On Katz Modular Forms

Gabor Wiese^{∗}

November 9, 2003

Abstract

These are slightly revised notes of a series of four talks given at the local geometry seminar in Leiden in December 2002 and January 2003. The subtitle “preliminary version” has vanished in this update, not because this text is in any form finished or complete, but because I do not intend to change it any further. For, since I learned some things since the first version, I feel that these notes actually ought to be completely reworked. However, since I got some positive feed-back on them, I decided to put them back on the web with only minor changes and corrections.

In the notes I take a very naive point of view on Katz modular forms, which is certainly at some points too naive. So certainly there are still some incorrect statements to be found.

In the first talk, the definition of Katz modular forms was given and motivated by showing that over the complex numbers they coincide with the well-known modular forms.

The second talk, consisting of two parts, presented a theorem due to Bas Edixhoven, which provides a method for the calculation of the space of cuspidal Katz modular forms of weight 1 over finite fields.

In the final talk, the so-called Serre conjecture in an extended form was presented, which - among other things - predicts the existence of a Katz eigenform of weight 1 in characteristic p belonging to a given irreducible odd representation ρ : G_{ℚ} → GL_{2}(\overline{𝔽_{p}}), which is unramified at p. Moreover, the results of some calculations of weight 1 modular forms over \overline{𝔽_{2}} (originally performed by Mestre) were presented.

These notes owe even their mere existence to long explanations by Bas Edixhoven.

The attentive reader will, however, most likely find many inaccuracies and even mistakes, very probably due to the author, who would be grateful to be informed about them.

∗gabor@math.leidenuniv.nl

Contents

1 Definition of Katz modular forms 2 1.1 The level structure diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Classical M-forms as invariant holomorphic sections . . . . . . . . . . . . . . . 7 1.3 Classical M-forms as invariant holomorphic differentials . . . . . . . . . . . . . 7 1.4 The case of Γ acting freely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Towards an algebraic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Definition of Katz M-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 q-expansions and Katz modular forms . . . . . . . . . . . . . . . . . . . . . . . 14 1.8 First properties of Katz modular forms . . . . . . . . . . . . . . . . . . . . . . . 17

2 Calculating characteristic p Katz modular forms of weight 1, after Bas Edixhoven 17

2.1 Different definitions of Katz modular forms and comparison with the classical

theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Parabolic cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Operators on modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Calculation of classical modular forms of weight k ≥ 2 from the Hecke algebra . 25 2.5 Calculation of Katz modular forms of weight 1 over finite fields . . . . . . . . . 28 2.6 Results on Γ_{0}(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Calculations of characteristic p modular forms of weight 1 and relation to Serre’s conjecture 34 3.1 Deligne’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Serre’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 Modularity of dihedral representations . . . . . . . . . . . . . . . . . . . . . . . 42 3.4 Explicit examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1 Definition of Katz modular forms

This talk was given on 10 December 2002.

My principal aim is to motivate and explain the idea behind the definition of Katz modular forms. We start with the classical theory and try to make the algebraisation appear naturally. Due to time limitations, I shall not be able to present but the most elementary properties of the Katz forms.

We will always consider the non-compactified version of the moduli spaces appearing and deal with the cusps separately. For to my mind the ideas then become more easily accessible. We shall thus naturally find functions transforming like modular forms, but that can even have essential singularities in the cusps. In this talk I will call them M-forms.

Since I have not (yet) read the original articles by Katz, I shall be careful always to speak of Katz modular forms and not of modular forms as defined by Katz. The reader should be aware that there might be differences.

The exposition is based on lecture notes and explanations by Bas Edixhoven.

1.1 The level structure diagram

In this section we will introduce a commutative diagram, which on the one hand encodes level structures, and on the other hand gives rise to M-forms as invariant holomorphic sections resp. differentials. The latter points will be dealt with in the following two sections.

By the level structure diagram I mean the following commutative diagram

\xymatrix{ℤ^{2} × ℍ \ar@{^{(}->}[r]^{(x,τ)↦(ϕ_{τ} (x),τ)} \ar@{->>}[dr]^{π} & ℂ × ℍ \ar@{->>}[r] \ar@{->>}[d]^{π} & 𝔼 \ar@{->>}[dl]^{π} \\  & ℍ,}

where the map ϕ_{τ} is defined by ( \begin{matrix}n \\ m \end{matrix} ) ↦ ( \begin{matrix}n \\ m \end{matrix} )^{T} ( \begin{matrix}τ \\ 1 \end{matrix} ) = nτ + m and π denotes the

obvious projection maps. We shall consider this diagram in the category of complex manifolds. Let us look at the fibre of a point τ ∈ ℍ under π, it is

0 → ℤ^{2} ⟶^{ϕ_{τ}} ℂ → E_{τ} → 0,

where E_{τ} = ℂ/Λ_{τ} with Λ_{τ} = ℤτ ⊕ ℤ. I.e. the fibre is an elliptic curve over ℂ, and we have kept track of the lattice, in a standard form, that gives rise to the curve.

Next we bring natural actions of the group SL_{2}(ℤ) into play. First of all, it (actually even the group SL_{2}(ℝ)) acts on the upper half plane ℍ by homographies, that is to say

γ.τ = \frac{aτ + b}{cτ + d}, with γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

We want to relate this action to the standard one on ℤ^{2}. First we note the obvious formula

(cτ + d) ( \begin{matrix}γ.τ \\ 1 \end{matrix} ) = γ ( \begin{matrix}τ \\ 1 \end{matrix} ) .

Furthermore, one immediately finds the commutative diagram

[figure: ℤ^{2}; γ^{T}; ϕ_{τ}; ℤ^{2}; ϕ_{γτ}; Λ_{τ}; ·(cτ+d); Λ_{γτ}.]

We will put these relations to two different uses. First, we define actions by the group SL_{2}(ℤ)

on the level structure diagram. So let γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) be a matrix in SL_{2}(ℤ). We make γ act on

ℤ^{2} × ℍ by γ.(( \begin{matrix}n \\ m \end{matrix} ) , τ) := (γ^{−1,T} ( \begin{matrix}n \\ m \end{matrix} ) , γ.τ) and on ℂ × ℍ by γ.(z, τ) := (\frac{z}{cτ+d}, γ.τ).

It is immediate to check, e.g. using the relations exhibitied above, that the left hand side of the level structure diagram is SL_{2}(ℤ)-equivariant. We transport the action to the right hand side, and could consequently pass to the quotient for any subgroup Γ < SL_{2}(ℤ). The quotient maps would also be analytic again. However, we avoid the use of quotients at this stage. Instead of speaking of a fibre of Γτ for the quotients, we can look at the family of exact sequences

0 → ℤ^{2} ⟶^{ϕ_{γτ}} ℂ → E_{γτ} → 0 for γ ∈ Γ.

We will now define subgroups of SL_{2}(ℤ) giving rise to families of elliptic curves having some common property related to their torsion groups.

For that it is convenient to consider such an exact sequence as a pair (E_{τ}, ϕ_{τ}) (here, of course, the second component determines the first one). We interpret ϕ_{τ} as the choice of a lattice basis.

Let N > 0 be an integer. The N-torsion group E_{τ}[N] of the elliptic curve E_{τ} is defined as the first term in the exact sequence

0 → \frac{1}{N} Λ_{τ}/Λ_{τ} → E_{τ} ⟶^{·N} E_{τ}.

The “choice of basis” isomorphism ϕ_{τ} descends to give the isomorphism

\overline{ϕ_{τ}} : (ℤ/N)^{2} → E_{τ}[N], x ↦ \frac{1}{N} ϕ_{τ}(x),

which should also be interpreted as a choice of basis of the torsion group. \overline{ϕ_{τ}} is called a level structure.

Let us now compare the exact sequence of τ with the one of γτ for γ ∈ SL_{2}(ℤ) as above, i.e. we want to relate the pair (E_{τ}, ϕ_{τ}) to (E_{γτ}, ϕ_{γτ}). From one of the diagrams above, we immediately obtain the commutative diagram of isomorphisms

[figure: (ℤ/N)^{2}; \overline{γ^{T}}; \overline{ϕ_{τ}}; (ℤ/N)^{2}; \overline{ϕ_{γτ}}; E_{τ}[N]; ·(cτ+d); E_{γτ}[N].]

We now consider three different families. We fix a τ ∈ ℍ.

• Let us define the group

Γ(N) := Ker(SL_{2}(ℤ) ↠ SL_{2}(ℤ/N)).

Thus Γ(N) is the normal subgroup of SL_{2}(ℤ) consisting of those matrices, the reduction modulo N of which is the unit matrix. Γ(N) gives rise to the family of the exact sequences represented by the pairs (E_{γτ}, ϕ_{γτ}) for γ ∈ Γ(N), which precisely have in common that the choice of basis of their torsion groups are the same (for the natural ismorphism between E_{τ} and E_{γτ}). Hence the family Γ(N)τ corresponds to the isomorphism class of a pair (E_{τ}, \overline{ϕ_{τ}}). By isomorphism of pairs we mean an isomorphism between the curves respecting the level structure, i.e. sitting in the commutative diagram

[figure: (ℤ/N)^{2}; \overline{ϕ_{τ}}; \overline{ϕ_{γτ}}; E_{τ}[N]; ·(cτ+d); E_{γτ}[N].]

• Next consider the group Γ_{1}(N) sitting in the cartesian diagram

[figure: Γ(N); SL_{2}(ℤ); SL_{2}(ℤ/N); □; Γ(N); Γ_{1}(N); Stab_{SL_{2}(ℤ/N)}(( \begin{matrix}1 \\ 0 \end{matrix} )).]

Thus, Γ_{1}(N) consists precisely of those matrices, the reduction modulo N of which is upper triangular with 1’s on the diagonal.

Consequently, Γ_{1}(N) gives rise to the family, where \overline{ϕ_{τ}}(( \begin{matrix}0 \\ 1 \end{matrix} )) = \overline{ϕ_{γτ}}(( \begin{matrix}0 \\ 1 \end{matrix} )). That

means that the natural isomorphism maps the point \frac{1}{N} of E_{τ} to the point \frac{1}{N} of E_{γτ}. We can also express that by introducing the injection

ψ_{τ} : ℤ/N ⟶^{1↦( \begin{matrix}0 \\ 1 \end{matrix} )} (ℤ/N)^{2} ⟶^{\overline{ϕ_{τ}}} E_{τ}[N].

Then this family corresponds to the isomorphism class of the pair (E_{τ}, ψ_{τ}) = (E_{τ}, 1/N).

• Finally, we consider the group Γ_{0}(N) in the cartesian diagram

[figure: Γ(N); SL_{2}(ℤ); SL_{2}(ℤ/N); □; Γ(N); Γ_{0}(N); Stab_{SL_{2}(ℤ/N)}(\overline{( \begin{matrix}1 \\ 0 \end{matrix} )}),]

where we consider \overline{( \begin{matrix}1 \\ 0 \end{matrix} )} as an element of the projective space ℙ^{1}(ℤ/N) over ℤ/N, on

which SL_{2}(ℤ) acts naturally. Hence, Γ_{0}(N) is the set of matrices in SL_{2}(ℤ) that reduce modulo N to an upper triangular matrix. The family corresponding to it can be characterised by saying that the subgroup of E_{τ}[N] generated by \frac{1}{N} is mapped isomorphically into the corresponding one of E_{γτ}[N]. We also have the interpretation as the isomorphism class of a pair (E_{τ}, < 1/N >).

I ought to remark that we have been quite restrictive considering only elliptic curves of the form E_{τ}. More generally one ought to regard pairs (E, ϕ), where E is an elliptic curve over ℂ and ϕ : ℤ^{2} → H_{1}(E(ℂ), ℤ) is a group isomorphism, in which case E = H_{1}(E(ℂ), ℝ)/H_{1}(E(ℂ), ℤ). Since, however, scaling the lattice by a non-zero complex number results in an isomorphic elliptic curve, the isomorphism class of (E, ϕ) always contains an element of the form (E_{τ}, ϕ_{τ}). In particular, there is an obvious way to broaden the definition of the pairs in the three points above, while the isomorphism classes stay the same.

1.2 Classical M-forms as invariant holomorphic sections

Given some k ∈ ℤ, let us consider a global holomorphic section F of the line bundle ℂ×ℍ ⟶^{π} ℍ, defined by the central vertical arrow in the level structure diagram. We shall - in this section - change the action of SL_{2}(ℤ) on ℂ×ℍ to be γ(z, τ) = ((cτ +d)^{k}z, γτ). (Earlier we had k = −1.)

By a global holomorphic section we mean a holomorphic function F : ℍ → ℂ × ℍ such that π ∘ F = id_{ℍ}. Hence F (τ) = (f(τ), τ), where f : ℍ → ℂ is holomorphic (it is F composed with the projection of ℂ × ℍ onto its first factor). Let us further assume that F is Γ-invariant, for a subgroup Γ ≤ SL_{2}(ℤ), i.e. F (γτ) = γF (τ). We now make this equation more explicit for γ ∈ Γ as above:

\begin{aligned}& F (γτ) & = & (f(γτ), γτ) \\ = & γF (τ) & = & (f(τ)(cτ + d)^{k}, γτ). \end{aligned}

We find that

(i) f : ℍ → ℂ is a holomorphic function and

(ii) f(γτ) = f(\frac{aτ+b}{cτ+d}) = (cτ + d)^{k}f(τ) for all γ ∈ Γ.

As we mentioned in the introduction, we shall call such a function f an M-form of weight k for Γ. We must impose some conditions on the cusps in order to obtain modular forms, which we shall postpone until later, since the decisive generalization can be done without them. We point out that f can have essential singularities at the cusps.

1.3 Classical M-forms as invariant holomorphic differentials

We shall now start with the algebraisation. That is, we are intending to express M-forms as sections of a sheaf (also) existing in algebraic geometry. It will be the pull-back of a sheaf of relative differentials.

To start with, we will introduce the sheaf of relative holomorphic differentials (i.e. differentials of the first kind) by the exact sequence

[figure: π^{∗}Ω^{1}_{ℍ}; Ω^{1}_{ℂ×ℍ}; Ω^{1}_{ℂ×ℍ/ℍ}; O_{ℂ×ℍ}dτ; O_{ℂ×ℍ}dz + O_{ℂ×ℍ}dτ; O_{ℂ×ℍ}dz,]

where z and τ are the standard coordinates on ℂ resp. ℍ. In the same way one finds Ω^{1}_{𝔼/ℍ} ≅ O_{𝔼}dz. Let us denote by 0 the sections in the level structure diagram mapping τ to (0, τ). Then we can pull back the sheaf of relative differentials to obtain a first version of the object playing the main role in this setting:

\underline{ω}_{𝔼/ℍ} := 0^{∗}Ω^{1}_{𝔼/ℍ} = 0^{∗}O_{𝔼}dz = O_{ℍ}dz.

For a Γ-invariant open subset U ⊆ ℍ the action of SL_{2}(ℤ) on the standard coordinates, coming from the action in the level structure diagram, gives rise to an action on \underline{ω}_{𝔼/ℍ}^{⊗k}(U) in the following way

γ(f(τ)(dz)^{⊗k}) = f(γτ)(dγz)^{⊗k} = f(γτ)\frac{1}{(cτ + d)^{k}} (dz)^{⊗k}.

We consequently see that f is an M-form of weight k for Γ if and only if f(τ)(dz)^{⊗k} is a Γinvariant global section of \underline{ω}_{𝔼/ℍ}^{⊗k}.

Appendix: the complex Kodaira-Spencer isomorphism

This seems to be an appropriate point to mention a first version of the Kodaira-Spencer isomorphism. We have the following two sheaves on ℍ

\underline{ω}_{𝔼/ℍ}^{⊗k} = O_{ℍ}(dz)^{⊗k} and Ω_{ℍ}^{⊗l} = O_{ℍ}(dτ)^{⊗l}.

Let us calculate the transformation of dτ^{⊗l} under an element γ ∈ SL_{2}(ℤ). Using the quotient rule we find

(dγτ)^{⊗l} = (d\frac{aτ + b}{cτ + d})^{⊗l} = \frac{1}{(cτ + d)^{2l}} (dτ)^{⊗l}.

Consequently, we obtain an isomorphism respecting the SL_{2}(ℤ)-action

\underline{ω}_{𝔼/ℍ}^{⊗k} ≅ Ω_{ℍ}^{⊗l}

with k = 2l. Thus M-forms of even weight can be defined as the f in the invariant sections of the holomorphic differentials on ℍ.

1.4 The case of Γ acting freely

We have seen that M-forms arise naturally as the elements of (\underline{ω}_{𝔼/ℍ}^{⊗k}(ℍ))^{Γ}, which is defined using analytic objects. In this section we assume that \underline{Γ acts freely on ℍ}, since we will then be able to give an algebraic analogue.

Proposition 1.4.1 Denote by p the quotient morphism ℍ → Γ\ℍ. Then we have

(p_{∗}\underline{ω}_{𝔼/ℍ})^{Γ} ≅ \underline{ω}_{(Γ\𝔼)/(Γ\ℍ)}.

Proof. I do not want to prove that in detail and just say that the freeness of the action of Γ on ℍ implies the freeness on 𝔼. Hence the quotient maps p : ℍ → Γ\ℍ and q : 𝔼 → Γ\𝔼 are unramified coverings. For an open cover ⋃ U_{i} of Γ\ℍ with U_{i} small enough, one consequently has canonical isomorphisms

(q_{∗}Ω^{1}_{𝔼})^{Γ}(π^{−1}(U_{i})) → Ω^{1}_{Γ\𝔼}(π^{−1}(U_{i}))

and similarly for Ω^{1}_{ℍ}. Therefore, one has to check that the defining exact sequences for the relative differential modules and the 0-sections are respected. □

One can find algebraic curves over Spec(ℂ), the complex points of which are precisely those curves. Hence, one can define an algebraic analogue of the sheaf \underline{ω}_{(Γ\𝔼)/(Γ\ℍ)}. Of course, that is the point, where we will use the moduli spaces of elliptic curves defined in Bas’ talks. We will be more precise on that in the following section.

1.5 Towards an algebraic definition

If Γ does not act freely, then (Γ\𝔼)/(Γ\ℍ) is not an elliptic curve. To make up for that, one instead looks at all (suitable) elliptic curves in order to recover the same information as in the preceding section. We will make that precise below.

Let us start by a calculation, which we will need several times in this section.

Lemma 1.5.1 Let E, E′, S and S′ be schemes fitting in the cartesian diagram

[figure: E′; h; E; □; S′; g; S,]

and assume that there are sections for the vertical arrows, both called 0, which also make the diagram commutative, i.e. h ∘ 0 = 0 ∘ g. Then

g^{∗}\underline{ω}_{E/S}^{⊗k} ≅ \underline{ω}_{E′/S′}^{⊗k}.

Proof. It is a well-known fact (Hartshorne, II.8.10) that the sheaf of relative differentials is stable under base change: h^{∗}Ω^{1}_{E/S} ≅ Ω^{1}_{E′/S′}. We get g^{∗}\underline{ω}_{E/S}^{⊗k} = (g^{∗} ∘ 0^{∗})Ω_{E/S}^{1⊗k} = (0 ∘ g)^{∗}Ω_{E/S}^{1⊗k} = (h ∘ 0)^{∗}Ω_{E/S}^{1⊗k} ≅ 0^{∗}Ω_{E′/S′}^{1⊗k} = \underline{ω}_{E′/S′}^{⊗k}. □

In the setting of the lemma we have the commutative diagram

\xymatrix{Ω^{1}_{E/S}(E) \ar[r]^{h^{∗}} \ar[d]^{0^{∗}} & h^{∗}Ω^{1}_{E/S}(E′) \ar[r]^{∼} \ar[d]^{0^{∗}} & Ω^{1}_{E′/S′}(E′) \ar[d]^{0^{∗}} \\ \underline{ω}_{E/S}(S) \ar[r] & h^{∗}\underline{ω}_{E/S}(S′) \ar[r]^{∼} & \underline{ω}_{E′/S′}(S′).}

Due to the section relation 0^{∗}∘π^{∗} = id_{\underline{ω}_{E/S}(S)}, the left bottom arrow can be defined as 0^{∗}∘h^{∗}∘π^{∗}. Composing it with the isomorphism on the right, we obtain a homomorphism

(h/g)^{∗} : \underline{ω}_{E/S}(S) → \underline{ω}_{E′/S′}(S′).

Let us now look at a special case. We have the cartesian diagram

[figure: E_{γτ}; ·(cτ+d); E_{τ}; □; Spec(ℂ); Spec(ℂ).]

Considering the differential modules again analytically, one finds that the homomorphism of abelian groups

((cτ + d)/id)^{∗} : \underline{ω}_{E_{τ} / Spec(ℂ)}^{⊗k}(Spec(ℂ)) → \underline{ω}_{E_{γτ} / Spec(ℂ)}^{⊗k}(Spec(ℂ)),

r(ds)^{⊗k} ↦^{π^{∗}} r(dz)^{⊗k} ↦^{h^{∗}} (cτ + d)^{k}r(dz)^{⊗k} ↦^{0^{∗}} (cτ + d)^{k}r(ds)^{⊗k}

is given by multiplication by (cτ + d)^{k}. Here r is an element of ℂ, considered as a function on Spec(ℂ), the standard coordinate of which we call s.

Now let F be a rule assigning to each E_{τ}/ Spec(ℂ) an element in \underline{ω}_{E_{τ} / Spec(ℂ)}^{⊗k}(Spec(ℂ)), subject to the condition that, whenever two such elements fit in a cartesian diagram as above, then the pull-back (h/g)^{∗} has to be respected, i.e. in general

(h/g)^{∗}F (E/S) = F (E′/S′)

In the case treated right before, this condition comes down to

f(γτ) = (cτ + d)^{k}f(τ),

if we put F (E_{τ}/ Spec(ℂ)) = f(τ)ds. We thus find another characterisation of the transformation rule of M-forms for Γ, if we precisely allow the cartesian diagrams for E_{τ} and E_{γτ} with γ ∈ Γ. This condition is naturally expressed by saying that the level structures in question must be respected.

Defining F only on the E_{τ}’s does, however, not imply the holomorphicity of f. In the case, where Γ acts freely, we can impose the additional condition that F must also take a value at the elliptic curve (Γ\𝔼)/(Γ\ℍ). If F also respects the cartesian diagram (in the sense as above)

[figure: E_{τ}; h; Γ\𝔼; □; Spec(ℂ); g; Γ\ℍ,]

then F (E_{τ}/ℂ) is uniquely determined by F ((Γ\𝔼)/(Γ\ℍ)) and f is holomorphic.

We cannot carry this last remark over word for word to the case, where Γ does not act freely, since then the object (Γ\𝔼)/(Γ\ℍ) is not an elliptic curve, which we would like. However, the way we got the modular forms in this section does generalize. We give the definition in the following section in detail.

1.6 Definition of Katz M-forms

We will now restrict the generality and deal only with the congruence subgroup Γ_{1}(N) for N > 0. It is, however, not difficult to replace the category defined below by the corresponding one for Γ_{0}(N) for instance.

We introduce the category [Γ_{1}(N)]_{R} of elliptic curves with a given torsion point for a ring R.

• \underline{Objects:} Pairs (E/S/R, ϕ). E/S/R is an elliptic curve, i.e. E/S a proper smooth scheme over Spec(R), whose geometric fibres are connected smooth curves of genus 1, and there is an S-valued point 0 of E. And ϕ : (ℤ/N)_{S} ↪ E[N] is an embedding of group schemes. We briefly recall that E[N] is the S-group scheme obtained from the cartesian diagram

\xymatrix{E \ar[rr]^{·N} &  & E \\  & □ \\ E[N] \ar[rr] \ar[uu] &  & S, \ar[uu]^{0}}

i.e. it is the kernel of the multiplication by N map, which results from E/S being an abelian group scheme.

• \underline{Morphisms:} Cartesian diagrams

[figure: E′; h; E; □; S′; g; S,]

such that the diagram

\xymatrix{E′ \ar[r]^{h} & E \\ S′ \ar[r]^{g} \ar[u]^{0} & S \ar[u]^{0}}

is commutative, and the embedding

ϕ′ : (ℤ/N)_{S′} ↪ E′[N]

is obtained by base change from

ϕ : (ℤ/N)_{S} ↪ E[N].

Next we describe the object H^{0}([Γ_{1}(N)]_{R}, \underline{ω}^{⊗k}). I think it is an example of the cohomology of stacks (= champs). There is a down to earth way to describe it. I shall, however, amuse myself by introducing this object slightly more categorially first, and then writing down what it comes down to.

Let C be a category and H^{0}(·, \underline{ω}) a functor from C to another category D. Let us moreover assume that there exists a forgetful functor forget : D → (Sets). Now the set (class?, probably need some assumption on the categories) H^{0}(C, \underline{ω}) is defined as the set of morphisms of functors from (trivial : C → (Sets), C ↦ \{0\}) to (forget ∘ H^{0}(·, \underline{ω})). This means that f ∈ H^{0}(C, \underline{ω}) is a rule assigning to each C ∈ C an element in the set (forget ∘ H^{0}(·, \underline{ω}))(C) compatible with the morphisms in the category C.

We shall now apply this to our situation. The category C will, of course, be replaced by [Γ_{1}(N)]_{R}. The functor H^{0}(·, \underline{ω}) is taken to be

(E/S/R, ϕ) ↦ H^{0}(S, \underline{ω}_{E/S}^{⊗k}) = \underline{ω}_{E/S}^{⊗k}(S).

Here we have set

\underline{ω}_{E/S}^{⊗k} = 0^{∗}Ω_{E/S}^{1⊗k},

which is a sheaf on S.

We repeat ourselves by pointing out that an element F ∈ H^{0}([Γ_{1}(N)]_{R}, \underline{ω}^{⊗k}) is the choice of elements F (E/S/R, ϕ) ∈ H^{0}(S, \underline{ω}_{E/S}^{⊗k}), subject to the rule

(h/g)^{∗}F (E/S/R, ϕ) = F (E′/S′/R, ϕ′)

for any cartesian diagram as above.

Definition 1.6.1 Let R be a ring and k and N integers with N > 0. The space of Katz M-forms of weight k for the group Γ_{1}(N) is defined as

M(Γ_{1}(N), k)_{R} = H^{0}([Γ_{1}(N)]_{R}, \underline{ω}^{⊗k}).

We point out that if the category has a final object (E_{Γ_{1}(N)}/Y_{Γ_{1}(N)}/R, ϕ_{Γ_{1}(N)}), then

H^{0}([Γ_{1}(N)]_{R}, \underline{ω}^{⊗k}) = H^{0}(Y_{Γ_{1}(N)}, \underline{ω}_{E_{Γ_{1}(N)}/Y_{Γ_{1}(N)}}^{⊗k}).

Let us now describe briefly, why the Katz M-forms coincide with the classical M-forms, which have no essential singularities at the boundary, if the ring R is ℂ.

Assume first that Γ = Γ_{1}(N) acts freely. Then Bas proved in his second talk of this series that there exists a final object (E_{Γ_{1}(N)}/Y_{Γ_{1}(N)}/R, ϕ_{Γ_{1}(N)}) in the category [Γ_{1}(N)]_{R} if N > 4 and N is invertible in R. If R = ℂ then by the universality, the complex points of Y_{Γ_{1}(N)} correspond to pairs (E_{τ}/ Spec(ℂ), 1/N). One actually finds (for that, some compatibilities ought to be checked) that E_{Γ_{1}(N)}(ℂ) ≅ Γ\𝔼 and Y_{Γ_{1}(N)}(ℂ) ≅ Γ\ℍ and that these isomorphisms respect the structural morphisms. Thus the difference between the analytically defined group \underline{ω}_{(Γ\𝔼)/(Γ\ℍ)}(Γ\𝔼) and its algebraic counterpart \underline{ω}_{E_{Γ_{1}(N)}/Y_{Γ_{1}(N)}}(Y_{Γ_{1}(N)}) lies on the boundary. Consequently, for freely acting Γ the Katz M-forms over ℂ coincide with the classical ones, which do not have essential singularities in the cusps.

If Γ_{1}(N) does not act freely, we do not dispose of such an easy description. Instead, we pass to a subgroup Γ_{1}(Nm), which acts freely on ℍ (Nm > 4). We will also assume that m is invertible in R. We then have an injection (of sets)

H^{0}([Γ_{1}(N)]_{R}, \underline{ω}^{⊗k}) ↪ H^{0}([Γ_{1}(Nm)]_{R}, \underline{ω}^{⊗k}), f ↦ f ∘ ψ,

coming from the (surjective) functor

ψ : [Γ_{1}(Nm)]_{R} → [Γ_{1}(N)]_{R}, (E/S/R, ϕ) ↦ (E/S/R, ϕ ∘ i),

where i is natural embedding of group schemes (ℤ/N)_{S} → (ℤ/Nm)_{S} defined by sending 1 to m.

Applying this with R = ℂ, we obtain that also Katz M-forms for Γ_{1}(N) are holomorphic. More precisely, they can be characterized as the subspace of Katz M-forms for Γ_{1}(Nm) satisfying the transformation rule for Γ_{1}(N).

Appendix: the Kodaira-Spencer isomorphism

In this section we mention that the Kodaira-Spencer morphism

\underline{ω}_{E/S}^{⊗2} → Ω^{1}_{S/T}

for E/S an elliptic curve and S → T any morphism can be defined in general (in the algebraic setting), and not only over ℂ, as we did before. It actually is an isomorphism if we take (E/S/T) to be E_{Γ}/Y_{Γ}/ℤ[1/N] with Γ = Γ_{1}(N) and N > 4. This, of course, allows us to consider Mforms of even weight for this group to be global differentials, just as we had in the complex case.

1.7 q-expansions and Katz modular forms

In this section we let N > 2 and Γ = Γ_{1}(N), since we do not want to be troubled with irregular cusps. This is not a conceptual restriction, it just saves a bit of work here.

We start by recalling the classical notion of q-expansions, which we will, of course, recover later in the Katz setting.

As γ := ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) is in Γ, every M-form f satisfies

f(γτ) = f(τ + 1) = f(τ).

So f is periodic with period 1, whence we can write

f(τ) = g(q) = ∑_{n≥−∞} a_{n}q^{n}

with q = e^{2πiτ}, where the last equality is the expansion around 0, which is called the q-expansion of f at the standard cusp ∞ (because τ → i∞ ⇔ q → 0).

We want to extend this notion to other cusps as well. Let x = σ∞ for σ ∈ SL_{2}(ℤ) be another cusp and let Γ_{x} be its stabilizer in the group Γ. We get

σ^{−1}Γ_{x}σ ≤ Γ_{∞} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )^{ℤ}, hence σ^{−1}Γ_{x}σ = ( \begin{matrix}1 & h \\ 0 & 1 \end{matrix} )^{ℤ}

for some h|N. Let us now consider the function

f|_{k}(σ)(τ) = (cτ + d)^{−k}f(στ), for σ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

(|_{k} is called the Petersson (slash-)operator.) We point out that f satisfies the transformation rule for σ if and only if f|_{k}(σ) = f. An easy calculation gives

f|_{k}(γ)|_{k}(γ′) = f|_{k}(γγ′).

Using this with σ^{−1}γσ = ( \begin{matrix}1 & h \\ 0 & 1 \end{matrix} ) for some γ ∈ Γ_{x}, one gets

f|_{k}(σ)(τ) = f|_{k}(σ)(τ + h).

This allows us again to develop f|_{k}(σ) into a Laurent series in the variable q^{1/h}. That is called the q-expansion at the cusp x.

In order to find a translation into the language of Katz forms, we consider the following cartesian diagram

[figure: (E_{τ}, \frac{−b ( mod N )}{N}τ + \frac{a ( mod N )}{N}); ·(cτ+d)^{−1}; (E_{στ}, \frac{1}{N}); □; Spec(ℂ); Spec(ℂ).]

If F is now a Katz M-form with the usual identification f(τ)(ds)^{⊗k} = F (E_{τ}/ℂ, 1/N), then we find from the transformation rule

f|_{k}(σ)(τ)(ds)^{⊗k} = (cτ + d)^{−k}f(στ)(ds)^{⊗k} = (cτ + d)^{−k}F (E_{στ}, 1/N)

= F (E_{τ}, \frac{−b ( mod N )}{N}τ + \frac{a ( mod N )}{N}).

That means that the function f|_{k}(σ) corresponds to the evaluation of F at the usual elliptic curves E_{τ}, but now with the torsion point \frac{−b ( mod N )}{N}τ + \frac{a ( mod N )}{N}.

We bring the Tate curve into the business and consider the isomorphism

(ℂ/ℤτ + ℤ, \frac{−b ( mod N )}{N}τ + \frac{a ( mod N )}{N}) ⟶^{e^{2πi}} (ℂ^{∗}/q^{ℤ}, ζ_{N}^{a}q^{−b/N}),

where ζ_{N} = e^{2πi/N}.

Let us recall briefly some properties of the Tate curve (it is 𝔾_{m}/q^{ℤ}):

• The Tate curve Tate(q)_{ℤ((q))} is an elliptic curve over ℤ((q)).

• \underline{ω}_{Tate(q)_{ℤ((q))}/ℤ((q))}^{⊗k}(ℤ((q))) = ℤ((q))(\frac{dt}{t})^{⊗k}.

• The N-torsion group of Tate(q)_{ℤ((q))[q^{1/N} ,ζ_{N}]} is given by ζ_{N}^{a}q^{b/N} for all pairs a, b, which generate ℤ/N.

Now we show that the Tate curve Tate(q)_{ℂ((q))[q^{1/N}]} provides us with a q-expansion. We use that its ℂ-points are precisely the elliptic curves of the form ℂ^{∗}/q^{ℤ}. We look at the big cartesian diagram

[figure: (E_{τ}, \frac{−b ( mod N )}{N}τ + \frac{a ( mod N )}{N}); e^{2πi}; ∼; □; (ℂ^{∗}/q^{ℤ}, ζ_{N}^{a}q^{−b/N}); □; (Tate(q)_{ℂ((q))[q^{1/N}]}, ζ_{N}^{a}q^{−b/N}); Spec(ℂ); Spec(ℂ); Spec(ℂ((q))[q^{1/N}]).]

We note that the left column describes f|_{k}(σ)(τ). We use that \underline{ω}_{Tate(q)_{ℂ((q))[q^{1/N}]}/ℂ((q))[q^{1/N}]}^{⊗k}(ℂ((q))[q^{1/N}]) is of the form ℂ((q))[q^{1/N}](\frac{dt}{t})^{⊗k}, whence

f|_{k}(σ)(τ)(ds)^{⊗k} = g(q^{1/N})(\frac{dt}{t})^{⊗k},

where g is a Laurent series in q. f|_{k}(σ) is also classically expressed in a q-expansion, as we have seen. These two q-expansions coincide because they both describe the same holomorphic function on a pointed disc. Now it is clear, how one should define q-expansions for Katz M-forms also when the ring is not ℂ.

The q expansion at the cusp σ∞ of a Katz M-form F of weight k for the group Γ_{1}(N) over a ℤ[1/N]-algebra R is defined as the Laurent series g ∈ R[ζ_{N}]((q^{1/N})) such that

F (Tate(q)_{R((q))[q^{1/N} ,ζ_{N}]}/R((q))[q^{1/N}, ζ_{N}]/R, ζ_{N}^{a}q^{−b/N}) = g(q^{1/N})(\frac{dt}{t})^{⊗k}.

We point out that for the cusp ∞ we have that h = 1, and hence the q-expansion is in the variable q (instead of q^{1/N}). At the cusp 0 one finds h = N, so q^{1/N} is really needed (in general), but on the other hand the adjoining of a primitive N-th root of unity is not.

Definition 1.7.1 A Katz M-form F of weight k for the group Γ_{1}(N) over a ℤ[1/N]-algebra R is called a

• Katz modular form (an element of M_{k}(Γ_{1}(N), R)) if the q-expansions are power series at all cusps, resp. a • Katz cusp form (an element of S_{k}(Γ_{1}(N), R)) if the q-expansions are power series without

constant term at all cusps.

If R = ℂ, we can conclude from what we developped above that the Katz modular forms are precisely the classical ones.

1.8 First properties of Katz modular forms

• Let R′ → R be a flat ring homomorphism and (E/S/R, ϕ) and elliptic curve with a level structure. Denote by (E′/S′/R′, ϕ′) the curve obtained by base change. Since exactness is preserved, one finds

M_{k}(Γ_{1}(N), R′) ⊗_{R′} R ≅ M_{k}(Γ_{1}(N), R)

and the same for the space of Katz cusp forms. For a general R′′ → R one still has a homomorphism of R-algebras.

• Let now R = k be a field. Then the space of Katz modular forms is a k-vector space of finite dimension. One can actually calculate this dimension using Riemann-Roch, if one consideres the compactified moduli space.

• Let A be a subring of ℂ, in which N is invertible, and f a classical modular form for Γ_{1}(N) of weight k, the coefficients of the standard q-expansion of which are all in A. Then f is also a Katz modular form for the ring A of weight k for Γ_{1}(N).

• There are no Katz modular forms of negative weight.

2 Calculating characteristic p Katz modular forms of weight 1, after Bas Edixhoven

This talk was given in two parts on 15 and 22 January 2003.

The aim of this talk is to present a theorem by Bas Edixhoven ([E]), which describes the space of cuspidal Katz modular forms of weight 1 with a given character over a finite field just in terms of the Hecke algebra of classical modular forms (over the complex numbers, for the group Γ_{1}(N), without a character). Moreover, effective bounds on the number of Hecke operators needed are given. Using standard software tools, the calculations can be performed using only linear algebra methods.

We shall also present a variant of the theorem (valid in certain cases) for the calculation of cuspidal Katz modular forms of weight 1 with trivial character, where one uses the classical Hecke algebra for modular forms for Γ_{0}(N).

Along the way we introduce Hecke operators, parabolic cohomology and also a way to calculate classical modular forms of weight k ≥ 2.

2.1 Different definitions of Katz modular forms and comparison with the classical theory

In the introduction to the first talk I warned the reader that our definition of Katz forms might differ from Katz’ original one. Since we need to work with different notions of modular forms, we present them in this section and also give relations between them.

Let us fix a ℤ[1/N]-algebra R, where N ≥ 5 is an integer (most works for arbitrary N). Katz worked in [K] in the more general setting of the category of elliptic curves with a full level N-structure, i.e. a fixed isomorphism of group schemes

α : (ℤ/N)_{S}^{2} ≅ E[N].

This is a more general case of our situation, where we consider the category [Γ_{1}(N)]_{R} of elliptic curves E/S/R together with embeddings of group schemes

β : (ℤ/N)_{S} ↪ E[N],

which correspond to fixing an N-torsion point (i.e. the image of 1 in E[N]). The Katz modular forms (resp. cusp forms) will, as previously, be denoted by M_{k}(Γ_{1}(N), R) (resp. S_{k}(Γ_{1}(N), R)).

In Gross’ article [G] a different notion is presented. Namely, he considers the category [Γ_{1}(N)]′_{R} of elliptic curves E/S/R as before, but now with an embedding of group schemes

γ : (μ_{N})_{S} ↪ E[N].

The theory of modular forms can be developped precisely as we presented it in the first talk. The (Katz) modular forms (resp. cusp forms) obtained in this setting will be denoted by M_{k}(Γ_{1}(N), R)′ (resp. S_{k}(Γ_{1}(N), R)′).

The advantage of the new setting is that one does not need to adjoin a primitive N-th root of unity to the ring in order to obtain the q-expansion at infinity. In the previously developped setting this becomes necessary just because we want to evaluate a modular form at the Tate curve 𝔾_{m,R}/q^{ℤ} together with the torsion point ζ_{N} (see the remark below, why we need to choose precisely that one). The [Γ_{1}(N)]′_{R}-setting is hence an elegant way to avoid this trouble! And it is clear that the two settings coincide if R contains the N-th roots of unity.

Let us now compare the two settings by quoting the (also for the sequel) very important

Theorem 2.1.1 Let S be an R-algebra with R an ℤ[1/N]-algebra for some integer N ≥ 5. Let k ∈ ℕ and suppose one of the following holds:

(i) k ≥ 2

(ii) R → S is flat.

Then the following natural maps are isomorphisms:

M_{k}(Γ_{1}(N), R) ⊗_{R} S ≅ M_{k}(Γ_{1}(N), S),
M_{k}(Γ_{1}(N), R)′ ⊗_{R} S ≅ M_{k}(Γ_{1}(N), S)′,
S_{k}(Γ_{1}(N), R) ⊗_{R} S ≅ S_{k}(Γ_{1}(N), S) and
S_{k}(Γ_{1}(N), R)′ ⊗_{R} S ≅ S_{k}(Γ_{1}(N), S)′.

Proof. For the “undashed” version we refer to [DI], Theorem 12.3.2, and for the other one to [G], Proposition 2.5, for k ≥ 2. The flat case works for both settings in the same way. □

From this we can conclude the

Proposition 2.1.2 The Hecke and diamond operators T_{n} and < a > for n ∈ ℕ and a ∈ (ℤ/N)^{∗} generate the same ring of endomorphisms 𝕋 in S_{k}(Γ_{1}(N), ℤ[1/N]) and S_{k}(Γ_{1}(N), ℤ[1/N])′, and these two 𝕋-modules are isomorphic.

Proof. Edixhoven proves this proposition in [E] (Proposition 4.11) without reference to the above theorem. Here we give a different proof.

Both spaces are free ℤ[1/N]-modules. Since ℤ[1/N] ↪ ℤ[1/N, ζ_{N}] is flat, it follows from the theorem and the remarks preceding it that

S_{k}(Γ_{1}(N), ℤ[1/N]) ⊗ ℤ[1/N, ζ_{N}] ≅ S_{k}(Γ_{1}(N), ℤ[1/N, ζ_{N}])
= S_{k}(Γ_{1}(N), ℤ[1/N, ζ_{N}])′ ≅ S_{k}(Γ_{1}(N), ℤ[1/N])′ ⊗ ℤ[1/N, ζ_{N}].

This isomorphism gives rise to an isomorphism (by conjugation)

ϕ : End_{ℤ[1/N,ζ_{N}]}(S_{k}(Γ_{1}(N), ℤ[1/N, ζ_{N}])) ≅ End_{ℤ[1/N,ζ_{N}]}(S_{k}(Γ_{1}(N), ℤ[1/N, ζ_{N}])′),

under which the T_{n} and < a > correspond on both sides. This implies the assertion. □

In the previous talk/section we have seen that the classical space of modular forms coincides with the Katz space over ℂ:

M_{k}(Γ_{1}(N), ℂ) ≅ M_{k}(Γ_{1}(N), ℂ).

It is customary to define the classical subspace of modular forms of weight k with coefficients in ℤ[1/N] as

M_{k}(Γ_{1}(N), ℤ[1/N]) = { f ∈ M_{k}(Γ_{1}(N), ℂ) | q−expansion at ∞ has coefficients inℤ[1/N] }.

Furthermore, one sets

M_{k}(Γ_{1}(N), A) = M_{k}(Γ_{1}(N), ℤ[1/N]) ⊗_{ℤ[1/N]} A

for any ℤ[1/N]-algebra A. We now use that our q-expansion coincides with the classical one in both cases (over ℂ). As we have pointed out, in [Γ_{1}(N)]′_{R} (with R = ℤ[1/N]) the coefficients of the q-expansions are in ℤ[1/N]. From this we immediately obtain for all k:

M_{k}(Γ_{1}(N), ℤ[1/N]) = M_{k}(Γ_{1}(N), ℤ[1/N])′.

For k ≥ 2 the theorem above implies by comparison with the Katz theory that

M_{k}(Γ_{1}(N), ℤ[1/N]) ⊗_{ℤ[1/N]} ℂ = M_{k}(Γ_{1}(N), ℂ),

which is a weak version of the theorem by Shimura (I think) that says that the classical space of modular forms (i.e. over ℂ) has a basis of forms with Fourier coefficients in ℤ.

Remark 2.1.3 The natural isomorphism

ℂ/ℤστ + ℤ ⟶^{·(cτ+d)} ℂ/ℤτ + ℤ

for a matrix σ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{1}(N)(ℤ) fixes precisely the N-torsion points \frac{r}{N} for r ∈ (ℤ/N)^{∗}

(as an immediate calculation shows).

So the only possible correspondences between Katz forms (in both settings) and classical modular forms over ℂ are given by

F (ℂ/ℤτ + ℤ, \frac{r}{N}) = f(τ)(ds)^{k},

where ds is a basis of the ℂ-vector space of global sections of \underline{ω}_{ℂ/ℤτ+ℤ}^{⊗k}. These correspondences differ precisely by the diamond operators.

In other words, if one wants the Katz forms to coincide with the classical forms, one has to choose any of the N-torsion points r/N and not rτ/N.

Remark 2.1.4 A different approach to solving the trouble of having to adjoin a primitive N-th root of unity in order to obtain a q-expansion (by the above remark one does not have the choice of an essentially different torsion point) could be to consider q-expansions at the cusp 0. Then an easy calculation shows that one gets an expansion in the variable q^{1/N} instead of q, not involving N-th roots of unity. The Hecke operators do, however, not fix the cusp 0. That means that the q-expansion of the transformed function involves q-expansions at some other cusps. At those cusps the adjoining of an N-th root of unity is also not necessary.

The elegant solution to use [Γ_{1}(N)]′_{R} presented in Gross’ paper seems preferable.

2.2 Parabolic cohomology

In the case of classical modular forms of weight k ≥ 2 for Γ_{1}(N) with N ≥ 5, one can describe the Hecke algebra by an action on a faithful module that is completely described in terms of group cohomology.

More precisely, one can identify (via an isomorphism of ℂ-vector spaces) the space S_{k}(Γ_{1}(N), ℂ) ⊕ \overline{S_{k}(Γ_{1}(N), ℂ)} with the vector space H_{par}^{1}(Γ_{1}(N), M_{k}(ℂ)), where M_{k}(ℂ) = Sym^{k−2}(ℂ × ℂ). This vector space has a natural ℤ-lattice, namely H_{par}^{1}(Γ_{1}(N), M_{k}(ℤ)). As we will see in the next section, one gets a natural action of Hecke operators on this space. Since the isomorphisms, which we will explain here below, respect the Hecke action, we obtain a faithful Hecke module (in simple terms!), which allows us to calculate the Hecke algebra (for classical modular forms). See also the propositions in the section on calculating classical modular forms.

2.3 Operators on modular forms

We describe a general setting, which allows us to treat Hecke and diamond operators at the same time.

Let α ∈ GL_{2}(ℚ)^{+} and set for abbreviation Γ := Γ_{1}(N), where we again impose that N be greater equal 5. Then the groups

Γ_{α} := α^{−1}Γα ∩ Γ and Γ^{α} := αΓα^{−1} ∩ Γ

have finite index in Γ. We consider the commutative diagram

\xymatrix{Γ_{α}\ℍ \ar[r]^{α}_{∼} \ar[d]^{π} & Γ^{α}\ℍ \ar[d]^{π} \\ Γ\ℍ & Γ\ℍ,}

where π denotes the natural projections. For the covering groups we obtain (recall the assumption N ≥ 5) the commutative diagram

\xymatrix{Γ_{α} \ar[r]^{γ↦αγα^{−1}}_{∼} \ar@{^{(}->}[d]^{ι} & Γ^{α} \ar@{^{(}->}[d]^{ι} \\ Γ & Γ,}

where ι denotes the natural inclusions.

We can now define operators T_{α} on modular forms and compatibly on group cohomology.

(i) For a modular form F (over ℂ) set

(T_{α}F)(τ) = (det(α)^{−1}) ∑_{x∈(πα)^{−1}(τ)} F (π(x)).

It has to be checked (in fact it is obvious on the q-expansions as we will see) that a modular form is transformed into a modular form and a cusp form into a cusp form.

(ii) On the group cohomology we define the homomorphism T_{α} by the commutative diagram:

[figure: H^{1}(Γ_{α}, M); conj. by α; ∼; cores; H^{1}(Γ^{α}, M); res; H^{1}(Γ, M); T_{α}; H^{1}(Γ, M),]

where M is a Γ-module. One should also calculate at this point that the morphism T_{α} fixes the parabolic subspace, and that the isomorphism of the preceding chapter between parabolic cohomology (over ℂ) and the space of holomorphic and antiholomorphic cusp forms respects the operators T_{α}.

Now we apply this abstract situation to two cases, in both of which we will give an equivalent description on the moduli spaces, which allows us to extend the operators in a natural way to Katz modular forms.

Diamond operators:

Let a be an integer coprime to N. Then there exists a matrix α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N). As

Γ_{1}(N) is a normal subgroup of Γ_{1}(N) the groups Γ_{α} and Γ^{α} are equal to Γ and the maps π are the identity. The operator corresponding to α is called the diamond operator and denoted < a >. It will become apparent that it indeed only depends on a and not on the choice of α.

Under α, the elliptic curve with given torsion point (ℂ/ℤτ + ℤ, 1/N) is mapped to (ℂ/ℤατ + ℤ, 1/N), which is isomorphic to (ℂ/ℤτ + ℤ, a/N).

This suggests the extension of the definition of the diamond operator to Katz modular forms as follows:

< a >: M_{k}(Γ_{1}(N), R) → M_{k}(Γ_{1}(N), R) such that
(< a > F)(E/S/R, P) = F (E/S/R, aP).

If we are working with embeddings μ_{N} ↪ E[N] instead of torsion points, we must, of course, just compose with the a-th power map μ_{N} → μ_{N}.

In all cases, one thus gets an action of the group (ℤ/N)^{∗} ≅ Γ_{0}(N)/Γ_{1}(N) on the space of modular forms (cusp forms). Let ϵ : (ℤ/N)^{∗} → R^{∗} be a character. Then we set

M_{k}(Γ_{1}(N), ϵ, R) = { f ∈ M_{k}(Γ_{1}(N), R) | < a > f = ϵ(a)f ∀a ∈ (ℤ/N)^{∗} }

and for the other spaces in question similarly.

If R = K is a field of characteristic not 2, one finds by considering < −1 > that there are no non-trivial (Katz) modular forms if ϵ(−1) ≠ (−1)^{k}.

Furthermore, one finds

M_{k}(Γ_{1}(N), 1, R) = M_{k}(Γ_{0}(N), R).

In fact, I have not defined the space on the right (although a definition suggests itself), so we can also work with this as a definition.

Hecke operators:

For simplicity, we give the definition of Hecke operators only for primes l and define the operators for composite numbers by the formulae that one usually obtains by calculation.

So let l be a prime. The l-th Hecke operator T_{l} is defined by the matrix α = ( \begin{matrix}1 & 0 \\ 0 & l \end{matrix} ).

We must point out that we divide by l in the definition of the operator. As we will see on qexpansions, the factor 1/l vanishes from the description. So one must take the action on the q-expansion as a definition in case of characteristic l. This requires some additional work, that we do not perform here (in fact we shall just overlook this problem). The reader is referred to [G], proposition 4.1.

Straightforward calculations yield (whether l divides N or not)

Γ_{α} = Γ_{1}(N) ∩ Γ_{0}(l) and Γ^{α} = Γ_{1}(N) ∩ (Γ_{0}(l))^{T} ,

where the superscript T stands for transpose. By identifying τ ↦ (ℂ/ℤτ + ℤ, 1/N, < τ/l >), one gets a one-to-one correspondence between Γ^{α}\ℍ and triples (E, P, H), where E is an elliptic curve with N-torsion point P and H ≤ E a (cyclic) subgroup of order l that does not contain the point P. For Γ_{α} one can proceed similarly (the third component must be replaced by < 1/l >). Direct inspection shows that on the moduli spaces the map α corresponds to

(E, P, H) ↦ (E/H, P mod H, E[l]/H).

Of course, the maps π just mean dropping the third component.

Now it should be obvious how to extend the definition of T_{l} to Katz modular forms (in both settings, since the embedding of μ_{N} is not affected), namely by setting

T_{l} : M_{k}(Γ_{1}(N), R) → M_{k}(Γ_{1}(N), R) such that

(T_{l}F)(E/S/R, P) = ∑_{H} F ((E/H)/S/R, P mod H),

where the sum runs over all (cyclic) subgroups H of E of order l, which do not contain the point P.

The Hecke operators T_{n} with n composite are defined by the formulae:

(i) T_{l^{r+1}} = T_{l} ∘ T_{l^{r}} − l^{k−1} < l > ∘T_{l^{r−1}} if l ∤ N,

(ii) T_{l^{r+1}} = T_{l} ∘ T_{l^{r}} if l | N and

(iii) T_{nm} = T_{n} ∘ T_{m} if n and m are coprime.

It is easy to determine all order l subgroups H of the Tate curve. Consequently, a simple calculation yields the formula for the action of the Hecke operator on the q-expansions. Here I just give the result (for l ∤ N, otherwise the second summand must be dropped):

a_{n}(T_{l}f) = a_{ln}(f) + l^{k−1}a_{n/l}(< l > f),

where a_{n/l} has to be interpreted as 0 if l ∤ n. From this one also obtains the for the sequel very important formula

a_{1}(T_{n}f) = a_{n}(f).

2.4 Calculation of classical modular forms of weight k ≥ 2 from the Hecke algebra

In this section we present a well known method for calculating a basis of eigenforms for the Hecke algebra (in the classical setting). We shall restrict to the case k ≥ 2 in all of this section because of the practical fact that using parabolic cohomology (resp. modular symbols) only gives the Hecke algebra in this case. As far as we have developped the theory here, we can also not conclude (at least I do not know how) that there is also a ℤ[1/N]-basis in weight 1. It should, however, be possible to prove that by extending theorem 2.1.1 to include k = 1 for rings contained in \overline{ℚ}.

In the sequel of this section we commit a slight inconsistency with the previous and the following settings: we shall use ℤ instead of ℤ[1/N] in several places. This is justified since we do not treat Katz modular forms here. See also the discussion of classical modular forms in the beginning of this talk/chapter.

Definition 2.4.1 By the Hecke algebra of weight k for the group Γ_{1}(N) (for classical modular forms) we understand the subring 𝕋 < End_{ℂ}(S_{k}(Γ_{1}(N), ℂ)) generated by the operators T_{n} n ∈ ℕ.

Proposition 2.4.2 Under the isomorphism between cusp forms and parabolic cohomology

H_{par}^{1}(Γ_{1}(N), M_{k}(ℤ)) ⊗ ℂ ≅ S_{k}(Γ_{1}(N), ℂ) ⊕ \overline{S_{k}(Γ_{1}(N), ℂ)},

the Hecke operators T_{n} on parabolic cohomology corresponds to the operator T_{n}⊕\overline{T_{n}}. Moreover, the Hecke algebra 𝕋 generated by the T_{n} as a subring of End_{ℂ}(S_{k}(Γ_{1}(N), ℂ)) is isomorphic with the algebra generated by the T_{n} as a subring of End_{ℂ}(H_{par}^{1}(Γ_{1}(N), M_{k}(ℤ)) ⊗ ℂ).

Proof. The second statement is an immediate consequence of the first, which one has to check. □

Proposition 2.4.3 Let Γ be any subgroup of SL_{2}(ℤ) of finite index. There is a free ℤ-module CMS_{k}(Γ), called the space of cuspidal modular symbols, together with an isomorphism

CMS_{k}(Γ) ⊗ ℂ ≅ S_{k}(Γ, ℂ) ⊕ \overline{S_{k}(Γ, ℂ)}.

One can define Hecke operators T_{n} on CMS_{k}(Γ), which correspond to T_{n} ⊕ \overline{T_{n}} via the isomorphism. Moreover, the Hecke algebra 𝕋 generated by the T_{n} as a subring of End_{ℂ}(S_{k}(Γ, ℂ)) is isomorphic with the algebra generated by the T_{n} as a subring of End_{ℂ}(CMS_{k}(Γ) ⊗ ℂ).

Proof. That is proved in [M]. □

Proposition 2.4.4 Let A be an abelian group. Then we have:

(a) 𝕋 contains all operators < a > for a ∈ (ℤ/N)^{∗}.

(b) 𝕋 is generated by the T_{n} as an abelian group (not only as a ring).

(c) 𝕋 is commutative.

(d) There is a natural isomorphism S_{k}(Γ_{1}(N), A) → Hom_{A−lin}(𝕋 ⊗ A, A) given by f ↦ (T ↦ a_{1}(T f)).

Proof. (a) Let l_{1} and l_{2} be two distinct primes such that l_{i} ≡ a(N). The result is immediate from the formula

l_{i}^{k−1} < a >= T_{l_{i}}^{2} − T_{l_{i}^{2}}.

(b,c) These are immediate consequences of the formulae defining Hecke operators T_{n} for composite n.

(d) A good way to rephrase the statement over a field, say ℂ, is to say that

<, >: 𝕋 × S_{k}(Γ_{1}(N), ℂ) → ℂ, < T, f >:= a_{1}(T f)

is a perfect pairing. This follows from the formula a_{1}(T_{n}f) = a_{n}(f) as follows. If for all n we have 0 = a_{1}(T_{n}f) = a_{n}(f), then f = 0 by the uniqueness of the q-expansion. Conversely, if a_{1}(T f) = 0 for all f, then a_{1}(T (T_{n}f)) = a_{1}(T_{n}T f) = a_{n}(T f) = 0 for all f and all n. As S_{k}(Γ_{1}(N), ℂ) is by definition a faithful 𝕋-module, we find T = 0, proving the perfectness.

Having established the result over ℂ, the statement with A = ℤ is just the definition of S_{k}(Γ_{1}(N), ℤ) as the subspace of forms with Fourier coefficients in ℤ. The general result follows by tensoring with A. We also note that 𝕋 does not have any torsion, since we defined it as a subset of a vector space in characteristic 0. □

Remark 2.4.5 Let K be a perfect field (of characteristic not dividing N). Then the freeness of 𝕋 implies (by the proposition above)

S_{k}(Γ_{1}(N), \overline{K}) = Hom_{\overline{K}−lin}(𝕋 ⊗ \overline{K}, \overline{K})
= Hom_{K−lin}(𝕋 ⊗ K, \overline{K}).

In words: The q-expansions of cusp forms with coefficients in \overline{K} are precisely the K-linear maps 𝕋 ⊗ K → \overline{K}.

Remark 2.4.6 Let us consider more generally 𝕋 just as a finite dimensional (i.e. Artinian) commutative K-algebra over a perfect field K.

By a system of eigenvalues of a 𝕋-module M we understand a map λ : 𝕋 → K such that there exists an element 0 ≠ m ∈ M satisfying T.m = λ(T)m for all T ∈ 𝕋.

In particular, let us consider the 𝕋-module

S := Hom_{K−lin}(𝕋, \overline{K}),

for the action (T.f)(T̃) = f(T T̃). It is in fact a faithful 𝕋-module because (T.f)(1) = f(T) = 0 for all f implies that T = 0.

Given a system of eigenvalues λ, an element f ∈ S is by definition a normalised eigenform for λ if

(i) (T.f)(T̃) = f(T T̃) = λ(T)f(T̃), which implies (T.f)(1) = λ(T)f(1), and

(ii) f(1) = 1.

This is immediately seen to be equivalent to f being a K-algebra homomorphism (and necessarily also λ).

Furthermore, we have

|Hom_{K−alg}(𝕋, \overline{K})| = |Hom_{\overline{K}−alg}(𝕋 ⊗ \overline{K}, \overline{K})|
≤ dim\overline{K}(𝕋 ⊗ \overline{K}) = dim_{K}(𝕋) = dim\overline{K}(Hom_{K−lin}(𝕋, \overline{K})).

One has equality for square-free N in characteristic 0. In that case we can hence conclude that S has a \overline{K}-basis consisting of eigenforms (with image in a finite extension of K because of the finite dimensionality of 𝕋).

Moreover, we have a left G_{K} = G(\overline{K}|K)-action S via

σ.f = σ ∘ f.

Remark 2.4.7 We now apply the above remark to the Hecke algebra 𝕋. A system of eigenvalues λ (with normalised eigenform f) is then uniquely determined by λ(T_{n}). This value is, however, by the proposition precisely the n-th Fourier coefficient of f. Thus, calculating Fourier coefficients just means calculating eigenvalues of the Hecke operators. Those, however, are provided as matrices via the desciption of parabolic cohomology resp. modular symbols.

We also remark that the G_{K}-action is just applying the Galois automorphism to the Fourier coefficients.

There remains the problem of keeping apart the different systems of eigenvalues. This is solved by the following proposition.

Proposition 2.4.8 In the setting of remark 2.4.6 there is a bijection

Hom_{K−alg}(𝕋, \overline{K})/_{G_{K}−conjugacy} ↔ maximal ideals of 𝕋,

given by sending a K-algebra homomorphism f : 𝕋 → \overline{K} to its kernel.

Proof. We point out that in an Artinian algebra every prime ideal is maximal. Since for every maximal ideal 𝔪 the ring 𝕋/𝔪 is contained (via f) in \overline{K} the kernel is a maximal ideal. If g = σ ∘ f for some field automorphism σ, the kernel of f is, of course, equal to the kernel of g. Hence the allocation is well defined.

The surjectivity is clear: given 𝔪, just take f : 𝕋 → 𝕋/𝔪 ≤ \overline{K}.

Suppose now that f and g have the same kernel 𝔪. That precisely means that f and g give two embeddings of the field 𝕋/𝔪 into \overline{K}. By well known field theory, one can be obtained from the other by composing with an element of G_{K}, which proves the injectivity. □

Remark 2.4.9 By the proposition one can localize 𝕋 at its maximal ideals in order to keep exactly one particular system of eigenvalues modulo G_{K}-conjugacy. Linear algebra techniques (calculation of characteristic polynomials) then directly produce the Fourier coefficients of one eigenform. It is not difficult to calculate its (finite) G_{K}-orbit.

2.5 Calculation of Katz modular forms of weight 1 over finite fields

In this section we present Edixhoven’s results ([E], section 4) on how to calculate Katz modular forms (in the “dashed” setting as presented in [G]) of weight 1 over finite fields.

Let us, for this section, fix a prime p, a finite extension 𝔽 of 𝔽_{p}, a character ϵ : (ℤ/N)^{∗} → 𝔽^{∗} and an integer N ≥ 5 coprime to p.

The result to be presented is based on the following homomorphisms of spaces of Katz modular forms.

(i) \underline{The Frobenius F:} The homomorphism

F : S_{1}(Γ_{1}(N), 𝔽_{p})′ → S_{p}(Γ_{1}(N), 𝔽_{p})′

is defined by sending f to f^{p}, meaning the following: If f(E/S, α) ∈ \underline{ω}_{E/S}(S), then f^{p}(E/S, α) = (f(E/S, α))^{⊗p} ∈ \underline{ω}_{E/S}^{⊗p}(S). For the action of F on q-expansions at infinity, one immediately finds:

a_{n}(F (f)) = a_{n/p}^{p}(f) = a_{n/p}(f),

where we have made use of working over 𝔽_{p} for the last equality (here one needs the modified definition of Katz modular forms). As before, the coefficient a_{n/p}(f) has to be interpreted as 0 if p ∤ n. Let us observe the compatibility of F with diamond operators:

F ∘ < a >=< a > ∘F.

As 𝔽 is flat over 𝔽_{p} we can extend F to forms over 𝔽 with the same action on q-expansions. The compatibility with the diamond operators hence gives rise to the homomorphism (also denoted by F)

F : S_{1}(Γ_{1}(N), ϵ, 𝔽)′ → S_{p}(Γ_{1}(N), ϵ, 𝔽)′,

which on q-expansions is also given by a_{n}(F (f)) = a_{n/p}(f).

(ii) \underline{The Hasse invariant A:} The Hasse invariant A is a Katz modular form for SL_{2}(ℤ) of weight p − 1, which is defined over 𝔽_{p} and hence also over 𝔽 (see e.g. [K], section 2.0). All the q-expansions of A are identically 1. Also by the letter A we shall denote the multiplication by A map:

A : S_{1}(Γ_{1}(N), ϵ, 𝔽)′ → S_{p}(Γ_{1}(N), ϵ, 𝔽)′, f ↦ Af,

in the sense (Af)(E/S, P) = A(E/S, P)⊗f(E, P) ∈ \underline{ω}_{E/S}^{⊗p−1}⊗\underline{ω}_{E/S}. That this map is well defined comes from the fact that A is a modular form for SL_{2}(ℤ) and hence in particular for Γ_{0}(N), so that the character remains fixed. Since A has all q-expansions identically 1, we see

a_{n}(A(f)) = a_{n}(f).

(iii) \underline{The derivation θ:} In [K2] Katz constructs a homomorphism

θ : S_{p}(Γ_{1}(N), ϵ, 𝔽)′ → S_{p+2}(Γ_{1}(N), ϵ, 𝔽)′,

the effect of which on the q-expansions is given by

a_{n}(θ(f)) = na_{n}(f),

i.e. it acts like the logarithmic derivative q \frac{d}{dq}, whence the name.

Lemma 2.5.1 (a) The homomorphsims A and F are injective.

(b) We have the commutative diagram

\xymatrix{S_{1}(Γ_{1}(N), ϵ, 𝔽)′ \ar@{^{(}->}[r]^{F} \ar@{^{(}->}[dr]^{A} & S_{p}(Γ_{1}(N), ϵ, 𝔽)′ \ar[d]^{T_{p}} \\  & S_{p}(Γ_{1}(N), ϵ, 𝔽)′.}

(c) F ∘ T_{l}^{(1)} = T_{l}^{(p)} ∘ F if l ≠ p is a prime.

(d) A ∘ T_{p}^{(1)} = T_{p}^{(p)} ∘ A + ϵ(p) ∘ F.

Proof. All statements are immediately verified on the q-expansions using the formulae presented. □

Proposition 2.5.2 There is the exact sequence of 𝔽-vector spaces

0 → S_{1}(Γ_{1}(N), ϵ, 𝔽)′ ⟶^{F} S_{p}(Γ_{1}(N), ϵ, 𝔽)′ ⟶^{θ} S_{p+2}(Γ_{1}(N), ϵ, 𝔽)′.

Moreover, the kernel of θ, hence also the image of F, is explicitly given by

{ f ∈ S_{p}(Γ_{1}(N), ϵ, 𝔽)′ | a_{n}(f) = 0 ∀n s.t. p ∤ n }.

Proof. An element f in the kernel of θ is characterised by na_{n}(f) = 0 for all n. As the characteristic is p, this is equivalent to a_{n}(f) = 0 for all n such that p ∤ n. Hence, the kernel of θ is indeed equal to the set in the assertion.

On the q-expansions the formula a_{n}(F (f)) = a_{n/p}(f) makes it obvious that the image of F is contained in the kernel of θ.

The other inclusion is more difficult and is proved in [K2]. Bas Edixhoven, however, also explained to me a more direct argument. □

Proposition 2.5.3 Let K be a perfect field. A modular form f ∈ S_{k}(Γ_{1}(N), ϵ, K)′ is uniquely determined by the first Bk coefficients of its q-expansion at infinity, where

B = \frac{1}{12}N ∏_{l|N,l prime}(1 + \frac{1}{l}).

Proof. This follows from the proof of [E], proposition 4.2. □

Corollary 2.5.4 Let f be an element of S_{p}(Γ_{1}(N), ϵ, 𝔽)′. Then f ∈ Im(F) if and only if a_{n}(f) = 0 for all n ≤ B(p + 2) such that p ∤ n.

Proof. The image of F is equal to the kernel of θ by proposition 2.5.2. f is in the kernel of θ if and only if θ(f) = 0, which is equivalent of a_{n}(θ(f)) = 0 = na_{n}(f). The result is now clear from the preceding proposition. □

Theorem 2.5.5 (Edixhoven) Let 𝔽 be a finite field of characteristic p, N ≥ 5 an integer coprime to p and ϵ : (ℤ/N)^{∗} → 𝔽^{∗} a character.

The Hecke algebra 𝕋 is defined as the subring of End_{ℂ}(S_{p}(Γ_{1}(N), ℂ)) generated by the Hecke operators T_{n}. It can be generated as an abelian group by all T_{n} with n ≤ \frac{pN^{2}}{24} ∏_{l|N,l prime}(1 − \frac{1}{l^{2}} ).

Let R ≤ 𝕋 ⊗ 𝔽 be the sub 𝔽-vector space generated by the images of

• T_{n} for all n ≤ B(p + 2) with B as before such that p ∤ n and

• ϵ(l)− < l > for all l ∈ (ℤ/N)^{∗}.

Then there is an isomorphism of 𝔽-vector spaces

S_{1}(Γ_{1}(N), ϵ, 𝔽)′ → ((𝕋 ⊗_{ℤ} 𝔽)/R)^{∨_{𝔽}}.

Under this isomorphism the weight 1 Hecke operator T_{l}^{(1)} for l ≠ p prime corresponds to (T_{l}^{(p)} mod R)^{∨}, and T_{p}^{(1)} corresponds to (T_{p}^{(p)}+ < p > ∘F mod R)^{∨}. The operator F is given by F (T_{n}) = T_{n/p} if p divides n and by F (T_{n}) = 0 otherwise.

Proof. The mentioned bound on the generators of the Hecke algebra follows the proof of [E], proposition 4.2.

Let us now describe the main isomorphism.

[figure: f ∈; S_{1}(Γ_{1}(N), ϵ, 𝔽)′; F; S_{p}(Γ_{1}(N), ϵ, 𝔽)′; ∼; ((S_{p}(Γ_{1}(N), ℤ[1/N])) ⊗ 𝔽)(ϵ); ∼; prop. 2.4.4; ((Hom_{ℤ[1/N]−lin}(𝕋, ℤ[1/N]) ⊗ 𝔽)(ϵ); ∼; freeness; (𝕋 ⊗ 𝔽)^{∨_{𝔽}}(ϵ); (T_{n} ⊗ 𝔽 ↦ a_{n/p}(f)); (𝕋 ⊗ 𝔽)^{∨_{𝔽}}[ ϵ(l)− < l > | l ∈ (ℤ/N)^{∗} ]]

Hence using corollary 2.5.4 we find that the image of F is

(𝕋 ⊗ 𝔽)^{∨}[R] ≅ (𝕋 ⊗ 𝔽/R)^{∨},

as claimed.

It remains to check the claims on the Hecke operators. But they follow immediately from lemma 2.5.1 (c) and (d). □

2.6 Results on Γ_{0}(N)

In the previous section we have seen how to calculate Katz modular forms of weight 1 over a finite field 𝔽 of characteristic p of level N (N ≥ 5 and (N, p) = 1) with respect to a character ϵ.

It is natural to ask whether in the case of the trivial character we may replace the Hecke algebra 𝕋 for Γ_{1}(N) by the one for Γ_{0}(N) in the theorem. In this section we shall establish a sufficient criterion for this.

Let us first of all note that we ought to choose p = 2 as otherwise the theory is empty (there being no non-trivial modular forms of weight 1 for Γ_{0}(N)). Let us continue to assume N ≥ 5 with N odd.

Lemma 2.6.1 Let N > 1 be an odd integer. Then the following three statements are equivalent:

(i) All stabilizers of the action of Γ_{0}(N)/{±1} on ℍ are of odd order.

(ii) For all a, c ∈ ℤ such that (a, c) = 1 we have: a^{2} + c^{2} ≢ 0 (N).

(iii) There exists a prime l dividing N such that l ≡ 3(4).

Proof. It is well known that the stabilizer of τ ∈ ℍ is 1 except in the cases τ ∈ SL_{2}(ℤ)i and τ ∈ SL_{2}(ℤ)ζ_{3}. In the latter case, the stabilizer is of order 3 and conjugate (in SL_{2}(ℤ)) to the

group generated by the matrix ( \begin{matrix}0 & −1 \\ −1 & 1 \end{matrix} ). The stabilizer of i is the group of order 2 generated

by the matrix ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ).

So condition (i) is equivalent to the statement

∀ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) : ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}a & b \\ c & d \end{matrix} )^{−1} ∉ Γ_{0}(N).

Performing the matrix multiplication we find immediately that (i) is equivalent to (ii).

We shall now study equation a^{2} + c^{2} ≢ 0 (N) in the principal ideal domain ℤ[i], in which we have

a^{2} + c^{2} = (a + ic)σ(a + ic) = (a + ic)(a − ic),

where σ is complex conjugation (the non-trivial Galois automorphism of ℚ(i) over ℚ). Let us further recall that a prime number l ≠ 2 splits in ℤ[i] if and only if l ≡ 1 (4).

\underline{not (iii) ⇒ not (ii):} In this case all primes l dividing N are split. From what we recalled above, we can write l = (x_{l} + iy_{l})(x_{l} − iy_{l}) with x_{l}, y_{l} ∈ ℤ. If N = ∏_{l} l^{r_{l}}, then we put a + ic = ∏_{l}(x_{l} + iy_{l})^{r_{l}}. This implies N = (a + ic)(a − ic).

\underline{(iii) ⇒ (ii):} Let l be a prime dividing N, which is l ≡ 3 (4) and let a, c ∈ ℤ with (a, c) = 1 be given. Suppose there is an r ∈ ℤ such that rN = a^{2} + c^{2} = (a + ic)(a − ic). The prime decomposition of a + ic can only consist of primes 𝔓 lying over split primes, because otherwise a and c would not be coprime. Since l divides N it must also divide one of those 𝔓, which is impossible since l is non-split. □

Proposition 2.6.2 Assume that there exists a prime l dividing N s.t. l ≡ 3 (4). Then we have isomorphisms

S_{2}(Γ_{0}(N), 𝔽) = S_{2}(Γ_{1}(N), 𝔽)^{Δ} = S_{2}(Γ_{0}(N), 𝔽)′,

where Δ = Γ_{0}(N)/Γ_{1}(N){±1} = (ℤ/N)^{∗}/{±1}.

Proof. This follows from the proof of [E], theorem 5.6. One of the ingredients quoted therein is that the stabilizers of the covering group of Y_{1}(N) ↠ Y_{0}(N) have odd order. This, however, follows directly from lemma 2.6.1. □

Theorem 2.6.3 Let 𝔽 be a finite field of characteristic 2 and N ≥ 5 an odd integer, which is divisible by a prime number q s.t. q ≡ 3 (4).

The Hecke algebra 𝕋_{0} is defined as the subring of End_{ℂ}(S_{2}(Γ_{0}(N), ℂ)) generated by the Hecke operators T_{n}. It can be generated as abelian group by all T_{n} with n ≤ 2B with B as before.

Let R ≤ 𝕋_{0} ⊗𝔽 be the sub 𝔽-vector space generated by the images of T_{n} for all odd n ≤ 4B. Then there is an isomorphism of 𝔽-vector spaces

S_{1}(Γ_{0}(N), 𝔽)′ → ((𝕋_{0} ⊗_{ℤ} 𝔽)/R)^{∨_{𝔽}}.

Under this isomorphism the weight 1 Hecke operator T_{l}^{(1)} for l ≠ 2 prime corresponds to (T_{l}^{(2)} mod R)^{∨}, and T_{2}^{(1)} corresponds to (T_{2}^{(2)}+ < 2 > ∘F mod R)^{∨}. The operator F is given by F (T_{n}) = T_{n/2} if n is even and by F (T_{n}) = 0 if n is odd.

Proof. Denote by 𝕋_{1} the subring of End_{ℂ}(S_{2}(Γ_{1}(N), ℂ)) generated by all Hecke operators T_{n} and by M the sub 𝔽-vector space of 𝕋_{1} ⊗ 𝔽 generated by the images of the T_{n} with n odd. Furthermore, let I = ((1− < l >) ⊗ 𝔽 | l ∈ (ℤ/N)^{∗}) ◁ 𝕋_{1} ⊗ 𝔽.

The claims on the bounds follow again from [E], proposition 4.2.

By theorem 2.5.5 we have an isomorphism

S_{1}(Γ_{0}(N), 𝔽)′ ≅ ((𝕋_{1} ⊗ 𝔽)/(M + I))^{∨}.

Hence it suffices to have an isomorphism

(𝕋_{1} ⊗ 𝔽)/I ≅ 𝕋_{0} ⊗ 𝔽,

which is compatible with the Hecke operators. This isomorphism is obtained from proposition 2.6.2 using the isomorphisms

S_{2}(Γ_{0}(N), 𝔽) ≅ (𝕋_{0} ⊗ 𝔽)^{∨} and
S_{2}(Γ_{1}(N), 𝔽)^{Δ} ≅ (𝕋_{1} ⊗ 𝔽)^{∨}[I] ≅ ((𝕋_{1} ⊗ 𝔽/I))^{∨},

where the latter is obtained from 2.4.4 and the former by analogous arguments. □

3 Calculations of characteristic p modular forms of weight 1 and relation to Serre’s conjecture

This talk was given on 29 January 2003.

Very vaguely speaking, we will try to outline links - partly conjectural - between the theory of modular forms and number theory, more precisely the study of the absolute Galois group of the rational numbers.

The starting point is a theorem by Deligne, which associates to an eigenform over an algebraic closure of 𝔽_{p} a 2-dimensional odd Galois representation over the same field. Serre conjectured that there is a converse to Deligne’s result: given an odd irreducible 2-dimensional Galois representation there exists an eigenform of a precisely specified level, weight and character giving rise to this representation. The conjecture was extended to suggest that the minimal weight ought to be 1 if and only if the representation is unramified at p.

Serre’s conjecture has spectacular consequences, among them a direct proof of Fermat’s last theorem, which we sketch.

We will present some of the known results, focussing on dihedral representations.

Finally, we will discuss some calculations giving evidence for Serre’s conjecture, which were originally carried out by Mestre, and which we verified.

As mentioned before Serre’s conjecture was slightly modified and extended. Here we only treat the “currently up to date” form, as presented in Edixhoven’s paper [E2].

3.1 Deligne’s theorem

We will keep the notations of the previous talk, except that the condition N ≥ 5 has to be dropped. However, N will always be coprime to the characteristic p.

Let k be an integer, 𝔽 a finite extension of 𝔽_{p} and ϵ : (ℤ/N)^{∗} → 𝔽^{∗} a character. Then we consider the space S_{k}(Γ_{1}(N), ϵ, 𝔽)′ of Katz modular forms and we denote by 𝕋 = 𝕋(S_{k}(Γ_{1}(N), ϵ, 𝔽)′) its Hecke algebra, which is a finite dimensional 𝔽-vector space. Then the Hecke algebra of S_{k}(Γ_{1}(N), ϵ, \overline{𝔽_{p}})′ = S_{k}(Γ_{1}(N), ϵ, 𝔽)′ ⊗ \overline{𝔽_{p}} is isomorphic to 𝕋 ⊗ \overline{𝔽_{p}}.

From the formula

a_{1}(T_{n}f) = a_{n}(f)

we obtain (by the uniqueness of the q-expansion) an injection

S_{k}(Γ_{1}(N), ϵ, \overline{𝔽_{p}})′ → Hom_{\overline{𝔽_{p}}−lin}(𝕋 ⊗ \overline{𝔽_{p}}, \overline{𝔽_{p}}) ≅ Hom_{𝔽−lin}(𝕋, \overline{𝔽_{p}}), f ↦ (T_{n} ↦ a_{n}(f)).

In fact, the map is an isomorphism as one sees considering (more conceptually) a_{1}(T, f) =< T, f > as a bilinear pairing, which is non-degenerate in both variables. It remains to check one non-degeneracy: Given T ∈ 𝕋 ⊗ \overline{𝔽_{p}}, assume a_{1}(T f) = 0 for all f. Then in particular a_{1}(T (T_{n}f)) = a_{1}(T_{n}T f) = a_{n}(T f) = 0 for all f and all n. By the uniqueness of the q-expansion, we conclude T f = 0 for all f. Due to the faithfulness of the Katz forms as a 𝕋 ⊗ \overline{𝔽_{p}} module, we conclude T = 0.

Hom_{𝔽−lin}(𝕋, \overline{𝔽_{p}}) is naturally equipped with a 𝕋-action via

(T_{n}.f)(T_{m}) = f(T_{n}T_{m}),

which is compatible with the 𝕋-action on the space of Katz modular forms.

Inside Hom_{𝔽−lin}(𝕋, \overline{𝔽_{p}}) we can consider the set Hom_{𝔽−alg}(𝕋, \overline{𝔽_{p}}). Given an 𝔽-linear homomorphism f : 𝕋 → \overline{𝔽_{p}}, we see immediately

f is a 𝔽-algebra hom. ⇔ f is a normalised eigenform with eigenvalues f(T_{n}).

Using the isomorphism above we find that the 𝔽-algebra homomorphisms correspond precisely to the normalised Katz modular forms, which are eigenfunctions for all Hecke operators T_{n} with eigenvalues a_{n}(f).

Remark 3.1.1 We should note at this point that eigenspaces in S_{k}(Γ_{1}(N), ϵ, \overline{𝔽_{p}})′ are necessarily 1-dimensional. Just because of the formula a_{1}(T_{n}f) = a_{n}(f): if (λ_{n})_{n} is a system of eigenvalues then a_{n}(f) = a_{1}(T_{n}f) = a_{1}(λ_{n}f) = λ_{n}a_{1}(f) for any eigenform corresponding to (λ_{n})_{n}.

Moreover, there is a natural G_{𝔽} = G(\overline{𝔽_{p}}|𝔽)-action on Hom_{𝔽−lin}(𝕋, \overline{𝔽_{p}}) given by composition. One has:

Proposition 3.1.2 In the setting above there is a bijection

Hom_{𝔽−alg}(𝕋, \overline{𝔽})/_{G_{𝔽}−conjugacy} ↔ maximal ideals of 𝕋,

given by sending an 𝔽-algebra homomorphism f : 𝕋 → \overline{𝔽} to its kernel.

Proof. We point out that in an Artinian algebra every prime ideal is maximal. Since for every maximal ideal 𝔪 the ring 𝕋/𝔪 is contained (via f) in \overline{𝔽} the kernel is a maximal ideal. If g = σ∘f for some field automorphism σ, the kernel of f is, of course, equal to the kernel of g. Hence the allocation is well defined.

The surjectivity is clear: given 𝔪, just take f : 𝕋 → 𝕋/𝔪 ≤ \overline{𝔽}.

Suppose now that f and g have the same kernel 𝔪. That precisely means that f and g give two embeddings of the field 𝕋/𝔪 into \overline{𝔽}. By well known field theory, one can be obtained from the other by composing with an element of G_{𝔽}, which proves the injectivity. □

Remark 3.1.3 By the proposition one can localize 𝕋 at its maximal ideals in order to keep exactly one particular system of eigenvalues modulo G_{𝔽}-conjugacy. Linear algebra techniques (calculation of characteristic polynomials) then directly produce the Fourier coefficients of one eigenform. It is not difficult to calculate its (finite) G_{𝔽}-orbit.

One can show that in characteristic 0 there exists a basis of eigenforms for the space of (Katz) cusp forms, at least if the level is square-free. This, however, is no longer true in general for positive characteristic. We shall see examples of that in the end of this talk.

Before we can state Deligne’s theorem we have to introduce some notation. Let L|ℚ be a Galois extension and let 𝔓 be a prime above l. Then the other primes above l are given by σ𝔓 for σ ∈ G(L|ℚ). We denote by G_{𝔓}(L|ℚ) the decomposition group of 𝔓, i.e. the subgroup of G(L|ℚ) consisting of those σ s.t. σ(𝔓) = 𝔓. One has a natural isomorphism

G_{𝔓}(L|ℚ) → G(L_{𝔓}|ℚ_{l})

given by extending the automorphisms by continuity to the completions. It is immediate that one has G_{σ𝔓} = σG_{𝔓}σ^{−1}. Furthermore, reduction modulo the maximal ideal of the ring of integers of L_{𝔓} gives a surjection

0 → G_{0}(L_{𝔓}|ℚ_{l}) → G(L_{𝔓}|ℚ_{l}) → G(λ_{𝔓}|𝔽_{l}) → 0,

where λ_{𝔓} is the residue field of L_{𝔓}. The kernel is called the inertia group.

We need a special case. Assume that L|ℚ is unramified at l, which by definition means that the right hand side map is an isomorphism (i.e. the inertia group is trivial). Then one has the element Frobenius at l Frob_{l} ∈ G(L_{𝔓}|ℚ_{l}) ⊆ G(L|ℚ), which is the usual map x ↦ x^{l} on the finite field side. A different choice of prime 𝔓 would have resulted in a conjugate Frobenius. All the statements below will only depend on the conjugacy class, so there is no need to specify the ideal chosen any further.

Theorem 3.1.4 (Deligne) Let p be a prime, N a positive integer prime to p, k ≥ 1, 𝔽 an extension of 𝔽_{p} and ϵ : (ℤ/N)^{∗} → 𝔽^{∗} a character.

For any normalised eigenform f ∈ S_{k}(Γ_{1}(N), ϵ, \overline{𝔽_{p}}) of the Hecke algebra, there is a unique semi-simple continuous representation

ρ_{f} : G_{ℚ} → GL_{2}(\overline{𝔽_{p}})

satisfying

(i) ρ_{f} is unramified outside pN and

(ii) Tr(ρ_{f}(Frob_{l})) = a_{l}(f) and det(ρ_{f}(Frob_{l})) = ϵ(l)l^{k−1} for all primes l not dividing pN.

Remark 3.1.5 Let us note some equivalent formulations.

Statement (ii) can be summarized by saying that the characteristic polynomial of ρ_{f}(Frob_{l}) is X^{2} − a_{l}(f) + ϵ(l)l^{k−1} for all primes l ∤ pN.

In a more representation theoretic language, one also defines the (degree 2) character of the representation

χ_{f} = Tr ∘ ρ_{f}

and the (degree 1) determinant character

δ_{f} = det ∘ρ_{f}.

Then (ii) has obvious reformulations. But there is more to it: one has

det ∘ρ_{f} = (ϵ ∘ ψ_{N}) · ψ_{p}^{k−1},

where ψ_{n} denotes the n-th cyclotomic character ψ_{n} : G_{ℚ} → (ℤ/n)^{∗}, which is defined by σ(ζ_{n}) = ζ_{n}^{ψ(σ)} for any primitive n-th root of unity ζ_{n}. In particular, one has ψ_{N}(Frob_{l}) = l ∈ (ℤ/N)^{∗} and ψ_{p}(Frob_{l}) = l ∈ (ℤ/p)^{∗} by the assumption on l. Hence, the claimed equality holds for a set of primes of density 1, and hence by Chebotarev always.

Let c now denote a complex conjugation (this is the “infinite Frobenius”). It is again welldefined up to conjugacy. It is obtained by completing \overline{ℚ} for an absolute value above the absolute value of ℚ, or, equivalently, by choosing an embedding of \overline{ℚ} into ℂ and using the complex conjugation on ℂ. In particular, c(ζ_{n}) = ζ_{n}^{−1} for any n, whence ψ_{N}(c) = ψ_{p}(c) = −1. Accordingly, det(ρ_{f}(c)) = ϵ(ψ_{N}(c)) · ψ_{p}^{k−1}(c) = ϵ(−1)(−1)^{k−1} = (−1)^{k}(−1)^{k−1} = −1, where we used that f is a non-zero modular form with character ϵ, and that is only possible if ϵ(−1) = (−1)^{k}.

A representation ρ with det(ρ(c)) = −1 is called an odd representation. Consequently, the representations obtained from Deligne’s theorem are all odd.

3.2 Serre’s conjecture

We shall start by associating to a given 2-dimensional representation

ρ : G_{ℚ} → GL_{2}(\overline{𝔽_{p}})

a triple (N(ρ), ϵ(ρ), k(ρ)). By χ = Tr ∘ ρ we denote the character of ρ. Call L|ℚ the Galois extension cut out by ρ, i.e. G(L|ℚ) = G_{ℚ}/Ker(ρ).

Ideally, if ρ = ρ_{f}, then this triple should correspond to the minimal level, character and the weight of f. We shall later see a theorem making this precise.

N(ρ) = the p-primary part of the Artin conductor: We will introduce the Artin conductor and try to motivate it a little bit. Fix a prime l (not

necessarily distinct from p) and a prime 𝔓 of L lying above l.

We shall first introduce the ramification groups:

• G_{−1}(L_{𝔓}|ℚ_{l}) = G_{𝔓}(L|ℚ) = G(L_{𝔓}|ℚ_{l}) is the decomposition group, which we have already defined above.

• G_{i}(L_{𝔓}|ℚ_{l}) = { σ ∈ G_{−1}(L_{𝔓}|ℚ_{l}) | σx ≡ x (𝔓^{i+1}) ∀x ∈ O_{L_{𝔓}} } is the i-th ramification group (in lower numbering) for an integer i ≥ 0.

• G_{0}(L_{𝔓}|ℚ_{l}) is the inertia group as defined above.

• G_{1}(L_{𝔓}|ℚ_{l}) is the unique l-Sylow subgroup of G_{0}(L_{𝔓}|ℚ_{l}) and is called the (wild) ramification group.

• G_{i}(L_{𝔓}|ℚ_{l}) ⊇ G_{i+1}(L_{𝔓}|ℚ_{l}) for all i ≥ −1.

• The subfield cut out by G_{0} is the maximal unramified subextension, and the subfield cut out by G_{1} is the maximal tamely ramified subextension of L_{𝔓}|ℚ_{l}.

• Let us write: g_{i,l} = |G_{i}(L_{𝔓}|ℚ_{l})|. This number does not depend on the choice of prime 𝔓 above l.

• The number c_{i,l} = codim((\overline{𝔽_{p}}^{2})^{G_{i}(L_{𝔓}|ℚ_{l})}) does not depend on the choice of prime 𝔓 over l.

Definition 3.2.1 Given a representation ρ as above, we associate to it the numbers:

f_{l}(ρ) = ∑_{i≥0} \frac{g_{i,l}}{g_{0,l}} c_{i,l}.

It is a theorem that these numbers are integers greater equal 0.

The local Artin conductor of ρ at l is defined to be the number

𝔣_{l}(ρ) = 𝔣_{l}(χ) = l^{f_{l}(ρ)}.

The global Artin conductor of ρ is defined to be the number

𝔣(ρ) = 𝔣(χ) = ∏_{l prime}𝔣_{l}(ρ).

Our number N(ρ) is defined as

𝔣(ρ) = ∏_{l prime, l≠p} 𝔣_{l}(ρ).

We recall the analogously defined injective function (s ≥ 1)

η_{L_{𝔓}|ℚ_{l}}(s) = ∑_{i=1}^{s} \frac{g_{i,l}}{g_{0,l}} ,

which defines the ramification groups in upper numbering:

G^{η_{L_{𝔓}|ℚ_{l}}(s)}(L_{𝔓}|ℚ_{l}) = G_{s}(L_{𝔓}|ℚ_{l}).

In particular, if the representation ρ is 1-dimensional, we have

f_{l}(ρ) = η_{L_{𝔓}|ℚ_{l}}(j) + 1,

where j is the maximal integer, s.t. ρ|_{G_{i}} ≠ 1_{G_{i}}. The ramification groups in upper numbering have important properties. First of all they are compatible with quotients. Secondly, they are the images (in the case of an abelian local extension) of the i-th group of principal units (1+p^{i}, “oneunits”, “Einseinheiten”) under the norm residue symbol of local class field theory. Certainly, one should at this philosophical point not forget to appreciate the conductor-discriminant formula

𝔡_{L|ℚ} = N_{L|ℚ}𝔇_{L|ℚ} = ∏_{l} l^{f_{L_{𝔓}|ℚ_{l}} ∑_{i≥0}(g_{i,l}−1)} = ∏_{χ} 𝔣(χ)^{χ(1)},

where the last product runs over all irreducible characters over the complex numbers of G(L|ℚ).

Let us note explicitly that N(ρ) is a measure for the ramification outside p and that ρ is unramified at l ≠ p if and only if l ∤ N(ρ).

ϵ(ρ) and k(ρ) modulo p − 1:

Let us denote by δ_{ρ} = det ∘ρ the determinant character. One can check that we have the commutative diagram

\xymatrix{G_{ℚ} \ar[rrrr]^{δ_{ρ}} \ar@{->>}[dr]^{ψ_{N(ρ)p}} &  &  &  & \overline{𝔽_{p}}^{∗} \\  & (ℤ/N(ρ)p)^{∗} \ar[r]^{∼} \ar@{-->}[urrr] & (ℤ/N(ρ))^{∗} \ar[urr]_{×}_{ϵ(ρ)} & (ℤ/p)^{∗}, \ar[ur]_{ϕ}}

where the homomorphism ϕ is given by raising to some power a = k(ρ) − 1, where the last equality defines k(ρ) modulo p − 1. This means precisely that the determinant character is given by

δ_{ρ} = (ϵ(ρ) ∘ ψ_{N(ρ)}) · ψ_{p}^{k(ρ)−1}.

\underline{k(ρ):}

There is a slightly complicated recipe for specifying k(ρ) in all cases. One should consult [E2] for that. We shall content ourselves with the following:

• k(ρ) depends only on the ramification of ρ at p.

• k(ρ) = 1 if and only if ρ is unramified at p.

Conjecture 3.2.2 (Serre) Let p be a prime number and ρ : G_{ℚ} → GL_{2}(\overline{𝔽_{p}}) a continuous irreducible odd representation.

Then there exists an eigenform f ∈ S_{k(ρ)}(Γ_{1}(N(ρ)), ϵ(ρ), \overline{𝔽_{p}})′ such that ρ is isomorphic with ρ_{f}.

Theorem 3.2.3 Let p ≠ 2 be a prime, g ∈ S_{k}(Γ_{1}(N), ϵ, \overline{𝔽_{p}})′ be a Hecke eigenform and ρ = ρ_{g} be the representation constructed by Deligne and assume additionally that it is irreducible.

Then there is an eigenform f ∈ S_{k(ρ)}(Γ_{1}(N(ρ)), ϵ(ρ), \overline{𝔽_{p}})′ such that ρ is isomorphic with ρ_{f}. Moreover, N is a multiple of N(ρ), k ≥ k(ρ) and ϵ is obtained from ϵ(ρ) by composing with

ℤ/N → ℤ/N(ρ).

Remark 3.2.4 Serre’s conjecture implies Fermat’s last theorem directly, i.e. without using the more difficult fact that it also implies Taniyama-Shimura.

The argumentation can be sketched quickly. Assume given a prime p ≥ 5, a, b, c ∈ ℤ satisfying abc ≠ 0 and

a^{p} + b^{p} = c^{p}.

Wlog. one can assume a ≡ 3 (4) and that b is even. Then one considers the elliptic curve

E : y^{2} = x(x − a^{p})(x + b^{p}).

One finds that E is semi-stable with discriminant

Δ_{E} = \frac{−1}{256}(abc)^{2p}.

To every elliptic curve over ℚ one can associate a mod p Galois representation

ρ : G_{ℚ} → Aut(E_{p}) ≅ GL_{2}(𝔽_{p}),

which is irreducible by a theorem by Mazur. The triple associated with ρ is

N(ρ) = 2, k(ρ) = 2, ϵ(ρ) = 1.

BUT there are no non-trivial eigenforms of weight 2 for Γ_{0}(2). Hence, assuming Serre’s conjecture, one finds a contradiction. (This section was copied from [D].)

In order to see another very useful consequence of Serre’s conjectures, we state the following consequence of Chebotarev’s density theorem.

Theorem 3.2.5 Let p be a prime, 𝔽 a finite extension of 𝔽_{p} and ρ : G_{ℚ} → GL_{2}(𝔽) a continuous, semi-simple representation.

(i) If \underline{p ≥ 3}, then ρ is uniquely determined by Tr(ρ(Frob_{l})) for any set of unramified primes l of density 1.

(ii) If \underline{p = 2}, then ρ is uniquely determined by Tr(ρ(Frob_{l})) and det(ρ(Frob_{l})) for any set of unramified primes l of density 1.

Proof. This is proved (in fact also cited) in [F], proposition 2.6 (b). We just have to note that the trace of the second exterior power of ρ is given by the determinant, which one can check by a simple calculation. □

Serre’s conjecture can also be seen as an analogue of the following conjecture over ℂ ([F], conjecture 3.7, implying a special case of Artin’s conjecture).

Conjecture 3.2.6 Let ρ : G_{ℚ} → GL_{2}(ℂ) a continuous irreducible odd representation. Then ρ is equivalent to ρ_{g} for some newform g of weight 1 (One can associate to such a newform a representation of the requested type, which is a result by Deligne-Serre.)

Serre’s conjecture says more about the structure of the absolute Galois group of ℚ than the above conjecture, because all the finite subgroups of PGL_{2}(ℂ) are

• C_{n} for n ≥ 1,

• D_{n} for n ≥ 1„

• A_{4}, S_{4} and A_{5},

whereas the finite subgroups of PGL_{2}(\overline{𝔽_{p}}) are

• conjugates of the upper triangular matrices,

• C_{n}, D_{n} for n ≥ 1 with p ∤ n,

• PSL_{2}(𝔽_{p^{r}}), PGL_{2}(𝔽_{p^{r}}) both for any r ≥ 1,

• A_{4}, S_{4} and A_{5}.

Knowing this, the following theorem is a nearly full proof of conjecture 3.2.6.

Theorem 3.2.7 (Langlands-Tunnel) Let ρ : G_{ℚ} → GL_{2}(ℂ) a continuous irreducible odd representation such that ρ(G_{ℂ}) is solvable. Then ρ is equivalent to ρ_{g} for some newform g of weight 1, which is a Hecke eigenform.

Remark 3.2.8 In the case of dihedral image, this was already known to Hecke.

3.3 Modularity of dihedral representations

In this section we rapidly develop the representation theory of dihedral groups and, using the theorem of Langlands-Tunnel, we conclude that any odd dihedral representation is modular. Furthermore, we shall consider a test case for weight 1 modular forms; namely the dihedral extensions obtained from a quadratic number field that ramifies precisely at a given prime q.

• (My working definition, might not be the standard one.) A finite group G is called dihedral (≅ D_{A}) if there is an abelian group A of odd order, so that we have a split exact sequence of finite groups

\xymatrix{0 \ar[r] & A \ar[r] & G \ar[r] & ℤ/2 \ar[r] \ar[l]_{ϕ} & 0,}

where the non-trivial element σ ∈ ϕ(ℤ/2) satisfies σaσ = a^{−1}. Then G is a semidirect product of A and ℤ/2.

• Given any character χ : A → \overline{𝔽_{p}}^{×}, we define the induced representation

Ind_{A}^{G}χ = \{ f : G → \overline{𝔽_{p}} map | f(ag) = χ(a)f(g) ∀a ∈ A \} ≅ \overline{𝔽_{p}}^{2},

where the isomorphism is given by sending f to ρ_{χ}(f) = (f(1), f(σ)). G acts on Ind_{A}^{G}χ by (g.f)(g̃) = f(g̃g). Via the isomorphism above we find immediately that

ρ_{χ}(σ) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) and ρ_{χ}(a) = ( \begin{matrix}χ(a) & 0 \\ 0 & χ(a^{−1}) \end{matrix} ) for all a ∈ A.

• ρ_{χ} is irreducible if and only if χ is not the trivial character.

This is easily proved by observing that the eigenvectors of ρ_{χ}(σ) are (1, 1)^{T} and (1, −1)^{T}. The matrix of ρ_{χ}(a) stabilizes them if and only if χ(a) = χ(a^{−1}) for all a ∈ A, implying the assertion.

• Two representations ρ_{χ} and ρ_{ψ} are conjugate (i.e. equivalent) if and only if either χ = ψ or χ = ψ^{−1}. This can be seen by considering Mρ_{χ}(a) = ρ_{ψ}(a)M for a matrix M and all a ∈ A.

• Any 2-dimensional representation ρ : G → GL_{2}(\overline{𝔽_{p}}) is induced from a character as above. To see this, one restricts ρ to A, where it becomes reducible as A is abelian. So, after a

suitable base change, one finds ρ(a) = ( \begin{matrix}ψ_{1}(a) & 0 \\ 0 & ψ_{2}(a) \end{matrix} ) for two characters ψ_{1}, ψ_{2} of A.

Writing down the condition that ρ(σ) acts as inversion on A, one sees ψ_{1}(a) = ψ_{2}(a^{−1}). Replacing the basis vectors by a suitable scalar multiple, one can further assume that

ρ(σ) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). Hence, we have precisely recovered the form above.

• Assume now that p does not divide the order of A. Then there are (|A| − 1)/2 distinct irreducible 2-dimensional representations of G. For that is the number of pairs (χ, χ^{−1}) of non-trivial characters of A. As the character table is square, we also know that we have thus described all irreducible representations of G.

Proposition 3.3.1 Any odd representation ρ : G_{ℚ} → GL_{2}(\overline{𝔽_{p}}) with dihedral image in PGL_{2}(\overline{𝔽_{p}}) comes from a modular form, which is the reduction modulo p of a form in characteristic 0.

Proof. We know that ρ is induced from a character χ. We can lift the character to characteristic 0. Choose a lift and call it ψ. It can be induced to give an odd representation Φ in characteristic 0, whose reduction is ρ. By the Langlands-Tunnel theorem (in fact already by results of Hecke), there is an eigenform in characteristic 0 corresponding to Φ. One can reduce this modular form to characteristic p, in order to obtain the desired modular form. □

In a similar manner one can prove the

Proposition 3.3.2 Any odd representation ρ : G_{ℚ} → GL_{2}(𝔽_{3}) comes from a modular form, which is the reduction modulo p of a form in characteristic 0.

Proof. One uses a split surjection GL_{2}(ℤ[\sqrt{−2}]) ↠ GL_{2}(𝔽_{3}) in order to lift the representation into characteristic 0. Then the Langlands-Tunnel theorem can be applied to give an eigenform in characteristic 0 with the desired reduction. □

We shall now construct for each prime q ∤ 2p dihedral representations of G_{ℚ}, which are of Artin conductor q. We already know their modularity, so they provide a good testing ground for any calculations of modular forms of weight 1 over \overline{𝔽_{p}}.

Let K be the quadratic extension of ℚ defined by

K = \begin{cases}ℚ(\sqrt{q}) & for q ≡ 1 (4) \\ ℚ(\sqrt{−q}) & for q ≡ 3 (4). \end{cases}

It has discriminant ideal 𝔡_{K|ℚ} = (q). Let now H_{K} be the Hilbert class field of K, and consider inside H_{K} the maximal field L such that the degree L|K is odd. This degree is then u, where h_{K} = 2^{r}u is the class number of K. If we write A = G(L|K), then G(L|ℚ) is the dihedral group D_{A}.

Let now χ be a character of A. We compute the Artin conductor of the induced representation ρ_{χ} using the formula

𝔣(L|ℚ, Indχ) = 𝔡_{K|ℚ}^{χ(1)}N_{K|ℚ}(𝔣(L|K, χ))

and find that it is (q). (In fact, I should have introduced the Artin conductor for any extension of number fields, and not just extensions of ℚ. But that is possible extending the definitions in an evident way.)

3.4 Explicit examples

We implemented the algorithm deriving from theorem 2.6.3 for the calculation of Katz cusp forms of weight 1 for Γ_{0}(N) over \overline{𝔽_{2}} for odd N ≥ 5 (recall that there are only non-zero modular forms if p = 2).

Let us recall some notation:

• The number B is defined as \frac{1}{12}N ∏_{l|N,l prime}(1 + \frac{1}{l}).

• The Hecke algebra 𝕋_{i} for i ∈ {0, 1} is defined as the subring of End_{ℂ}(S_{2}(Γ_{i}(N), ℂ)) generated by the Hecke operators T_{n}. It can be generated as an abelian group by all T_{n} with n ≤ 2B for i = 0.

• The sub 𝔽-vector space R ≤ 𝕋_{0} ⊗ 𝔽 is generated by the images of T_{n} for all odd n ≤ 4B.

Explicitly, we calculate in MAGMA the 𝔽_{2}-vector space

((𝕋_{0} ⊗_{ℤ} 𝔽_{2})/R)^{∨_{𝔽_{2}}},

on which we have described the action of the Hecke operators.

Theorem 2.6.3 states that there is an isomorphism

S_{1}(Γ_{0}(N), \overline{𝔽_{2}})′ → ((𝕋_{0} ⊗_{ℤ} \overline{𝔽_{2}})/R)^{∨\overline{𝔽_{2}}}

compatible with the Hecke operators if there is a prime q|N that is q ≡ 3 (4). Otherwise, there are certainly counter examples.

Let us recall that we have seen, when deriving theorem 2.6.3 from theorem 2.5.5, that we needed the surjection

𝕋_{1} ⊗ \overline{𝔽_{2}}/(R + I) ↠ 𝕋_{0} ⊗ \overline{𝔽_{2}}/R

to be an isomorphism, which we found to be true if there is such a q dividing N with q ≡ 3 (4). Dualizing, we obtain in any case an injection (from theorem 2.5.5)

((𝕋_{0} ⊗_{ℤ} \overline{𝔽_{2}})/R)^{∨\overline{𝔽_{2}}} ↪ S_{1}(Γ_{0}(N), \overline{𝔽_{2}})′.

So, we always get Katz cusp forms of the good weight and level, but if q ≡ 1 (4) we are not certain to find them all.

Let us note also, that in the case of q|N with q ≡ 3 (4), we know that all Katz modular forms are reductions of characteristic 0 modular forms (proposition 2.6.2). We shall below see an example of level N = 1429, where we have a form not coming from characteristic 0 in weight 1.

This method was already used by Mestre in 1987 to produce evidence for Serre’s conjecture ([MS]). In fact, Serre thanks Mestre in the introduction to his paper presenting the conjectures ([S]) for his computational work. Apparently, Mestre was only able to do prime levels, and he did not prove the method (maybe because the condition on the q was missing).

We redid most of the calculations of [MS] and verified them with one exception (p = 1367). Our aim for the future is, of course, to go beyond that.

The computational results, which I will present in the rest of this talk/section, were hence originally obtained by Mestre.

Let us look at some examples. Mestre’s table:

\begin{matrix}N = q & 229 & 257 & 283 & 331 & 491 & 563 & 577 & 643 & 653 \\ ≡ (4) & 1 & 1 & 3 & 3 & 3 & 3 & 1 & 3 & 1 \\ d & 2 & 2 & 3 & 3 & 6 & 6 & 3 & 3 & 4 \\ h & 3 & 3 & 3 & 3 & 9 & 9 & 7 & 3 & 1 \\ #dih & 1 & 1 & 1 & 1 & 4 & 4 & 3 & 1 & 0 \\ #ef & 1 & 1 & 1 & 1 & 4 & 4 & 3 & 1 & 2 \\ B(m) & 2 & 2 & 3 & 3 & 3 & 3 & 1 & 3 & 2 \\ 𝔽 & 𝔽_{2} & 𝔽_{2} & 𝔽_{2} & 𝔽_{2} & 𝔽_{2^{3}} & 𝔽_{2^{3}} & 𝔽_{2^{3}} & 𝔽_{2} & 𝔽_{2^{2}} \end{matrix}

\begin{matrix}N = q & 751 & 761 & 1129 & 1229 & 1367 & 1381 & 1399 & 1423 & 1429 \\ ≡ (4) & 3 & 1 & 1 & 1 & 3 & 1 & 3 & 3 & 1 \\ d & 9 & 2 & 5 & 2 & 16 & 4 & 15 & 6 & 8 \\ h & 15 & 3 & 9 & 3 & 25 & 1 & 27 & 9 & 5 \\ #dih & 7 & 1 & 4 & 1 & 12 & 0 & 13 & 4 & 2 \\ #ef & 7 & 1 & 4 & 1 & 12 & 2 & 13 & 4 & 5 \\ B(m) & 3 & 2 & 2 & 2 & 3 & 2 & 3 & 3 & 2 \\ 𝔽 & 𝔽_{2^{4}} & 𝔽_{2} & 𝔽_{2^{3}} & 𝔽_{2} & 𝔽_{2^{10}} & 𝔽_{2^{2}} & 𝔽_{2^{9}} & 𝔽_{2^{3}} & 𝔽_{2^{6}} \end{matrix}

d = dim 𝕋_{0} ⊗ 𝔽_{2}/R  h = class number of K
K = ℚ(\sqrt{q}) for q ≡ 1 (4)  K = ℚ(\sqrt{−q}) for q ≡ 3 (4)
#dih = #{dihedral reps}  #ef = #{eigenforms}
B(m) = max T_{l} − a_{l} nilp. deg. m  𝔽 = minimal field

We see that there are three cases, where there are non-dihedral representations, i.e. also fields ramifying only at q.

The condition (B(m)) can be rephrased to say that there is a Hecke operator having a Jordan block of size m and that m is maximal with that property. Hence, B(m) with m ≥ 2 means that there is no basis of eigenforms for the space of Katz cusp forms.

An important example is p = 1429. Looking at the eigenvalues of the Hecke operators, one finds that the image of the associated Galois representation is the group SL_{2}(𝔽_{8}), which is not a subgroup of GL_{2}(ℂ). Hence, there are eigenforms that are not reductions of forms in characteristic 0. This means that there are more Katz modular forms than classical ones in weight 1.

References

[D] Darmon, H.: Serre’s Conjectures in Murty, K. (Ed.): Seminar on Fermat’s last theorem CMS Conference Proceedings Vol. 17, AMS Publ., Providence

[DI] Diamond, F., Im, J.: Modular forms and modular curves in Murty, K. (Ed.): Seminar on Fermat’s last theorem CMS Conference Proceedings Vol. 17, AMS Publ., Providence

[E] Edixhoven, B.: Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight 1, submitted.

[E2] Edixhoven, B.: Serre’s conjecture, in Cornell, Silverman (Ed.): Fermat’s Last Theorem, Springer

[F] Darmon, H., Diamond, F., Taylor, R.: Fermat’s Last Theorem in Elliptic Curves and Modular Forms, International Press, 1997

[G] Gross, B.H.: A Tameness Criterion for Galois Representations Associated to Modular Forms (MOD p), in Duke Mathematical Journal, Vol. 61, No. 2, pp. 445-517

[K] Katz, N.: p-adic properties of modular schemes and modular forms, in Modular Functions of One Variable III, LNM 350, Springer

[K2] Katz, N.: A Result on Modular Forms in Characteristic p in Modular Functions of One Variable V, LNM 601, Springer

[M] Merel, L.: Universal Fourier Expansions of Modular Forms, in Frey, G. (Ed.) On Artin’s conjecture for odd 2-dimensional representations, pp. 59–94, Lecture Notes in Math., 1585, Springer, Berlin, 1994

[MS] Mestre, J.-F.: Lettre à Serre dated 8 October 1987

[S] Serre, J.P.: Sur les représentations modulaires de degré 2 de Gal(\overline{ℚ}/ℚ) in Duke Mathematical Journal, Vol. 54, 1987, pp. 179-230