz (see [12, 35, 23]), R. Coleman showed that modular forms arise in p-adic analytic families, from which the classical forms can be extracted (see [7, 8, 10]). Coleman and Mazur gave a geometric interpretation of these results in their construction of the eigencurve (see [11]), a rigid analytic curve which “parameterizes” finite slope overconvergent p-adic modular eigenforms.

Recent joint work by K. Buzzard and myself [4] has revealed some surprising structure to the eigencurve, which sheds light on various open questions in [11], as well as various problems regarding classical modular forms, such as the Gouvêa–Mazur conjecture [18, 20]. This work also provides some of the first evidence for more general conjectures such as Buzzard’s slope conjectures [3], conjectures of Gouvêa [21] on the distribution of classical slopes, and questions of D. Wan relating to a possible p-adic “Riemann hypothesis” [39]. It also relates to several conjectures concerning classical Hecke algebras, which are described in section 4.2.

2 Dissertation Research

2.1 Reduction of Abelian Varieties

Fontaine proved ([15]) that there are no non-zero abelian varieties over ℤ. This result has been extended in various ways, in particular by work of R. Schoof ([30, 31, 32, 33]). One problem of recent interest is the existence of semistable abelian varieties with limited ramification. For N square-free and not contained in the set {1, 2, 3, 5, 6, 7, 10, 13}, the modular variety J_{0}(N) is a semistable non-zero abelian variety over ℤ[1/N]. The question then remains to determine whether such varieties exist over ℤ[1/N] in the remaining cases. A. Brumer and K. Kramer [2] showed that no such varieties exist for N ∈ {2, 3, 5, 7}. By using different techniques, Schoof [32] extended this result to N ∈ {2, 3, 5, 7, 13}. By combining the techniques of Brumer–Kramer and Schoof, I have extended this result still further [6], to give the best possible result: Theorem 1. Let S := {1, 2, 3, 5, 6, 7, 10, 13}. Let N be square-free. Assuming the GRH discriminant bounds of Odlyzko, there exists a non- semistable abelian variety over ℤ[1/N] if and only if N ∉ S. Schoof and I believe that reliance (for N = 6 and 10) on the GRH discriminant bounds of Odlyzko [27] can not be removed without using radically different techniques.

2.2 Almost rational torsion points

Let X be an algebraic curve of genus greater than one. Let J(X) be the Jacobian variety of X, and embed X in J(X). Let X_{tors} denote the set of torsion points on X, ie. X_{tors} := X ∩ J_{tors}. This set is finite (the Manin–Mumford conjecture), a result of M. Raynaud [28]. Certain work has gone in to determining X_{tors} for special X. For example, M. Baker [1] and (independently) A. Tamagawa [37] explicitly determined the set of torsion points of X_{0}(N) for N prime. Developing these ideas further, Ribet defined the notion of an almost rational torsion point, and used this concept to derive the Manin-Mumford conjecture [29] using some unpublished results of Serre [34]. The method of Baker and Tamagawa was in essence to determine the set of almost rational points on J_{0}(N). I have shown [5] that one can completely classify which torsion points can occur as almost rational torsion points for semistable elliptic curves over ℚ, proving (for example) that they must all be defined over ℚ(\sqrt{−3}).

3 Aspects of the Eigencurve

We use here the terminology of [11]. The eigencurve is a rigid analytic curve C_{p} whose ℂ_{p}- valued points parameterize p-adic overconvergent eigenforms of finite slope (and fixed tame level). The set of “classical points” (i.e., points which correspond to classical modular forms) are dense in C_{p}, and so the study of C_{p} provides a geometric approach to questions about classical modular forms. Study of the eigencurve is still in its infancy. In particular, several basic geometrical questions remain unresolved. In the last few months, K. Buzzard and I have made substantial progress towards understanding several of these questions, although much work remains to be done.

3.1 Explicit Slope Conjectures

Gouvêa and Mazur conjectured, and Coleman proved, that modular forms arise in p-adic families, parameterized by their p-adic weight; in other words, there is a natural projection from C_{p} to “weight space” W_{N}. Here W_{N} is a rigid analytic space whose ℂ_{p} points are continuous characters:

\underleftarrow{lim}_{n}(ℤ/Np^{n}ℤ) ⟶ ℂ_{p}^{∗}.

The curve C_{p} has infinitely many irreducible components over W_{N}, although components can generally be distinguished over a particular point of weight space by their slope. Let us recall the definition of slope:

Definition 1. Fix a prime p, a tame level N, and an integer k ∈ ℤ. Let M_{κ} denote the space of of overconvergent p-adic forms of tame level N and weight κ ∈ W_{N}(ℂ_{p}). Let

f = ∑_{n=1}^{∞} a_{f}(n)q^{n} ∈ M_{κ}

be a normalized eigenform. Then the slope of f is equal to v_{p}(a_{f}(p)), where v is the valuation on ℂ_{p} such that v(p) = 1.

The mere existence of the eigencurve implies the result (first proved by Coleman) that finite slope eigenforms come in p-adic families. One question then, is to control how the slope varies within a family. Some very explicit conjectures were formulated about the variation of slopes by Gouvêa and Mazur. Although these conjectures were originally stated for classical modular forms ([18]), control theorems of Coleman [7],[8], allow an equivalent reformulation in terms of overconvergent forms [20]. We may now state one form of the Gouvêa–Mazur conjectures. By abuse of notation we shall denote the space of overconvergent modular forms of integral weight (τ^{k}, k) by M_{k} (see [11], pg. 28).

Conjecture 1. Let k ∈ W_{N} be an integral weight. Let d(k, α) be the dimension of the subspace of M_{k} spanned by eigenforms of slope α. Then d(k, α) depends only on the congruence class of k modulo (p − 1)p^{⌈α⌉}.

Based on computational evidence provided W. Stein, Buzzard conjectured that in many cases it was possible to predict all the slopes for a particular integral weight and level [3]. Buzzard’s conjectures are very combinatorial, although they are (recursively) effective. Working with N = 1 and p = 2, Buzzard and I have formulated a more precise version of these conjectures [4], and proved the following special case:

Theorem 2 (Buzzard,Calegari). Let M_{0} be the space of 2-adic overconvergent modular forms of weight 0 and tame level 1. Then the slopes of eigenforms in M_{0} are all distinct and are given by the following numbers:

λ_{n} = 1 + 2 · ord_{2} (\frac{2n!}{n!}) for n ≥ 1, λ_{n} = 3, 7, 13, 15, …

Our conjectures imply the Gouvêa–Mazur conjectures for p = 2 and N = 1 (indeed they predict a much stronger result), and moreover they seem to be approachable.

3.2 Does the eigencurve have holes?

One open question asked by Coleman and Mazur ([11], pg. 5) is the following: Is C_{p} proper over weight space? In other words, given a family of overconvergent finite slope eigencurves parameterized by a punctured disc, so that the puncture can be filled in with a p-adic modular form, is this p-adic modular form necessarily overconvergent of finite slope? K. Buzzard and I have an approach to answering this question in the affirmative. So far, our methods have proved the following (again using notation from [11], pg. 28):

Theorem 3. Let C_{2}/ℂ_{2} be the eigencurve of tame level 1. Then C_{2} is proper over B^{∗} ⊆ W_{1}.

(Recall that the connected component of W_{N} is isomorphic to a rigid analytic disc, B, and B^{∗} denotes a particular smaller disc within B.) Although our results so far are limited to a special case, our approach is not limited in this way.

4 Galois Representations

4.1 Crystalline representations modulo p^{n}

Let f be a (classical) cuspidal modular eigenform of level N and weight k over \overline{ℚ}_{p}. Let O be the field of coefficients of f. Attached to f is an irreducible 2-dimensional representation

ρ_{f} : G_{ℚ,S} ⟶ GL_{2}(O)

for the complete local ring O, whose fraction field is a finite extension of ℚ_{p}. Let D_{p} := Gal(\overline{ℚ}_{p}/ℚ_{p}) be a choice of decomposition group at p. A theorem of G. Faltings [14] says that the local representation ρ|_{D_{p}} is crystalline (in the sense of J-M. Fontaine (see [16]) if and only if p does not divide N. On the other hand, the residual representation \overline{ρ}_{f} is always the reduction of some crystalline representation, although not necessarily one with Hodge–Tate weights [0, k − 1]. This result follows from work surrounding Serre’s conjecture [36]; see in particular B. Edixhoven [13]. A natural question is then: given f and n > 1, is the representation

\overline{ρ}_{f,n} : G_{ℚ,S} ⟶ GL_{2}(O/𝔪^{n})

the reduction of some modular representation ρ_{g} which is crystalline at p? If p does not divide N, then the answer is trivially yes, with g = f. If p exactly divides N then the answer is also yes. This follows from the fact that such forms have finite slope at p, and so correspond to points of integral weight on the eigencurve, around which crystalline points are dense. If p^{2}|N, then the answer is presumably (in general) no, although there is no known method for determining this given any particular representation. One problem is the lack of a suitable integral theory of crystalline representations. The theory of Fontaine–Laffaille [17] applies only to representations with Hodge–Tate weights in [0, p − 1], and presumably, as with residual representations, the cost of removing powers of p from the level is to increase the weight.

One may also ask the same question about local representations, in other words, when does a local representation:

\overline{ρ} : Gal(\overline{ℚ}_{p}/ℚ_{p}) ⟶ GL_{2}(O/𝔪^{n})

lift to a characteristic zero crystalline representation?

4.2 Boundedness conjectures on Hecke algebras

I will preface this section with a quotation from Gouvêa and Mazur’s article [19]:

To underline a curious aspect of this …, consider, given a residual representation \overline{ρ}, the set of distinct eigenforms of level p, all weights, and coefficients in ℤ_{p}. Before the recent work of Coleman, it was not known (to us) that this set was infinite. Going a bit further back, we do not believe that before the work of Hida, it was not even known that there are, in all, an infinite number of eigenforms of level p with Fourier coefficients in ℤ_{p}.

This local situation is in stark contrast to what is expected to happen globally. For example, if 𝕋_{k} denotes the Hecke algebra of weight k and level 1 it is believed that

[𝕋_{k} ⊗_{ℚ} \overline{ℚ} : ℚ] = dim(S_{k})

The infinite number of ℤ_{p}-eigenforms alluded to in the quotation above arise from specialization of Coleman’s families, passing from one family to another by “twinning”. It seems, however, that classical eigenforms having small fields of definition over ℚ_{p} is the norm rather than the exception. Fix a level N and a prime p. Let T_{p} be the classical Hecke algebra of modular forms of weight k on Γ_{1}(N). T_{p} is a complete Noetherian semilocal ring, which admits a decomposition

T_{p} ⊗ \overline{ℚ}_{p} = ∏ K_{i}

As Gouvêa and Mazur note, for certain N and p, as the weight k varies, there are infinitely many fields K_{i} such that K_{i} ≃ ℚ_{p}. In general, one has the following conjecture:

Conjecture 2. The degrees [K_{i} : ℚ_{p}] are bounded independently of k.

Even the much weaker following conjecture is still (I believe) completely open:

Conjecture 3. With the same notation as above, let O_{i} be the integral closure of ℤ_{p} in K_{i}. Then the residual degrees [O_{i}/𝔪_{i}, 𝔽_{p}] are bounded independently of k.

Note that this conjecture does not follow from the fact that their are a finite number of residual modular representations of a fixed level, since this only controls the residual degree of the Hecke algebra 𝕋, not its integral closure.

References

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    http://www.math.uiuc.edu/Algebraic-Number-Theory/0264/index.html

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