Finite Flatness of Torsion Subschemes of Hilbert-Blumenthal Abelian Varieties

Jordan S. Ellenberg

January 31, 2000

Contents

1 HBAV’s and Hilbert modular forms 4 1.1 Elliptic curves with multiplicative reduction . . . . . . . . . . . . 4 1.2 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Semi-HBAV’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 The Tate HBAV and q-expansion . . . . . . . . . . . . . . . . . . 9 1.5 Uniformization of HBAV’s . . . . . . . . . . . . . . . . . . . . . . 12

2 Discriminantal sets of Hilbert modular forms 15 2.1 Vertices and edges . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Existence of discriminantal sets . . . . . . . . . . . . . . . . . . . 23 2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Abstract

Let E be a totally real number field of degree d over ℚ. We give a method for constructing a set of Hilbert modular cuspforms f_{1}, …, f_{d} with the following property. Let K be the fraction field of a complete dvr A, and let X/K be a Hilbert-Blumenthal abelian variety with multiplicative reduction and real multiplication by the ring of integers of E. Suppose n is an integer such that n divides the minimal valuation of f_{i}(X) for all i. Then X[n′]/K extends to a finite flat group scheme over A, where n′ is a divisor of n with n′/n bounded by a constant depending only on f_{1}, …, f_{d}. When E = ℚ, the theorem reduces to a well-known property of f_{1} = Δ. In the cases E = ℚ(\sqrt{2}) and E = ℚ(\sqrt{5}), we produce the desired pairs of Hilbert modular forms explicitly and show how they can be used to compute the group of Néron components of a Hilbert-Blumenthal abelian variety with real multiplication by E.

Introduction

Let A be a complete dvr with fraction field K, and valuation ord : K^{∗} → ℤ. Let X/K be a Hilbert-Blumenthal abelian variety, or HBAV (Definition 1.3) with multiplicative reduction over K. Let n be an integer. We are interested in the following question:

Does the torsion subscheme X[n]/K extend to a finite flat group scheme over A?

Suppose X is an elliptic curve, with minimal discriminant Δ_{min}. It follows from the theory of Tate [22] that the above question has an affirmative answer whenever ord(Δ_{min}) is divisible by n. In the present work, we will generalize this fact, replacing the modular form Δ with a finite set of Hilbert modular forms.

More precisely: we define a discriminantal set of Hilbert modular forms to be a set of modular forms whose q-expansions satisfy a certain convex-geometric condition (Definition 2.12.) In the following theorem, a totally real field E and a fractional ideal 𝔠 ⊂ E are fixed, and all HBAV’s have real multiplication by the ring of integers of E.

Theorem (Corollary 2.16). Let f_{1}, …, f_{r} be a discriminantal set of 𝔠-Hilbert modular forms of level N (Definition 1.9.) Then there exists an integer m_{{f_{i}}} such that the following is true. Suppose X/K is an HBAV with multiplicative reduction over A, and

• λ is a 𝔠-polarization for X; (Definition 1.5)

• ω is a Néron non-vanishing differential on X; (Definition 1.22)

• ι is a Néron N-level structure on X. (Definition 1.22)

Let n be an integer such that n|ord(f_{i}(X, λ, ω, ι)) for all i, and let n′ be the numerator of n/m_{{f_{i}}} expressed in lowest terms.

Then X[n′]/K extends to a finite flat group scheme over A.

In fact, given the values of the ord(f_{i}(X, λ, ω, ι)), the valuation of the Tate parameter of X can be confined to a finite set of possibilities (Remark 2.17.) So the work here may be thought of as a small step in the direction of a “Tate’s algorithm for HBAV’s.”

We also prove an existence theorem for discriminantal sets.

Theorem (Theorem 2.18). Let E be a totally real number field of degree d over ℚ, and 𝔠 ⊂ E a fractional ideal. Then there exists a discriminantal set of 𝔠-Hilbert modular forms for E, with cardinality d.

Our ultimate aim is to investigate Diophantine equations associated to moduli spaces of HBAV’s. The current work can be seen as the geometric portion of this arithmetic-geometric problem. To illustrate the ideas involved, consider the following situation. Suppose that X/ℚ is a modular elliptic curve, with semistable reduction over ℤ[1/m]. Let p |̸ m be a prime, and let

ρ : Gal(ℚ̄/ℚ) → GL_{2}(𝔽̄_{p})

be the Galois representation on the p-torsion points of X. Then the primes ℓ not dividing pm such that X[p]/ℚ_{ℓ} extends to a finite flat group scheme over ℤ_{ℓ} are exactly those which do not divide the Artin conductor of ρ. Moreover, the weight k(ρ) is equal to 2 if and only if X[p] extends to a finite flat group scheme over ℤ_{p}. [20, Prop. 4].

It follows ([20, Prop. 7]) that an elliptic curve semistable over ℤ (and therefore modular) with ρ irreducible cannot have minimal discriminant a perfect pth power; for in that case X[p] would extend to a finite flat group scheme over every ℤ_{ℓ} (including ℤ_{p}), so that ρ would have weight 2 and conductor 1, an impossibility by Ribet’s theorem. One motivation for our work here is to extend arguments like the above to the general case of modular HBAV’s, and to study Diophantine equations that stand in relation to Hilbert modular varieties as Fermat’s equation does to the modular curve X(2). In [3], we apply these ideas to show that solutions to the generalized Fermat equation A^{4} + B^{2} = C^{p}, under certain 2-adic conditions on A, B, C, would produce non-modular HBAV’s. The HBAV’s involved, however, are shown to be modular in [4]. This result and related ones will appear in a later paper.

The article is organized as follows. In §1, we review the necessary theory of Hilbert-Blumenthal abelian varieties and Hilbert modular forms. In §2.2, we define discriminantal sets and prove Corollary 2.16 above. In §2.3, we prove Theorem 2.18 above. Finally, in §2.4 we produce explicit discriminantal sets of modular forms for the cases E = ℚ[\sqrt{5}] and E = ℚ[\sqrt{2}].

Acknowledgments

It is my pleasure to thank Barry Mazur for introducing me to the problems discussed in this paper. I also thank David Carlton and Brian Conrad for useful discussions, and the referee for helpful suggestions. Much of the work recorded here was supported by an NSF Graduate Fellowship.

Notation

Throughout this article, we will fix

• E a totally real number field of degree d over ℚ;

• ρ_{1}, …, ρ_{d} the embeddings of E into ℝ;

• O the ring of integers of E;

• 𝔬 the unit ideal in O; • 𝔡 the different of O/ℤ;

• U the group of units of O;

• U^{++} the group of totally positive units of O;

• A a complete dvr;

• K the fraction field of A;

• K̄ an algebraic closure of K;

• “ord” the valuation homomorphism K^{∗} → ℤ;

If 𝔠 is a fractional ideal in O, we denote by 𝔠^{++} the set of totally positive elements of 𝔠. More generally, if 𝔪 is any projective rank 1 O-module, we define a positivity structure on 𝔪 to be a subset 𝔪^{++} which is closed under addition and multiplication by 𝔬^{++}.

If 𝔞 and 𝔟 are projective rank 1 O-modules, we mean by 𝔞𝔟 the projective rank 1 O-module 𝔞 ⊗_{O} 𝔟.

The superscript ^{∨} signifies the dual abelian variety when applied to an abelian variety, and the ℤ-dual when applied to an O-module.

If X/S is any scheme, and s = Spec k is a point of S, we will often write X_{k} to denote the fiber product X ×_{S} s.

If X/S is an group scheme, we denote the identity section by

e_{X} : S → X.

If ν : X → X is an endomorphism of X, we denote the kernel of ν by X[ν]. In particular, X[n] is the subgroup scheme of n-torsion points of X.

If X/S is a smooth group scheme, we denote the locally free O_{S}-module e_{X}^{∗}Ω_{X/S}^{1} by \underline{ω}(X/S). The dual of \underline{ω}(X/S) (as O_{S}-module) will be called Lie(X/S).

1 HBAV’s and Hilbert modular forms

1.1 Elliptic curves with multiplicative reduction

The goal of this paper is to generalize a rather innocuous fact about elliptic curves with multiplicative reduction over non-archimedean local fields.

Let X/K be an elliptic curve with split multiplicative reduction over A. We know from the theory of Tate [22] that X can be thought of as a “quotient” of a torus by a discrete group:

X ≅ 𝔾_{m}/q_{X}^{ℤ}

for some parameter q_{X} ∈ K^{∗} with ord(q_{X}) > 0. For our purposes, we will not need to delve too closely into the meaning of the isomorphism written above.

Suffice it to say that there exists an elliptic curve over ℤ((q)), called the Tate curve, or Tate(q), fitting into a cartesian diagram

X ⟶ Tate(q)
↓  ↓
Spec K ⟶^{ϕ_{X}} Spec ℤ((q))

where ϕ_{X} is induced by the ring homomorphism sending q to q_{X}.

In view of the above, we are naturally interested in computing the parameter q_{X}. Indeed, the integer ord(q_{X}) is already an important invariant of X; for instance, the component group of the special fiber of the Néron model of X is a cyclic group of order ord(q_{X}).

Theorem 1.1. Let Δ_{min}(X) be the minimal discriminant of X. Then

ord(Δ_{min}(X)) = ord(q_{X}).

Proof. The minimal discriminant of Tate(q) is the usual q-expansion

q − 24q^{2} + …

Thus, the minimal discriminant of X is

q_{X} − 24q_{X}^{2} + …

and the theorem is immediate.

Corollary 1.2. If n is a rational integer dividing ord(Δ_{min}(X)), the group scheme X[n]/K extends to a finite flat group scheme over A. In particular, let X /A be the Néron model of X. Then the group scheme X [n] is finite and flat over A.

Proof. Since multiplication by n is flat, only finiteness needs to be proved.

Let X^{0} be the identity component of X; let X_{s} be the fiber of X over the closed point of Spec A, and let X_{s}^{0} be the identity component of X_{s}. Then X_{s}^{0}[n] is a finite flat group scheme over s of degree n.

Let A′ be an unramified extension of A such that the residue field k′ of A′ is separably closed. Fix an embedding of k′ into an algebraic closure k̄ of k.

It follows from Tate’s algorithm that

Φ := X (A′)/X^{0}(A′) ≅ ℤ/mZ

where m = ord(q_{X}) is a multiple of n. In particular, Φ[n] has order n.

Let x_{α} ∈ X (A′) be a point representing an element α of Φ[n]. Then nx_{α} ∈ X^{0}(A′). Let x̄_{α} be the restriction of x_{α} to the geometric fiber Xs̄ = X_{s} ×_{k} k̄.

Multiplication by n is surjective on k̄-points of X 0 ; s̄; thus, we may choose y ∈ X 0 ; s̄(k̄) such that ny = nx̄_{α}.

Now let x′_{α} = x̄_{α} − y. Then x′_{α} + X 0 ; s̄[n] is a subscheme of Xs̄[n], of degree n over s̄. Moreover, x′_{α} + X 0 ; s̄[n] and x′_{β} + X 0 ; s̄[n] are disjoint for α ≠ β, since they lie in distinct components of Xs̄. We conclude that Xs̄[n] is a group scheme of degree at least n^{2} over s̄, whence X_{s}[n] is a group scheme of order at least n^{2} over s.

Now X [n] is a quasi-finite scheme over a complete dvr, and as such it can be written as the disjoint union of a finite scheme Y′ and a scheme Y′′ which is void over the special point [6, (6.2.6)]. Since X [n] is flat, Y′ has equal degree over the generic and special points. But we have shown that the degree of X [n] over the special point at least n^{2}, which is the degree of X [n] over the generic point. Therefore, Y′′ is void, and X [n] is finite, as desired.

Our aim is to generalize the above corollary to the case where X is a Hilbert-Blumenthal abelian variety; that is, we want to give a criterion, in terms of the values of certain modular forms evaluated at X, for X[n]/K to extend to a finite flat group scheme over A.

1.2 Definitions and examples

The basic theory of Hilbert-Blumenthal abelian varieties (HBAV’s), also called abelian varieties with real multiplication, can only be summarized in this small space. For more detail and justification, we refer the reader to Katz [10, ch. 1], whose definitions and notation we follow closely in this paper. Another excellent reference is [23].

Definition 1.3. A Hilbert-Blumenthal abelian variety (HBAV) over a ring R is a pair (X, ϕ), where

• X is an abelian scheme of dimension d over Spec R;

• ϕ : O ↪ End(X) is an injection making Lie(X/R) a locally (on R) free rank-1 O ⊗_{ℤ} R module.

We will refer to X as “an HBAV with real multiplication by O” if there is any ambiguity about what totally real ring of integers is acting. We will denote the endomorphism ϕ(α) by [α]. When no confusion is likely, an HBAV (X, ϕ) will be referred to simply as X. Remarks 1.4. 1. If X/K is an abelian scheme of dimension d over a field

of characteristic 0, endowed with an injection ϕ : O ↪ End(X), the condition on Lie(X/K) is automatically satisfied [16, Prop. 1.4].

2. An elliptic curve is an HBAV with real multiplication by ℤ.

3. Brumer has shown [2] that the Jacobian of the genus-2 curve

C : y^{2} = x^{6} + 2cx^{5} + (c^{2} + 2c + 2 − bd)x^{4}  (1.2.1)
+ (2c^{2} + 2c + 2 + b − 2bd − 4d)x^{3}
+ (c^{2} + 4c + 5 + 3b − bd)x^{2} + (2c + 6 + 3b)x + (b + 1)

is an HBAV over ℚ(b, c, d) with real multiplication by ℤ[\frac{1+\sqrt{5}}{2}].

4. The endomorphism [α] is an isogeny [1, 7.3,Def. 4] for all α = 0, since it factors through [n] for some rational integer n, and [n] is an isogeny.

Following Katz [10], we begin by defining some extra structures on HBAV’s.

Definition 1.5. Let X be an HBAV and 𝔠 a fractional ideal of O. Let X ⊗_{O} 𝔠 be the HBAV representing the functor T ↦ X(T) ⊗_{O} 𝔠, and let

ϵ : 𝔠 → Hom_{O}(X, X ⊗_{O} 𝔠)

be the evident homomorphism. Let P (X) be the O-module of O-linear symmetric morphisms from X to X^{∨}. We say an element of P (X) is positive if it is a polarization of X as an abelian scheme.

Then a 𝔠-polarization of X (as an HBAV) is an isomorphism

λ : X ⊗_{O} 𝔠 → X^{∨}

such that the map 𝔠 → P (X) given by

c → λ ∘ ϵ(c)

is an isomorphism which associates the totally positive elements of 𝔠 to the totally positive elements of P (X).

Remark 1.6. If 𝔠′ is a fractional ideal in the same narrow ideal class as 𝔠, then any 𝔠-polarization of an HBAV X naturally gives rise to a 𝔠′-polarization. So the choice of 𝔠 is essentially only a choice of a narrow ideal class of O.

Since \underline{ω}_{X/R} is R-dual to Lie(X/R), it is locally isomorphic to 𝔡^{−1} ⊗ R, and is hence also a locally (on R) free O ⊗ R-module.

Definition 1.7. A non-vanishing differential on a HBAV X is a generator for \underline{ω}_{X/R} as O ⊗ R-module.

Equivalently, we may say that a non-vanishing differential is an isomorphism between Lie(X/R) and 𝔡^{−1} ⊗_{ℤ} R.

Finally, we stipulate a notion of level structure.

Definition 1.8. An N-level structure (or Γ_{00}(N)-structure) for a HBAV X is an O-linear closed immersion of group schemes over R

ι : 𝔡^{−1} ⊗ μ_{N} ↪ X,

where μ_{N}/R is the (finite) group scheme of N-th roots of unity.

In case N = 1, we write ι_{1} for the trivial level structure.

Now we are ready to define Hilbert modular forms.

Definition 1.9. Suppose d > 1. A 𝔠-Hilbert modular form of level N and weight k over R_{0} is a function on quadruples (X, λ, ω, ι), where

• X is an HBAV over an R_{0}-algebra R;

• λ is a 𝔠-polarization of X;

• ω is a non-vanishing differential on X;

• ι is an N-level structure for X,

such that

• f(X/R, λ, ω, ι) is an element of R depending only on the R-isomorphism class of (X, λ, ω, ι);

• f commutes with any extension of scalars R → R′;

• For any x in (O ⊗ R)^{∗},

f(X, λ, xω, ι) = N_{ℚ}^{E}(x)^{−k}f(X, λ, ω, ι).

The R_{0}-module of such functions is denoted by M_{O}^{k}(𝔠, N; R_{0}). The ring of modular forms of arbitrary integral weight is called M_{O}^{∗}(𝔠, N; R_{0}).

Remark 1.10. If d = 1 (that is, if O = ℤ) another condition on f is necessary, namely that it be “holomorphic at the cusps.” That this condition is unnecessary when d > 1 is the content of the Koecher principle (Proposition 1.17 below.)

Example 1.11. Suppose O = ℤ[\frac{1+\sqrt{5}}{2}], and let 𝔠 = 𝔡. Since the narrow class group of O is trivial, this is essentially the only choice (see Remark 1.6); we choose 𝔡 rather than 𝔬 for notational convenience later on.

Then

M_{O}^{∗}(𝔡, 1; ℂ) ≅ ℂ[ϕ_{2}, χ_{5}, χ_{6}, χ_{15}]/R

where the generators are modular forms of weights 2, 5, 6, 15, and R is a relation in weight 30. (See Hirzebruch [8]). Nagaoka [15] proved that the forms χ_{6} and χ_{12} := (1/4)(χ_{6}^{2} − ϕ_{2}χ_{5}) are in fact base changes of forms in M_{O}^{∗}(𝔡, 1; ℤ).

1.3 Semi-HBAV’s

We will often want to consider a class of group schemes somewhat more general than HBAV’s. Recall that a semi-abelian scheme X over a base S is a smooth commutative group scheme over S, each of whose fibers is an extension of an abelian variety by a torus. If the fiber over s ∈ S is an extension of an abelian variety by a split torus, we say X is split at s.

Definition 1.12. A semi-HBAV over Spec R is a pair (X, ϕ), where

• X is a semi-abelian scheme of dimension d over R;

• ϕ : O ↪ End(X) is a homomorphism making Lie(X/R) a locally (on R) free rank-1 (O ⊗_{ℤ} R)-module.

Note that it follows from the second condition in Definition 1.12 that each fiber of a semi-HBAV is either an abelian variety or a torus. We say a semi-HBAV is split if each of its geometric fibers is either an abelian variety or a split torus.

If K is the fraction field of a complete dvr A, and Y/K is an abelian variety, we say that Y has multiplicative reduction (resp. split multiplicative reduction) if the identity component of its Néron model is a semi-abelian variety whose special fiber is a torus (resp. semi-abelian variety whose special fiber is a split torus.)

By analogy with the HBAV case, we define a non-vanishing differential on a semi-HBAV X/R to be a generator for the (O ⊗_{ℤ} R)-module \underline{ω}_{X/R}, and an N-level structure to be an O-linear closed immersion

ι : 𝔡^{−1} ⊗ μ_{N} ↪ X.

We have already seen that elliptic curves with split multiplicative reduction over A can be expressed as pullbacks of a Tate curve over ℤ[[q]]. The situation for HBAV’s with split multiplicative reduction (indeed, for general abelian varieties with split multiplicative reduction) is very much the same; all HBAV’s with split multiplicative reduction will be pullbacks of a certain “Tate HBAV.”

1.4 The Tate HBAV and q-expansion

In order to define the Tate HBAV, we must first describe the power series ring over which it is defined. To this end we make the following definitions.

Let 𝔪 be a projective rank 1 O-module endowed with a positivity structure 𝔪^{++}.

Definition 1.13. A linear form L : 𝔪 → ℤ is positive if it sends all positive elements of 𝔪 to positive integers.

Definition 1.14. Let S be a set of d linearly independent positive linear forms. An element m ∈ 𝔪 is S-semipositive if L(m) ≥ 0 for every L ∈ S. The set of S-semipositive elements of 𝔪 is denoted 𝔪^{S}.

Let 𝔞 be a projective rank 1 O-module, 𝔠 a fractional ideal of O, and set 𝔟 = 𝔞𝔠^{−1}. We define a positivity structure on 𝔞𝔟 by defining 𝔞𝔟^{++} to be the semigroup generated by all elements of the form a ⊗ ac^{−1}, with a ∈ 𝔞 and c ∈ 𝔠^{++}.

Now define

ℤ[𝔞𝔟, S] = ℤ[…, q^{α}, …]/(q^{0} − 1, q^{α_{1}+α_{2}} − q^{α_{1}}q^{α_{2}})

where α, α_{1}, α_{2} vary over (𝔞𝔟)^{S}. Let I_{S} be the ideal generated by q^{α} for all α ∈ (𝔞𝔟)^{S} − {0}, and let ℤ[[𝔞𝔟, S]] be the completion of ℤ[𝔞𝔟, S] in the I_{S}-adic topology. Let I be the ideal in ℤ[[𝔞𝔟, S]] generated by q^{α} for all totally positive α. We denote by ℤ((𝔞𝔟, S)) the localization of ℤ[[𝔞𝔟, S]] away from I.

Now ℤ[[𝔞𝔟, S]] is a normal Noetherian domain complete with respect to I. Moreover, the pairing

q : 𝔟 ⊗ 𝔞 → 𝔾_{m}(ℤ((𝔞𝔟, S)))

defined by

q(β ⊗ α) = q^{βα}.

is a polarized set of periods in the sense of the construction of Mumford [14]. Therefore, there exists an abelian variety Tate_{𝔞,𝔟}(q) over ℤ((𝔞𝔟, S)) which we may think of as the quotient

𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m}/q_{𝔞}(𝔟),

where

q_{𝔞} : 𝔟 → 𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m}(ℤ((𝔞𝔟, S)))

is the homomorphism induced by q.

We record below some basic facts about the Tate HBAV, which can readily be derived from Mumford’s work in [14]. (See [3] for more details.)

1. Tate_{𝔞,𝔟}(q) is an HBAV, which extends to a split semi-HBAV T̃ate_{𝔞,𝔟}(q)/ℤ[[𝔞𝔟; S]].

2. Tate_{𝔞,𝔟}(q) has a canonical 𝔠-polarization λ_{can}.

3. There is a canonical isomorphism

ω_{𝔞} : Lie(Tate_{𝔞,𝔟}(q)) → Lie(𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m}/ℤ((𝔞𝔟, S)))

which extends to an isomorphism between Lie(T̃ate_{𝔞,𝔟}(q)) and Lie(𝔡^{−1}𝔞^{−1}⊗_{ℤ} 𝔾_{m}/ℤ[[𝔞𝔟, S]]).

In particular, if R is a ring, any isomorphism

j : 𝔞 ⊗_{ℤ} R →^{∼} 𝔬 ⊗_{ℤ} R

induces a non-vanishing differential on Tate_{𝔞,𝔟}(q) ×_{ℤ} R by composition with ω_{𝔞}. We denote this non-vanishing differential by ω(j).

4. Let N be a rational integer. There exists an exact sequence of group schemes over ℤ((𝔞𝔟, S))

0 → 𝔡^{−1}𝔞^{−1} ⊗ μ_{N} →^{ι_{𝔞}} Tate_{𝔞,𝔟}(q)[N] → ((1/N)ℤ/ℤ) ⊗_{ℤ} 𝔟 → 0,

and ι_{𝔞} extends to a morphism of group schemes over ℤ[[𝔞𝔟, S]]:

ι_{𝔞} : 𝔡^{−1}𝔞^{−1} ⊗ μ_{N} → T̃ate_{𝔞,𝔟}(q)[N].

So any isomorphism

ϵ : 𝔬/N𝔬 →^{∼} 𝔞/N𝔞

induces a level N structure on Tate_{𝔞,𝔟}(q) by composition with ι_{𝔞}. We denote this level N structure by ι(ϵ). Note that ι(ϵ) extends to a level N structure on T̃ate_{𝔞,𝔟}(q).

Suppose f is a 𝔠-Hilbert modular form of level N over a ring R_{0}, and (𝔞, 𝔟, j, ϵ) is a quadruple where

• 𝔞 is a projective rank 1 O-module and 𝔟 = 𝔞𝔠^{−1};

• j : 𝔞 ⊗_{ℤ} R →^{∼} 𝔬 ⊗_{ℤ} R is an isomorphism, for some R_{0}-algebra R;

• ϵ : 𝔬/N𝔬 →^{∼} 𝔞/N𝔞 is an isomorphism.

Such a quadruple is called a cusp (or a cusp of level N) over R.

In case N = 1, we denote by ϵ_{1} the canonical isomorphism from 𝔬/𝔬 to 𝔞/𝔞.

Definition 1.15. The q-expansion of f at the cusp (𝔞, 𝔟, j, ϵ) is the value

f(Tate_{𝔞,𝔟}(q) ×_{ℤ} R, λ_{can}, ω(j), ι(ϵ)).

A priori, a q-expansion of a modular form at a cusp over R is an element of R_{0}((𝔞𝔟, S)) ⊗_{R_{0}} R. In fact, q-expansions are constrained to lie in a smaller ring. We record three standard facts below.

Proposition 1.16. The q-expansion of f at (𝔞, 𝔟, j, ϵ) is independent of the choice of S.

Proof. Let S_{1}, S_{2} be two choices of S. Suppose

(𝔞𝔟)^{S_{2}} ⊂ (𝔞𝔟)^{S_{1}}.

Then the Tate HBAV computed using S_{1} is just a pullback of the Tate HBAV computed using S_{2}, via the natural inclusion ℤ[[𝔞𝔟; S_{2}]] ⊂ ℤ[[𝔞𝔟; S_{1}]]. Therefore, the q-expansions computed using S_{1} are the same as those using S_{2}.

The general result follows from choosing S_{3} such that

(𝔞𝔟)^{S_{3}} ⊂ (𝔞𝔟)^{S_{i}}(i = 1, 2).

Proposition 1.17 (Koecher principle). Suppose d > 1. Let f be a modular form whose q-expansion at a cusp (𝔞, 𝔟, j, ϵ) is

∑_{α∈𝔞𝔟} a_{α}(f)q^{α}.

Then a_{α} = 0 unless α = 0 or α ∈ 𝔞𝔟^{++}.

Proof. See [16, Prop. 4.9].

Proposition 1.18. Let f be a modular form whose q-expansion at (𝔞, 𝔟, j, ϵ) is

∑_{α∈𝔞𝔟} a_{α}(f)q^{α},

and let v ∈ U be congruent to 1 (mod N). Then a_{v^{2}α}(f) = (N_{ℚ}^{E}v)^{k}a_{α}(f) for all α ∈ 𝔞𝔟.

Proof. Consider the base change of Tate_{𝔞,𝔟}(q) induced by the automorphism of ℤ[[𝔞𝔟; S_{1}]] sending q to q^{u}. The result then follows from invariance of Hilbert modular forms under base change.

1.5 Uniformization of HBAV’s

As described above, HBAV’s with split multiplicative reduction are pullbacks of the Tate HBAV, just as is the case for elliptic curves.

We say a homomorphism q : 𝔞𝔟 → 𝔾_{m}(K) is positive if it takes 𝔞𝔟^{++} to A. Any such q yields a homomorphism

q_{𝔞} : 𝔟 → 𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m}(K)

and, for suitably chosen S, a map of schemes

q̃: Spec A → Spec ℤ[[𝔞𝔟, S]].

Theorem 1.19 (Uniformization Theorem). Let X/K be a 𝔠-polarized H-BAV with split multiplicative reduction over A. Then there exists a projective rank 1 O-module 𝔞, a set of d linearly independent positive linear forms S, and a positive homomorphism

q_{X} : 𝔞𝔟 → 𝔾_{m}(K)

(where 𝔟 = 𝔞𝔠^{−1}) such that X/K is isomorphic (as a 𝔠-polarized HBAV) with the pullback of Tate_{𝔞,𝔟}(q) by q̃_{X}. The choice of q_{X} is unique up to multiplication by squares of units of O.

This theorem follows from a more general theorem on abelian varieties with split purely multiplicative reduction, proved independently by Mumford [14] and Raynaud [18]. A thorough treatment of this theorem, and still more general ones, can be found in the book of Faltings and Chai [5].

Remark 1.20. If X is isomorphic to a pullback of a Tate HBAV Tate_{𝔞,𝔟}(q) by q̃_{X}, as in the theorem, we will write

X ≅ (𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m})/q_{X;𝔞}(𝔟).

Corollary 1.21. Let X/K be an HBAV with multiplicative reduction, and let X^{0}/A be the identity component of the Néron model of X. Then X^{0} is a semi-HBAV.

Proof. After passing to an étale extension, we may assume that X has split multiplicative reduction. Then X is the pullback of Tate_{𝔞,𝔟}(q) by some q̃_{X}. The pullback of T̃ate_{𝔞,𝔟}(q) by q̃_{X} is then a semi-HBAV over A whose generic fiber is X; such a semi-abelian variety is isomorphic to X^{0} by [1, 7.4, Prop. 3].

Definition 1.22. Let X, X^{0} be as above. A Néron differential on X is a nonvanishing differential which is the restriction to K of a non-vanishing differential on X^{0}/A. Likewise, a Néron N-level structure on X is an N-level structure which is the restriction to K of an N-level structure on X^{0}/A.

Proposition 1.23. Suppose X/K is an HBAV such that

X ≅ (𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m})/q_{X;𝔞}(𝔟).

Suppose ω is a Néron non-vanishing differential on X and ι is a Néron N-level structure on X. Then there exists a cusp (𝔞, 𝔟, j, ϵ) over A such that ω = q̃_{X}^{∗}ω(j) and ι = q̃_{X}^{∗}ι(ϵ).

Proof. The formal completion of X^{0} over Spf(A) is isomorphic to that of 𝔡^{−1}𝔞^{−1}⊗_{ℤ} 𝔾_{m} [14, §3]. So the N-level structures

ι : 𝔡^{−1} ⊗_{ℤ} μ_{N} ↪ X^{0}

are in bijection with maps

ι : 𝔡^{−1} ⊗_{ℤ} μ_{N} ↪ 𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m}

(by [7, 5.4.1]), which are in turn classified by isomorphisms

ϵ : 𝔬/N𝔬 →^{∼} 𝔞/N𝔞.

Likewise, [7, 5.1.6] yields an isomorphism

Lie(X^{0}/A) ≅ Lie(𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m}/A) = 𝔡^{−1}𝔞^{−1} ⊗_{ℤ} A

which associates to every ω : Lie(X^{0}/A) ≅ 𝔡^{−1}⊗_{Z} A an isomorphism j : 𝔞⊗_{ℤ}A ≅ 𝔬 ⊗_{ℤ} A such that ω = q̃_{X}^{∗}ω(j).

The power of the Uniformization Theorem is that it allows us to reduce many questions about the geometry of HBAV’s to questions about the parameter q_{X}. In particular, we will see in Proposition 1.25 below that the question of whether the n-torsion subscheme of X can be extended to a finite flat group scheme over A can sometimes be answered immediately via knowledge of q_{X}.

Definition 1.24. Let

q : 𝔞𝔟 → K^{∗}

be a positive homomorphism. Then ord(q) is the element x ∈ (𝔡^{−1}𝔞^{−1}𝔟^{−1})^{++} such that

ord(q(y)) = Tr(y · ord(q))

for all y in 𝔞𝔟.

Note that if X/K is a 𝔠-polarized HBAV with split multiplicative reduction over A, the quantity ord(q_{X}) is defined up to multiplication by squares of units.

For the following proposition, note that, by [1, 7.3.6], the isogeny ν : X → X extends uniquely to an isogeny ν : X → X. In particular, ν : X → X is flat.

Proposition 1.25. Let X/K be a 𝔠-polarized HBAV with multiplicative reduction over A. Let A′ be an etale extension of A, with fraction field K′, such that X′ = X ×_{K} K′ has split multiplicative reduction. Write X′ as

X′ ≅ (𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m})/q_{X′;𝔞}(𝔟),

and let ν be an element of O such that

ord(q_{X′}) ∈ ν𝔡^{−1}𝔞^{−1}𝔟^{−1}.

Then X[ν]/K extends to a finite flat group scheme over A. In particular, let X /A be the Néron model of X. Then the group scheme X [ν] is finite and flat over A.

Proof. We follow closely the proof of Corollary 1.2.

Since ν is flat, only finiteness needs to be proved.

Let X′/A′ be the Néron model of X′. Since formation of Néron model commutes with etale base change, X′[ν] is isomorphic to X [ν] ×_{A} A′. Finiteness satisfies fpqc descent; thus, if X′[ν]/A′ is finite, so is X [ν]/A. So it suffices to consider the case A = A′. Moreover, we may suppose the residue field k of A to be separably closed.

Note that, by [12, 12.12], the generic fiber of X [ν] has degree N(ν)^{2} over Spec K.

Let s be the closed point of Spec A, and let X_{s}, X_{s}^{0} be the special fibers of X , X^{0}. Recall that ν : X^{0} → X^{0} was produced in [14] as the algebrization of a certain formal morphism; from that description, it follows that

X_{s}^{0}[ν] ≅ (𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m})[ν],

and in particular X_{s}^{0}[ν] is a finite group scheme of degree Nν over s. It follows from [5, III.8.2] that

Φ := X_{s}/X_{s}^{0} = 𝔡^{−1}𝔞^{−1}/ord(q_{X})𝔟  (1.5.2)

By the functoriality of Mumford’s construction, (1.5.2) is an equality, not only of groups, but of O-modules. In particular, Φ[ν] has order Nν.

The finiteness of X [ν] now follows exactly as in Corollary 1.2.

2 Discriminantal sets of Hilbert modular forms

The main part of this paper has as its goal to derive information about ord(q_{X}) from the values of Hilbert modular forms, and thereby, via Proposition 1.25, to obtain criteria guaranteeing that a torsion subscheme X[n]/K extends to a finite flat group scheme over A.

2.1 Vertices and edges

In this section we collect some elementary results about the convex geometry of subsets of projective rank 1 O-modules.

To begin with, let Λ ⊂ ℝ^{d} be the image of an embedding ℤ^{d} ↪ ℝ^{d}; that is, Λ is a lattice in ℝ^{d}. We denote by (ℝ^{d})^{++} the subset of ℝ^{d} consisting of points with all coordinates positive, and by Λ^{++} the intersection of Λ with (ℝ^{d})^{++}.

For the following definitions, let S be a non-empty subset of Λ^{++}.

Definition 2.1. An element α ∈ S is a vertex of S if there exists a linear form L : Λ → ℝ such that

L(α) < L(α′), ∀α′ ∈ S − {α}  (2.1.3)

A form L satisfying (2.1.3) is said to be minimized at α.

Definition 2.2. A pair {α, β} of vertices of S is an edge if there exists a linear form L : Λ → ℝ such that

L(α) = L(β)

and

L(α) ≤ L(α′), ∀α′ ∈ S.

In this case, we say L is minimized at {α, β}.

A linear form L : Λ → ℝ is said to be positive if it maps Λ^{++} to ℝ^{+}.

Proposition 2.3. Every positive linear form L is minimized at either an edge or a vertex.

Proof. Let α_{0} be an element of S. Define a subset R of (ℝ^{d})^{++} by

R = {x ∈ (ℝ^{d})^{++} : L(x) ≤ L(α_{0})}.

Let

y_{i} = L(0, …, 1, …, 0),

where the 1 above is in the ith coordinate. Then R lies inside the region

{x = (x_{1}, …, x_{d}) : 0 ≤ x_{i} ≤ y_{i}^{−1}L(α_{0})}.

So R is bounded, and since S ⊂ Λ is discrete, the set

S ∩ R = {α ∈ S : L(α) ≤ L(α_{0})}

is finite. S ∩ R is also non-empty, by virtue of containing α_{0}. In particular, there is a unique non-empty finite subset {α_{1}, …, α_{r}} of S such that

L(α_{i}) = L(α_{j}), ∀i, j

and

L(α_{i}) < L(α), ∀α ∈ S − {α_{1}, …, α_{r}}.

If r = 1, then α_{1} is a vertex of S. Suppose r > 1. Then by perturbing L slightly to some L + ϵ, one can arrange that, for some i,

(L + ϵ)(α_{i}) < (L + ϵ)(α), ∀α ∈ S − {α_{i}}.

So α_{i} is a vertex. Moreover, ϵ can be chosen so that L − ϵ is also minimized at a vertex. Since ϵ(α_{i}) < ϵ(α_{j}) for all j ≠ i, the vertex α_{j} where L − ϵ is minimized cannot be α_{i}. Now {α_{i}, α_{j}} is the desired edge on which L is minimized.

We will now restrict to the case where Λ admits an action of O. Let 𝔞 be a projective rank 1 O-module and 𝔠 a fractional ideal of O. Let 𝔟 = 𝔞𝔠^{−1}. Then 𝔞𝔟 is a lattice in 𝔞𝔟 ⊗_{ℤ} ℝ.

If 𝔦 is any fractional ideal of O, the real embeddings ρ_{1}, …, ρ_{d} of E induce an isomorphism

ρ_{𝔦} : 𝔦 ⊗_{ℤ} ℝ → ℝ^{d}.

A trace form

Tr : 𝔦 ⊗_{ℤ} ℝ → ℝ

is obtained by composing the trace form on ℝ^{d} (sum of coordinates) with ρ_{𝔦}. Likewise, a norm form

N : 𝔦 ⊗_{ℤ} ℝ → ℝ

is obtained from the norm form on ℝ^{d} (product of coordinates.)

A linear form L : 𝔞𝔟 → ℝ may be thought of as an element x_{L} of 𝔡^{−1}𝔞^{−1}𝔟^{−1}⊗_{ℤ} ℝ satisfying

L(α) = Tr(x_{L}α).

A positive linear form corresponds to an element of (𝔡^{−1}𝔞^{−1}𝔟^{−1} ⊗_{ℤ} ℝ)^{++}.

Fix a finite-index subgroup V of U^{++}, and let S be a subset of 𝔞𝔟^{++} which is stable under multiplication by V. The vertices and edges of such sets are rather manageable, thanks to the boundedness statement below and its corollaries.

Proposition 2.4. Let D_{V} be the region

{x ∈ (𝔡^{−1} ⊗_{ℤ} ℝ)^{++} : Tr(x) ≤ Tr(xu), ∀u ∈ V }

and let A be a positive real number. Then the set

{x ∈ D_{V} : N(x) ≤ A}

is bounded.

Proof. Since D_{V} is invariant under multiplication by positive reals, it suffices to prove that

D_{V}^{(1)} = {x ∈ D_{V} : N(x) = 1}

is bounded.

Composing ρ_{𝔡^{−1}} with the map

(x_{1}, …, x_{d}) ↦ (log |x_{1}|, …, log |x_{d}|)

yields an isomorphism

ψ : (𝔡^{−1} ⊗_{ℤ} ℝ)^{++} → ℝ^{d}

which induces a homeomorphism between the set of x in (𝔡^{−1} ⊗_{ℤ} ℝ)^{++} with N(x) = 1 and the hyperplane

H = {(x_{1}, …, x_{d}) ∈ ℝ^{d} : ∑_{i} x_{i} = 0}.

If ψ(x) = (y_{1}, …, y_{d}), we have

Tr(x) = ∑_{i} e^{y_{i}}
ψ(xu) = (y_{1} + log ρ_{1}(u), …, y_{d} + log ρ_{d}(u)), ∀u ∈ V.

By Dirichlet’s Unit Theorem, the points

{λ_{u} = (log ρ_{1}(u), …, log ρ_{d}(u)) : u ∈ V }

form a lattice in H, which we call Λ_{V}.

Choose a real number M large enough so that the region

{(y_{1}, …, y_{d}) ∈ H : max y_{i} ≤ M}  (2.1.4)

contains a fundamental domain for Λ_{V}. Now suppose ψ(x) = (y_{1}, …, y_{d}) lies outside the bounded region

{(y_{1}, …, y_{d}) ∈ H : max y_{i} ≤ M + log d}.

Then

Tr(x) > e^{max y_{i}} ≥ de^{M}.

On the other hand, there exists u ∈ V such that h+λ_{u} lies inside (2.1.4), whence Tr(xu) ≤ de^{M}. We conclude that x ∉ D_{V}, which proves the proposition.

Note that D_{V} is a fundamental domain for the action of V on (𝔡^{−1} ⊗_{ℤ} ℝ)^{++}.

Corollary 2.5. Let α_{0} ∈ 𝔞𝔟^{++} and A ∈ ℝ^{+}. There exist only finitely many α ∈ 𝔞𝔟^{++} such that

• N(α_{0}^{−1}α) ≤ A;

• There exists a linear form L satisfying

L(α_{0}) ≤ L(α_{0}u), ∀u ∈ V
L(α) ≤ L(αu), ∀u ∈ V.

Proof. The second condition says that the intersection

α_{0}^{−1}D_{V} ∩ α^{−1}D_{V} ∈ (𝔡^{−1}𝔞^{−1}𝔟^{−1} ⊗ ℝ)^{++}

is non-empty (by virtue of containing x_{L}), which is to say that

α ∈ α_{0}D_{V} (D_{V})^{−1}.

Adding the first condition, we have

α_{0}^{−1}α ∈ {x ∈ D_{V} (D_{V})^{−1} : N(x) ≤ A}.

Let P be the set of all x in D_{V} (D_{V})^{−1} with norm at most A. Every element x ∈ P can be written as ry_{1}y_{2}^{−1}, where r ∈ (0, A] and y_{1}, y_{2} ∈ D_{V}^{(1)}. By Proposition 2.4, D_{V}^{(1)} is bounded, whence so is P. Since 𝔞𝔟^{++} is discrete, there are only finitely many α in 𝔞𝔟^{++} such that α_{0}^{−1}α lies in P. The desired result follows.

Corollary 2.6. Let α_{0} be an element of (𝔞𝔟 ⊗ ℝ)^{++}. There exists A ∈ ℝ such that N(α_{0}^{−1}α) ≤ A for all vertices α of S.

Proof. Without loss of generality, assume α_{0} ∈ S.

By Proposition 2.4, the set

α_{0}^{−1}D_{V}^{(1)} ∈ 𝔡^{−1}𝔞^{−1}𝔟^{−1} ⊗ ℝ

is bounded, and in particular L(α_{0}) is bounded above by some constant C as L ranges over α_{0}^{−1}D_{V}^{(1)}.

Now let α be a vertex of S, minimizing some linear form L. Since D_{V} is a fundamental domain for the action of V on (𝔡^{−1} ⊗_{ℤ} ℝ)^{++}, we can replace L by Lu, for some u ∈ V, in such a way that x_{L} ∈ α_{0}^{−1}D_{V}.

Now, multiplying L by a positive real number, we may suppose x_{L} ∈ α_{0}^{−1}(D_{V})^{(1)}. We replace α with αu^{−1}, so that α is still minimized at L.

Now L(α) ≤ L(α_{0}) ≤ C. But now

C ≥ L(α) = Tr(x_{L}α) ≥ dN(x_{L}α)^{1/d} = dN(α_{0}^{−1}α)^{1/d},

whence the desired result.

Corollary 2.7. Let α_{0} be a vertex of S. Then there are only finitely many edges of S containing α_{0}.

Proof. Let {α_{0}, α} be an edge minimizing a linear form L. Then the second condition of Corollary 2.5 is met. By Corollary 2.6, the first condition is also met (for some A depending only on α_{0}.) Therefore, there are only finitely many possibilities for α.

Corollary 2.8. The actions of V on the vertices and on the edges of S have only finitely many orbits.

Proof. Fix α_{0} ∈ 𝔞𝔟^{++}, and choose a positive real number A such that N(α_{0}^{−1}α) ≤ A for all vertices α of S. Now let α be a vertex of S. There exists u ∈ V such that αu ∈ α_{0}D_{V}. But it follows from Proposition 2.4 that the set of x ∈ α_{0}D_{V} such that N(α_{0}^{−1}x) ≤ A is bounded. So the set

T = {x ∈ 𝔞𝔟^{++} ∪ α_{0}D_{V} : N(α_{0}^{−1}x) ≤ A}

is finite, and we have shown that every vertex α is in the V -orbit of an element of T.

Moreover, every edge {α_{1}, α_{2}} is in the orbit of some edge of the form {α′_{1}, α′_{2}}, with α′_{1} ∈ T. By Corollary 2.7, there are only finitely many such edges. The desired statement on the edges of S follows.

Definition 2.9. Let S_{1}, …, S_{r} be subsets of 𝔞𝔟^{++} stable under V. A compatible family for (S_{1}, …, S_{r}) is an r-tuple (ϕ_{1}, …, ϕ_{r}), where ϕ_{i} is either a vertex or an edge of S_{i}, and there exists a linear form L which is minimized on every ϕ_{i}.

Proposition 2.10. Let S_{1}, …, S_{r} be subsets of 𝔞𝔟^{++} invariant under V. Then the set of compatible families for (S_{1}, …, S_{r}) is acted on by V, and this action has only finitely many orbits.

Proof. The existence of the action is apparent.

By Corollary 2.8, there are only finitely many choices for ϕ_{1} up to the action of V. Fix such a choice. By Corollary 2.6, there exist A_{2}, …, A_{r} such that, if α_{i} is a vertex of S_{i}, we have N(α_{1}^{−1}α_{i}) ≤ A_{i}. Let α_{0} be a vertex contained in ϕ_{1} (e. g. α_{0} = ϕ_{1} if ϕ_{1} is a vertex.) Similarly, let α be a vertex contained in ϕ_{i}. It now follows from Corollary 2.5 that there are only finitely many choices for α, whence only finitely many choices for ϕ_{i}. The proposition follows.

2.2 The main theorem

We maintain the definitions of 𝔞, 𝔟, 𝔠 from the previous section. Let f be a 𝔠-cusp form of level N, defined over an integral domain R. Let Spec R_{1}, …, Spec R_{m} be an open affine cover of Spec R such that each ring R_{k} admits an isomorphism

j_{k} : 𝔞 ⊗_{ℤ} R_{k} →^{∼} 𝔬 ⊗_{ℤ} R_{k}.

Let

ϵ : 𝔬/N𝔬 →^{∼} 𝔞/N𝔞

be an isomorphism.

Then f has a q-expansion at the cusp (𝔞, 𝔟, j_{k}, ϵ)/R_{k} of the form

f(Tate_{𝔞,𝔟}(q), λ_{can}, ω(j_{k}), ι(ϵ)) = ∑_{α∈(𝔞𝔟)^{++}} a_{k;α}q^{α}.

Define a set

S_{𝔞,ϵ}(f) = {α ∈ 𝔞𝔟^{++} : a_{k;α} ≠ 0}.

This set is evidently independent of the choice of k and j_{k}. By Proposition 1.18, S_{𝔞,ϵ}(f) is invariant under the action of units of the form v^{2}, where v ∈ U is congruent to 1 mod N and N_{ℚ}^{E}(v) = 1. Such units form a subgroup V of finite index in U^{++}. Thus, the definitions and results in the previous section may be applied to S_{𝔞,ϵ}(f).

Definition 2.11. Let f, R, ϵ be as above. We say f has property (U) at (𝔞, ϵ) if a_{k;α} ∈ R_{k}^{∗} for all k and all vertices α of S_{𝔞,ϵ}(f).

Suppose ϕ is a vertex or edge of a set S ⊂ 𝔞𝔟^{++}, where 𝔞 and 𝔟 are defined as in the previous section. Define

δ(ϕ) = \{ \begin{matrix}α & ϕ is a vertex α \\ α − α′ & ϕ is an edge \{α, α′\} \end{matrix}

The ambiguity in sign in the latter case will not concern us.

Definition 2.12. Suppose (f_{1}, …, f_{r}) is an r-tuple of 𝔠-cusp forms such that the following properties hold.

• f_{m} has property (U) at (𝔞, ϵ), for all m in {1 … r}.

• Suppose (ϕ_{1}, …, ϕ_{r}) is a compatible family for (S_{𝔞,ϵ}(f_{1}), …, S_{𝔞,ϵ}(f_{r})). Then the values (δ(ϕ_{1}), …, δ(ϕ_{r})) generate 𝔞𝔟 ⊗_{ℤ} ℚ as a ℚ-vector space.

Then we say {f_{1}, …, f_{r}} is a discriminantal set of cusp forms at (𝔞, ϵ). If {f_{1}, …, f_{r}} is a discriminantal set for every choice of 𝔞 and ϵ, we say simply that {f_{1}, …, f_{r}} is a discriminantal set of cusp forms.

Remark 2.13. To check whether a given {f_{1}, …, f_{r}} is a discriminantal set is a finite computation, since by Proposition 2.10 there are only finitely many V - orbits of compatible families for (S_{𝔞,ϵ}(f_{1}), …, S_{𝔞,ϵ}(f_{r})). (Here V is chosen so that multiplication by u ∈ V stabilizes S_{𝔞,ϵ}(f_{i}) for all i.)

We are now ready for our main results.

Let {f_{1}, …, f_{r}} be an arbitrary set of modular forms. For each compatible family ϕ = (ϕ_{1}, …, ϕ_{r}) for (S_{𝔞,ϵ}(f_{1}), …, S_{𝔞,ϵ}(f_{r})), we can define

Mϕ = ⨁_{i} ℤδ(ϕ_{i}) ⊂ 𝔞𝔟.

Let Mϕ^{∨} ⊃ 𝔡^{−1}𝔞^{−1}𝔟^{−1} be the ℤ-dual of Mϕ.

Theorem 2.14. Let {f_{1}, …, f_{r}} be a set of 𝔠-Hilbert modular forms of level N over an integral domain R. Suppose

• A is a complete dvr with an R-algebra structure and fraction field K;

• (𝔞, 𝔟, j, ϵ) is a cusp over A;

• q : 𝔞𝔟 → 𝔾_{m}(K) is a positive homomorphism.

Let X/K be the HBAV

(𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m})/q_{𝔞}(𝔟),

and let n be a rational integer such that

n|ord(f_{i}(X, λ_{can}, ω(j), ι(ϵ)))

for all i ∈ 1, …, r. Then

ord(q) ∈ nMϕ^{∨}

for some compatible family ϕ for (S_{𝔞,ϵ}(f_{1}), …, S_{𝔞,ϵ}(f_{r})).

Proof. Choose one of the R_{k} such that the image of Spec A in Spec R lies in Spec R_{k}. Without loss of generality, suppose R = R_{k}.

We have

f_{i}(X, λ_{can}, ω(j), ι(ϵ)) = ∑_{α∈S_{𝔞,ϵ}(f_{i})} a_{α}q(α)

where the Fourier coefficients a_{α} are understood to be base changed from R to A.

Let L : 𝔞𝔟 → ℤ be the positive linear form such that x_{L} = ord(q). Let ϕ_{i} be a vertex or edge of S_{𝔞,ϵ}(f_{i}) minimizing L; then (ϕ_{1}, …, ϕ_{r}) is a compatible family for (S_{𝔞,ϵ}(f_{1}), …, S_{𝔞,ϵ}(f_{r})).

Suppose ϕ_{i} is a vertex. Then ord(a_{ϕ_{i}}) = 0, by property (U), and

ord(q(ϕ_{i})) = L(ϕ_{i}) < L(α) = ord(q(α))

for all α ∈ S_{𝔞,ϵ}(f_{i}) − {ϕ_{i}}. So

ord(f_{i}(X, λ_{can}, ω(j), ι(ϵ))) = L(ϕ_{i}) = L(δ(ϕ_{i})).

Now suppose ϕ_{i} is an edge {α, α′}. Then L(α) = L(α′), and we have L(δ(ϕ_{i})) = 0.

We have shown that n|L(δ(ϕ_{i})) for all i. Therefore, n|Tr(α · ord(q)) for all α ∈ Mϕ. The proposition follows.

Corollary 2.15. Let {f_{1}, …, f_{r}} be a discriminantal set of 𝔠-Hilbert modular forms over level N over an integral domain R. Then there exists an integer m_{{f_{i}}} such that, for every choice of A, n, q, X as above,

ord(q) ∈ n′𝔡^{−1}𝔞^{−1}𝔟^{−1}

where n′ is the numerator of n/m_{{f_{i}}} expressed in lowest terms.

Proof. Since {f_{1}, …, f_{r}} is discriminantal, every Mϕ is a finite-index subgroup of 𝔞𝔟. So we can choose m_{{f_{i}}} such that

m_{{f_{i}}}𝔞𝔟 ⊂ Mϕ

for each compatible family ϕ. The result then follows immediately from Theorem 2.14.

The following corollary is an HBAV analogue of Corollary 1.2.

Corollary 2.16. Let {f_{1}, …, f_{r}} be a discriminantal set of 𝔠-Hilbert modular forms of level N over an integral domain R. Suppose

• A is a complete dvr with an R-algebra structure and fraction field K;

• X/K is an HBAV with multiplicative reduction over A, endowed with a 𝔠polarization λ, a Néron non-vanishing differential ω, and a Néron N-level structure ι.

Let n be an integer such that

n|ord(f_{i}(X, λ, ω, ι))

for all i ∈ 1, …, r, and let n′ be the numerator of n/m_{{f_{i}}}, expressed in lowest terms. Then the group scheme X[n′]/K extends to a finite flat group scheme over A.

Proof. First of all, after an etale extension we may assume the reduction to be split multiplicative.

Now, by Theorem 1.19, we have

X ≅ (𝔡^{−1}𝔞^{−1} ⊗_{ℤ} 𝔾_{m})/q_{X;𝔞}(𝔟),

as 𝔠-polarized HBAV’s, for some positive homomorphism q_{X} and some choice of 𝔞, 𝔟 with 𝔟 = 𝔞𝔠^{−1}. By Proposition 1.23, there is a cusp (𝔞, 𝔟, j, ϵ) over A such that ι = ι(ϵ) and ω = ω(j).

We are now in the situation of Corollary 2.15. The desired result follows from Proposition 1.25.

Remark 2.17. For any particular O and f_{1}, …, f_{r}, one can carry out a finer analysis. If the compatible family at which the linear form Tr(α · ord(q_{X})) is minimized is known, then the ideal ord(q_{X})O can be calculated exactly in terms of the values of the ord(f_{i}). Since there are, up to units, only finitely many possibilities for the compatible family, there are only finitely many possibilities for ord(q_{X})O given the values ord(f_{i}). We will give examples of such computations in section 2.4.

2.3 Existence of discriminantal sets

Since 𝔞𝔟⊗_{ℤ} ℚ is a ℚ-vector space of dimension d, a discriminantal set must contain at least d modular forms. In this section we will show that a discriminantal set of this cardinality actually exists.

We begin by recalling some facts about HBAV’s over ℂ. A lattice L in E ⊗_{ℚ} ℂ is a locally free rank 2 O-submodule, and a 𝔠-polarization for L is an isomorphism

Λ : ∧_{O}^{2}L →^{∼} 𝔡^{−1}𝔠^{−1}

which is positive in the following sense: there is an A ∈ (E ⊗_{ℚ} ℝ)^{++} such that

Λ(u, v) = Im(ūv)/A.

There is a bijective correspondence between 𝔠-polarized lattices (L, Λ) ∈ E⊗_{ℚ}ℂ and triples (X/ℂ, λ, ω), where X is an HBAV, λ a 𝔠-polarization, and ω a non-vanishing differential [10, (1.4.6)]. An N-level structure ι on X corresponds to an injection

I : (1/N)𝔡^{−1}/𝔡^{−1} ↪ (1/N)L/L.

Now let f be a modular form in M_{O}^{k}(𝔠, 1; ℂ), and let ν be a totally positive element of O. Suppose (L, Λ, I) is a 𝔠-polarized lattice with N-level structure, ν|N. Choose ℓ′ in the image of I such that νℓ′ = 0. Let L′ be the lattice generated by L and ℓ′; note that L′ is independent of the choice of ℓ′. Then extending Λ by bilinearity yields a 𝔠-polarization Λ′ for L′. Now define a function B_{ν}f on the space of 𝔠-polarized lattices with N-level structure by the rule

B_{ν}f(L, Λ, I) = ν^{−k}f(L′, Λ′).

Then B_{ν}f lies in M_{O}^{k}(𝔠, N; ℂ) by [10, (1.6.3)]. If the q-expansion of f at a cusp (𝔞, 𝔟, j, ϵ_{1}) is

a_{0} + ∑_{α∈(𝔞𝔟)^{++}} a_{α}q^{α},

it follows from [10, (1.7.6)] that the q-expansion of B_{ν}f at (𝔞, 𝔟, j, ϵ) is

a_{0} + ∑_{α∈(𝔞𝔟)^{++}} a_{α}q^{να}

for any choice of ϵ.

Suppose f is in M_{O}^{k}(𝔠, 1; R), where R is a ring with an injection R ↪ ℂ. Then we can define B_{ν}f as a modular form over ℂ, which then descends to M_{O}^{k}(𝔠, N; R) by the q-expansion principle [10, (1.2.16)].

Note that B_{ν}f has property (U) if and only if f does, and S_{𝔞,ϵ}(B_{ν}f) = νS_{𝔞,ϵ_{1}}(f) for all 𝔞, ϵ.

Theorem 2.18. Let 𝔠 be a fractional ideal of O. Then there exists an integer m, a 𝔠-cusp form f of level 1, and a d-tuple (ν_{1}, …, ν_{d}) of elements of O^{++} such that (B_{ν_{1}}f, …, B_{ν_{d}}f) is a discriminantal set of modular forms over ℤ[1/m].

Proof. First, there exists a 𝔠-cusp form f of level 1 defined over ℤ. This follows, for instance, from [23, IV,4.4].

Let {f_{1}, …, f_{r}}, r < d, be a set of modular forms such that

(P) for each choice of (𝔞, ϵ), and for each compatible family (ϕ_{1}, …, ϕ_{r}) for (S_{𝔞,ϵ}(f_{1}), …, S_{𝔞,ϵ}(f_{r})), the values (δ(ϕ_{1}) … δ(ϕ_{r})) are linearly independent over ℚ.

Note that a set of d modular forms with property (P) and property (U) is a discriminantal set.

Let V be a finite-index subgroup of U^{++} under which S_{𝔞,ϵ}(f_{i}) is invariant for all 𝔞, ϵ, and i. Let D_{V} ∈ (𝔡^{−1} ⊗_{ℤ} ℝ)^{++} →^{∼} (E ⊗_{ℤ} ℝ)^{++} be the be the region defined in Proposition 2.4. We wil show there is some ν in D_{V} ∩ O^{++} such that (f_{1}, …, f_{r}, B_{ν}f_{1}) satifies (P).

Suppose ν is an element of D_{V} ∩ O^{++} such that {f_{1}, …, f_{r}, B_{ν}f_{1}} does not satisfy (P). Then there exists a compatible family {ϕ_{1}, …, ϕ_{r}, ψ} for some

(S_{𝔞,ϵ}(f_{1}), …, S_{𝔞,ϵ}(f_{r}), S_{𝔞,ϵ}(B_{ν}f_{1}))

such that the values (δ(ϕ_{1}), …, δ(ϕ_{r}), ψ) are not linearly independent over ℚ.

Let β be a vertex belonging to ψ, and for each i let α_{i} be a vertex belonging to ϕ_{i}. Since ψ and ϕ_{1} simultaneously minimize some linear form, we know as in the proof of Corollary 2.5 that

β ∈ α_{1}(D_{V})^{−1}D_{V} .

We have shown above that S_{𝔞,ϵ}(B_{ν}f_{1}) = νS_{𝔞,ϵ}(f_{1}). In particular, ψ = νϕ′_{1} for some vertex or edge ϕ′_{1} of S_{𝔞,ϵ}(f_{1}). Let α′_{1} be the vertex belonging to ϕ′_{1} such that β = να′_{1}. Then

α′_{1} = βν^{−1} ∈ α_{1}(D_{V})^{−1}D_{V} (D_{V})^{−1}.

Now, by assumption, δ(ψ) = νδ(ϕ′_{1}) lies in the ℚ-vector space spanned by {δ(ϕ_{1}) … δ(ϕ_{r})}. Equivalently, ν lies in the ℚ-vector space W ⊂ O⊗_{ℤ}ℚ spanned by

{δ(ϕ_{1})δ(ϕ′_{1})^{−1} … δ(ϕ_{r})δ(ϕ′_{1})^{−1}}.

We have shown in Proposition 2.10 that there are only finitely many choices, up to the action of V, for (ϕ_{1}, …, ϕ_{r}). Suppose such a choice is fixed. Let A ∈ ℝ be chosen such that N(α_{1}^{−1}α) ≤ A for all vertices α of S_{𝔞,ϵ}(f_{1}). By Proposition 2.4,

{x ∈ (D_{V})^{−1}D_{V} (D_{V})^{−1} : N(α^{−1}x) < A}

is bounded; therefore, there are only finitely many choices for α′_{1}, whence only finitely many choices for ϕ′_{1}. There are also only finitely many choices for ϵ and the isomorphism class of 𝔞. Putting this all together, we have found that the set of ν for which (f_{1}, …, f_{r}, B_{ν}f_{1}) does not have property (P) is the intersection of D_{V} ∩ O^{++} with a finite union of subspaces of O ⊗_{Z} ℚ of dimension i < d. Thus we can choose ν such that {f_{1}, …, f_{r}, B_{ν}f_{1}} satisfies (P), as claimed.

Now proceed by induction. Evidently the set consisting solely of f satisfies (P), so we can produce a set of d cusp forms satisfying (P), defined over ℤ. By passing to some ℤ[1/m], we can ensure that f_{1}, whence every B_{ν}f_{1}, has property (U). We have now produced the desired discriminantal set of modular forms.

2.4 Examples

In this section, we will consider the special cases O = ℤ[\frac{1+\sqrt{5}}{2}] and O = ℤ[\sqrt{2}]. In each case, we give an explicit discriminantal set of modular forms of level 1. The following proposition, a more explicit form of Corollary 2.6 in case d = 2, will be useful in the computations to follow.

Proposition 2.19. Let O be the ring of integers of a real quadratic field. Let 𝔞 be a projective rank 1 O-module, 𝔠 a fractional ideal of O, and let 𝔟 = 𝔞𝔠^{−1}. Finally, let S ⊂ 𝔞𝔟^{++} be invariant under the action of V, a finite-index subgroup of U^{++} generated by v.

Let α_{0} be an element of S, and let α be a vertex of S. Then

N(α_{0}^{−1}α) < (1/4)(2 + Tr(v)).

Proof. Let ρ_{1}, ρ_{2} be the two embeddings of O into ℝ. Without loss of generality, suppose

ρ_{2}(v) > 1 > ρ_{1}(v).

Any element β of (𝔞𝔟 ⊗_{ℤ} ℚ)^{++} can be written uniquely as

β = x_{1} + x_{2}v^{−1},

with x_{1}, x_{2} ∈ ℚ. We know that

ρ_{2}(v)/ρ_{1}(v) > 1
ρ_{2}(1)/ρ_{1}(1) = 1
ρ_{2}(v^{−1})/ρ_{1}(v^{−1}) < 1.

So ρ_{2}(β)/ρ_{1}(β) < 1 if and only if x_{2} is positive. Likewise, ρ_{2}(vβ)/ρ_{1}(vβ) > 1 if and only if x_{1} is positive.

Now let β_{n} = v^{n}α_{0}^{−1}α. For large n, we have ρ_{2}(β_{n})/ρ_{1}(β_{n}) > 1, while for small n, the opposite is the case. So we may choose n such that

ρ_{2}(β_{n})/ρ_{1}(β_{n}) < 1 < ρ_{2}(β_{n+1})/ρ_{1}(β_{n+1}).  (2.4.5)

Now write

β_{n} = x_{1} + x_{2}v^{−1};

by (2.4.5), x_{1} and x_{2} are positive. Now if x_{1} + x_{2} > 1, then α_{0}β_{n} = v^{n}α is in the interior of the positive half-plane bounded by the line through α_{0} and v^{−1}α_{0}, and thus cannot be a vertex.

We conclude that

β_{n} = x_{1} + x_{2}v^{−1}, x_{1}, x_{2} ∈ ℚ_{>0}, x_{1} + x_{2} ≤ 1;

the maximal norm on this region is (1/4)(2 + Tr(v)), attained when x_{1} = x_{2} = 1/2.

Now let O = ℤ[\frac{1+\sqrt{5}}{2}], and let 𝔠 = 𝔡. Recall from Example 1.11 the cusp forms χ_{6} and χ_{12}, defined over ℤ, of level 1.

Proposition 2.20. {χ_{6}, χ_{12}} is a discriminantal set of Hilbert modular forms over ℤ, with m_{{χ_{6},χ_{12}}} = 2.

Proof. Since the class group of O is trivial, we may set 𝔞 = 𝔬 and 𝔟 = 𝔡^{−1}. Since the level of both forms is 1, we may set ϵ = ϵ_{1}. We therefore suppress subscripts and write S(χ_{i}) for S_{𝔞,ϵ}(χ_{i}). Finally, take j to be the identity map on 𝔬. The q-expansions of χ_{6} and χ_{12} can be

The computation of the q-expansions of χ_{6} and χ_{12} at (𝔬, 𝔡^{−1}, j, ϵ_{1}) is straightforward. First of all, it suffices to compute the Fourier expansions of χ_{6} and χ_{12} as holomorphic Hilbert modular forms [10, (1.7.6)]. Then one can use either the well-known expressions of χ_{6} and χ_{12} in terms of Eisenstein series or the method of Resnikoff [19] to compute these Fourier coefficients. One finds that

χ_{6} = q^{\frac{1}{2} − \frac{\sqrt{5}}{10}} + q^{\frac{1}{2} + \frac{\sqrt{5}}{10}} + q^{1− \frac{2\sqrt{5}}{5}} + 20q^{1− \frac{\sqrt{5}}{5}}
−90q + 20q^{1+ \frac{\sqrt{5}}{5}} + q^{1+ \frac{2\sqrt{5}}{5}} + …

and

χ_{12} = q + …,

where in each case the terms a_{α}q^{α)} with Tr(α) > 2 are omitted.

Since both forms have even weight, S(f_{i}) is invariant under all of U^{2} = U^{++} by Proposition 1.18. This group is generated by u = \frac{3+\sqrt{5}}{2}.

It follows from Proposition 2.19 that every vertex of S(χ_{6}) is an element of (1/2 − \sqrt{5}/10)U^{++}. Similarly, every vertex of S(χ_{12}) lies in U^{++}. From there, it is a simple matter to verify that the compatible families (ϕ_{1}, ϕ_{2}) for (S(χ_{6}), S(χ_{12})) split into the following four orbits under U^{++}:

(1/2 − \sqrt{5}/10, 1)
(1/2 − \sqrt{5}/10, 3/2 − \sqrt{5}/2)
(1/2 − \sqrt{5}/10, \{1, 3/2 − \sqrt{5}/2\})
(\{1/2 − \sqrt{5}/10, 1/2 + \sqrt{5}/10\}, 1)

Each of these families constitutes a ℚ-basis for ℚ[\sqrt{5}]. By inspection of the qexpansions above, χ_{6} and χ_{12} have property (U). So {χ_{6}, χ_{12}} is a discriminantal set. Since the ℤ-span of δ(ϕ_{1}) and δ(ϕ_{2}) contains 2𝔡^{−1} for each family above, we have m_{χ_{6},χ_{12}} = 2.

Corollary 2.21. Let X/K be a 𝔠-polarized HBAV with real multiplication by O = ℤ[\frac{1+\sqrt{5}}{2}] and multiplicative reduction, λ its polarization, and ω a Néron non-vanishing differential. Let n be an integer such that

n | ord(χ_{6}(X, λ, ω, ι_{1}))
n | ord(χ_{12}(X, λ, ω, ι_{1})).

Then the group scheme X[n′]/K extends to a finite flat group scheme over A, where n′ is the numerator of n/2 expressed in lowest terms.

Proof. Immediate from Proposition 2.20 and Corollary 2.16.

Proposition 2.22. Let X, λ, ω be as in Corollary 2.21, and set

• n_{6} = ord(χ_{6}(X, λ, ω, ι_{1}));

• n_{12} = ord(χ_{12}(X, λ, ω, ι_{1})).

Suppose furthermore that X has split multiplicative reduction, so that X is uniformized as

X ≅ (𝔡^{−1} ⊗_{ℤ} 𝔾_{m})/q_{X;𝔬}(𝔡^{−1}).

Then ord(q_{X}) is one of the values

−\sqrt{5}n_{6} + [(1 + \sqrt{5})/2]n_{12},
\sqrt{5}n_{6} + [(1 − \sqrt{5})/2]n_{12},
[(5 + \sqrt{5})/4] n_{6},
(1/2)n_{12}

up to multiplication by a totally positive unit.

Proof. We divide the proposition into three cases. Let L : 𝔡^{−1} → ℤ be the linear form such that L(α) = Tr(ord(q_{X})α).

Case 1: ord(q_{X}) is a rational integer, up to multiplication by a totally positive unit.

Without loss of generality, we may take ord(q_{X}) to be a rational integer m. It follows that L is minimized at the vertex α = 1 of S(χ_{12}). So

n_{12} = ord(χ_{12}(X, λ, ω, ι_{1})) = Tr(ord(q_{X})) = 2m.

Case 2: ord(q_{X}) = (\frac{5}{2} + \frac{\sqrt{5}}{2})mu, where m is a rational integer and u is a totally positive unit.

Again, we may assume ord(q_{X}) = (\frac{5}{2} + \frac{\sqrt{5}}{2})m. Then L is minimized at the vertex \frac{1}{2} − \frac{\sqrt{5}}{10} of S(χ_{6}). So

n_{6} = ord(χ_{6}(X, λ, ω, ι_{1})) = Tr ((\frac{1}{2} − \frac{\sqrt{5}}{10})(\frac{5}{2} + \frac{\sqrt{5}}{2}) m) = 2m.

Case 3: Neither Case 1 nor Case 2 obtains.

One then checks that L is not minimized on any edge of S(χ_{6}) or S(χ_{12}). Thus, L is minimized on some vertex α_{6} of S(χ_{6}) and on some vertex α_{12} of S(χ_{12}). By the list of compatible families for (S(χ_{6}), S(χ_{12})) given in the proof of Proposition 2.20, this implies that, after modification by a totally positive unit, (α_{6}, α_{12}) is either (1/2 − \sqrt{5}/10, 1) or (1/2 − \sqrt{5}/10, 3/2 − \sqrt{5}/2). Suppose (α_{6}, α_{12}) = (1/2 − \sqrt{5}/10, 1). We then have

n_{6} = ord(χ_{6}(X, λ, ω, ι_{1})) = Tr ((\frac{1}{2} − \frac{\sqrt{5}}{10}) · ord(q_{X}))
n_{12} = ord(χ_{12}(X, λ, ω, ι_{1})) = Tr(ord(q_{X})).

It follows that

ord(q_{X}) = −\sqrt{5}n_{6} + [(1 + \sqrt{5})/2]n_{12}.

Observe that (1/2 − \sqrt{5}/10, 3/2 − \sqrt{5}/2) differs by a totally positive unit from (1/2 + \sqrt{5}/10, 1). Assuming that (α_{6}, α_{12}) = (1/2 + \sqrt{5}/10, 1) then yields

ord(q_{X}) = \sqrt{5}n_{6} + [(1 − \sqrt{5})/2]n_{12}.

The desired result is now proven.

Example 2.23. Let X/ℚ be the Jacobian of the genus 2 curve

C : y^{2} = f(x) = (x^{3} − 4x^{2} − 3x − 1)(x^{3} + x + 3).

Then X is an HBAV with real multiplication by ℤ[\frac{1+\sqrt{5}}{2}], since it is the specialization to b = −4, c = −2, d = −1 of Brumer’s family (1.2.2). The only primes dividing the discriminant of f are 2, 13 and 19, so X has good reduction away from these primes.

The reduction of the given model of C at 13 is the stable curve

y^{2} = (x − 2)^{2}(x − 4)^{2}(x − 9)^{2},

so X has semistable reduction at 13. We can compute the values of the Hilbert modular forms χ_{6} and χ_{12} at X as follows. First, we use Igusa’s formulas in [9] to compute the values of Siegel modular functions in terms of the projective invariants of the sextic f. Then, we use Resnikoff’s calculations in [19] to compute the values of Hilbert modular functions of weight 0. (Note that in the weight 0 case, we need not specify a choice of ω.) We find

χ_{6}ϕ_{2}^{−3}(X, λ) = 5 · 13^{2} · 19 · 79^{−3}
χ_{12}ϕ_{2}^{−6}(X, λ) = 2 · 13^{5} · 19^{2} · 79^{−6}

where ϕ_{2} is the modular form over ℤ described in Example 1.11, and λ is the unique polarization (up to isomorphism) on X. One can compute that every q-expansion of ϕ_{2} has nonzero constant coefficient [19]; thus,

ord_{13}(ϕ_{2}(X, λ, ω, ι_{1})) = 0

for every Néron non-vanishing differential on X. So

ord_{13}(χ_{6}(X, λ, ω, ι_{1})) = 2
ord_{13}(χ_{12}(X, λ, ω, ι_{1})) = 5.

Let q_{X} be the Tate parameter of X/ℤ_{13}. It follows from Proposition 2.22 that

ord_{13}(q_{X}) = (5 + \sqrt{5})/2.  (2.4.6)

The model

C′ : y^{2} + y(x^{3} + x + 1) = −x^{5} − x^{4} − x^{3} − 4x^{2} − 3x − 1

for C is smooth over ℤ_{2}, so X has good reduction at 2. Also, one observes that X is semistable at 19.

Suppose the Galois representation on X[\sqrt{5}]

ρ : Gal(ℚ̄/ℚ) → GL_{2}(𝔽_{5})

is irreducible. Then, by [21, Th. 4.2], X is modular. There is a unique modular form f of weight 2 and level 13 · 19 = 247 whose Fourier coefficients are defined over ℚ[\sqrt{5}]^{1}; therefore, X is isogenous to the corresponding factor X_{f} of J_{0}(247).

1This can be seen from the tables of Fourier coefficients recently produced by William Stein, available at http://www.math.berkeley.edu/~was.

Moreover, it follows from (2.4.6) that ρ is unramified at 13; thus, ρ has conductor 19. So ρ must be the Galois representation associated to g, where g is the unique cuspform of weight 2 and level 19. But one observes by comparing Fourier coefficients that f and g are not congruent mod 5. Therefore, ρ is reducible, which is to say that X[\sqrt{5}] possesses a rational cyclic subgroup of order 5. By counting points over finite fields we can verify that the semisimplification of ρ is the direct product of the trivial character with the cyclotomic character; in other words, either X or an HBAV \sqrt{5}-isogenous to X contains a rational \sqrt{5}-torsion point.

Now suppose O = ℤ[\sqrt{2}]. Once again, the narrow class group is trivial, so we may set 𝔠 = 𝔡. There are 𝔡-cusp forms over ℂ of level 1 and weights 4 and 6 ([8]), which were shown by Nagaoka [15] to descend to forms χ_{4} and χ_{6} over ℤ.

Proposition 2.24. {χ_{4}, χ_{6}} is a discriminantal set of modular forms over ℤ, with m_{{χ_{4},χ_{6}}} = 2.

Proof. As in the proof of Proposition 2.20, we may set 𝔞 = 𝔬, 𝔟 = 𝔡^{−1}, j = id, and ϵ = ϵ_{1}, and refer simply to S(χ_{4}) and S(χ_{6}). Müller [13] gave expressions for χ_{4} and χ_{6} in terms of theta functions, from which we can derive the q-expansions at (𝔞, 𝔟, j, ϵ)

χ_{4} = q^{1/2−\sqrt{2}/4} − 2q^{1/2} + q^{1/2−\sqrt{2}/4}
− 4q^{1−\sqrt{2}/2} − 8q^{1−\sqrt{2}/4} + 24q − 8q^{1+\sqrt{2}/4} − 4q^{1+\sqrt{2}/2} + …
χ_{6} = q^{1/2} + 2q^{1−\sqrt{2}/2} − 176q^{1−\sqrt{2}/4} − 684q − 176q^{1+\sqrt{2}/4} + 2q^{1+\sqrt{2}/2} + …

The omitted terms are of the form a_{α}q^{α} with Tr(α) > 2.

Both S(χ_{4}) and S(χ_{6}) are invariant under the action of U^{++}, by Proposition 1.18. It follows from Proposition 2.19 that the vertices and edge-elements of S(χ_{4}) are the elements of (1/2−\sqrt{2}/4)U^{++}, and the vertices and edge-elements of S(χ_{6}) the elements of (1/2)U^{++}. It is then immediate that χ_{4} and χ_{6} have property (U). The compatible families (ϕ_{1}, ϕ_{2}) for (S(χ_{4}), S(χ_{6})) split into the following four orbits under the action of U^{++};

(1/2 − \sqrt{2}/4, 1/2)
(1/2 − \sqrt{2}/4, 3/2 − \sqrt{2})
(1/2 − \sqrt{2}/4, \{3/2 − \sqrt{2}, 1/2\})
(\{1/2 − \sqrt{2}/4, 1/2 + \sqrt{2}/4\}, 1/2)

In each case, (δ(ϕ_{1}), δ(ϕ_{2})) is a ℚ-basis for 𝔡^{−1} ⊗_{ℤ} ℚ, and the ℤ-span of δ(ϕ_{1}) and δ(ϕ_{2}) contains 2𝔡^{−1}.

Corollary 2.25. Let X/K be a 𝔠-polarized HBAV with real multiplication by O and multiplicative reduction, λ its polarization, and ω a Néron non-vanishing differential. Let n be an integer such that

n | ord(χ_{4}(X, λ, ω, ι_{1}))
n | ord(χ_{6}(X, λ, ω, ι_{1})).

Then the group scheme X[n′]/K extends to a finite flat group scheme over A, where n′ is the numerator of n/2 expressed in lowest terms. Proof. Immediate from Proposition 2.24 and Corollary 2.16.

From our description of the compatible families for {χ_{4}, χ_{6}}, we produce an analogue of Proposition 2.22. Proposition 2.26. Let X, K, λ, ω′ be as above, and let

• n_{4}|ord(χ_{4}(X, λ, ω′, ι_{1}))

• n_{6}|ord(χ_{6}(X, λ, ω′, ι_{1})). Then ord(q_{X}) is one of the values

−\sqrt{2}n_{4} + (1 + \sqrt{2})n_{6},

(4 + 3\sqrt{2})n_{4} + (−1 − \sqrt{2})n_{6},
(2 + \sqrt{2})n_{4},
n_{6}

up to multiplication by a totally positive unit. Proof. The proof is a computation exactly analogous to that in Proposition 2.22.

References

[1] S. Bosch, W. Lütkebohmert, and M. Raynaud. Nèron models. Springer-Verlag, 1990.

[2] A. Brumer. Personal communication.

[3] J. Ellenberg. Hilbert modular forms and the Galois representations associated to Hilbert-Blumenthal abelian varieties. PhD thesis, Harvard, 1998.

[4] J. Ellenberg and C. Skinner. On the modularity of ℚ-curves. In preparation.

[5] G. Faltings and C.-L. Chai. Degeneration of Abelian Varieties. Springer-Verlag, 1990.

[6] A. Grothendieck. Elements de geometrie algebrique, II. Publ. Math. IHES, 8, 1961.

[7] A. Grothendieck. Elements de geometrie algebrique, III. Publ. Math. IHES, 11, 1961.

[8] F. Hirzebruch. The ring of Hilbert modular forms for real quadratic fields of small discriminant. In Modular Functions of One Variable VI, number 627 in Lecture Notes in Mathematics, pages 288–323. Springer-Verlag, 1977.

[9] J.-I. Igusa. Modular forms and projective invariants. Am. J. Math., 89:817– 855, 1967.

[10] N. Katz. p-adic L-functions for CM fields. Invent. Math., 49:199–297, 1978.

[11] N. Katz and B. Mazur. Arithmetic moduli of elliptic curves. Princeton University Press, 1985.

[12] J. S. Milne. Abelian varieties. In G. Cornell and J. Silverman, editors, Arithmetic Geometry, pages 103–150, Springer-Verlag, 1986.

[13] R. Müller. Hilbertsche Modulformen und Modulfunktionen zu ℚ[\sqrt{8}]. Math. Ann., 266(1):83–103, 1983.

[14] D. Mumford. An analytic construction of degenerating abelian varieties over complete rings. Compositio Math., 24:239–272, 1972.

[15] S. Nagaoka. On the ring of Hilbert modular forms over Z. J. Math. Soc. Japan, 35:547–555, 1983.

[16] M. Rapoport. Compactifications de l’espace de modules de Hilbert-Blumenthal. Compositio Math., 36:255–335, 1978.

[17] M. Raynaud. Passage au quotient par une relation d’èquivalence plate.

[18] M. Raynaud. Variétés abéliennes et géométrie rigide. Actes, Congres. Intern. Math., pages 473–477, 1970.

[19] H. L. Resnikoff. On the graded ring of Hilbert modular forms associated with ℚ(\sqrt{5}). Mathematische Annalen, 208:161–170, 1974.

[20] J.P. Serre. Sur les représentations modulaires de degré 2 de Gal(ℚ̄/ℚ). Duke Math. Jour., 54(1):179–230, 1987.

[21] N. Shepherd-Barron and R. Taylor. Mod 2 and mod 5 icosahedral representations. J. Amer. Math. Soc, 10(2):283–298, 1997.

[22] J. Tate. A review of non-archimedean elliptic functions. In J. Coates and S.- T. Yau, editors, Elliptic curves, modular forms, and Fermat’s last theorem, pages 162–184, 1995.

[23] G. van der Geer. Hilbert modular surfaces. Springer-Verlag, 1988.