CNTA 99

MODULAR FORMS MOD p

or

GEOMETRY OF HILBERT MODULAR VARIETIES IN POSITIVE CHARACTERISTIC, HILBERT MODULAR FORMS MODULO p^{m}, AND APPLICATIONS TO CONGRUENCES

BETWEEN SPECIAL VALUES OF L-FUNCTIONS

Eyal Z. Goren McGill University

Lecture available from

http://www.math.mcgill.ca/ g̃oren/publications.html

Let L be a totally real field of degree g. Let p be a rational prime. \underline{To Simplify:} Assume p is inert, h_{L}^{+} = 1.

C.L. Siegel: ζ_{L}(1 − n) ∈ ℚ for n ≥ 1 integer, and is zero if and only if n is odd.

Such values, and more generally the values, say, L(1 − n, χ), where χ is a Dirichlet character, appear “everywhere”:

• Stable Homotopy group of spheres.

• In K-theory: R_{n} · |K_{2n−2}(O_{L})|/|K_{2n−1}(O_{L})_{t}|.

• Cohomology of formal groups.

• Counting problems in characteristic p (e.g., supersingular points)

• Volumes of symmetric spaces, hyperbolic manifolds.

• p-Sylow subgroups of cyclotomic ℤ_{p}-extensions.

• Special values of modular forms.

We shall derive the following congruences (following Kummer, Kubota-Leopoldt, Serre, Deligne-Ribet, Katz, Mazur,...):

Corollary 1. Let k ≥ 2.

1. Let p ≠ 2, then if k ≡ 0 (mod p − 1) val_{p}(ζ_{L}(1 − k)) ≥ −1 − val_{p}(k),

and ζ_{L}(1 − k) is p-integral if k ≢ 0 (mod p − 1).

2. If p = 2, then

val_{2}(ζ_{L}(1 − k)) ≥ g − 2 − val_{2}(k).

Corollary 2. Let k, k′ ≥ 2 and k ≡ k′ (mod (p − 1)p^{m}).

1. If k ≢ 0 (mod p − 1) then

(1 − p^{g(k−1)})ζ_{L}(1 − k) ≡ (1 − p^{g(k′−1)})ζ_{L}(1 − k′) (mod p^{m+1}).

2. If k ≡ 0 (mod p − 1) but p ≠ 2, then

(1 − p^{g(k−1)})ζ_{L}(1 − k) ≡ (1 − p^{g(k′−1)})ζ_{L}(1 − k′) (mod p^{m−1−val_{p}(k·k′)}). 3. If p = 2, then

(1 − 2^{g(k−1)})ζ_{L}(1 − k) ≡ (1 − 2^{g(k′−1)})ζ_{L}(1 − k′) (mod 2^{m+g−2−val_{2}(k·k′)}).

Hilbert modular morms over ℂ: A Hilbert Modular Form (HMF) over ℂ of weight r = (r_{1}, …, r_{g}) and level Γ_{1}(N) is a holomorphic (also at infinity) function,

f : ℌ^{g} ⟶ ℂ,

that satisfies

f(γ · τ) = j(γ, τ)^{r}f(τ),

for all γ ∈ Γ_{1}(N). Here

Γ_{1}(N) = \{γ = (\begin{matrix}a & b \\ c & d \end{matrix}) : γ ∈ SL_{2}(O_{L}), γ ≡ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix})\} (mod N),

and

j(γ, τ)^{r} = ∏_{i=1}^{g}(σ_{i}(c)τ_{i} + σ_{i}(d))^{r_{i}},

where we let Emb(L, ℂ) = {σ_{1}, ⋯, σ_{g}}.

One may think of f as a section of a line bundle over Γ_{1}(N)\ℌ^{g} defined by the factor of automorphy j(γ, τ)^{r}. The modular form f has a q-expansion

f(τ) = a_{0} + ∑_{ν∈D_{L}^{−1 +}}a_{ν} · e^{2πi·T r ντ}.

C.L. Siegel: For k ≥ 2 and even, there exists a modular form E_{k}^{∗} of weight (k, k, …, k) and level 1, such that its q-expansion (alas, w.r.t. another cusp) is

E_{k}^{∗} = 2^{−g}ζ_{L}(1 − k) + ∑_{ν∈O_{L}^{+}} (∑_{𝔠|(ν)}Norm(𝔠)^{k−1}) e^{2πi·T r ντ}.

By a Hilbert-Blumenthal abelian variety (HBAV) we mean a principally polarized abelian variety of dimension g together with an embedding

ι : O_{L} ⟶ End(A).

We shall use the notation \underline{A} for (A, ι, λ).

Then, HBAVs correspond to points on SL_{2}(O_{L})\ℌ^{g}, and also to polarized O_{L}-lattices up to homothety. HBAVs with Γ_{1}(N) structure would correspond to points on Γ_{1}(N)\ℌ^{g} and to polarized O_{L} lattices up to homothety together with some equivalence class of bases.

This modular interpretation can be extended to the cusps that stand for “totally degenerate” objects. Those are analogues of the Tate curve, and are of the form 𝔾_{m} ⊗ D_{L}^{−1}/periods.

Under this modular interpretation, one can derive the following facts (that generalizes the familiar notions for g = 1):

• a_{0} = special value of f at the point corresponding to a particular Tate object.

• Modular forms of weight (r_{1}, ⋯, r_{g}) correspond to (r_{1}, ⋯, r_{g})- homogeneous functions of polarized O_{L}-lattices.

Now, giving a polarized O_{L}-lattice in ℂ^{g} is the same as giving a polarized O_{L}-lattice in ℂ^{g} up to homothety together with a generator of ℂ^{g} as an O_{L} ⊗ ℂ module, and hence is the same as giving a HBAV \underline{A} = (A, ι, λ) together with a non-vanishing differential, i.e., a differential ω which generates 𝔱_{A}^{∗} as an O_{L} ⊗ ℂ-module.

Algebraic modular forms (Katz): Recall that we have fixed an inert prime p. Let 𝔽 be a fixed field of p^{g} elements, and fix an ordering

Emb(L, W (𝔽)) = {σ_{1}, …, σ_{g}},

such that

σ ∘ σ_{i} = σ_{i+1}.

Let 𝕋 be the split torus over W (𝔽) such that

𝕋(R) = (O_{L} ⊗ R)^{×}, R ∈ W (𝔽) − Algebras.

Then the σ_{i} furnish an identification 𝕋 = ⊕_{i=1}^{g}𝔾_{m}.

Let X(T) be the characters of the torus 𝕋. It is a free abelian group with identity element 1 – the trivial character.

If we let χ_{i} be the projection on the i-th component, then

X(T) = {χ_{1}^{r_{1}} … χ_{g}^{r_{g}} : r_{1}, …, r_{g} ∈ ℤ}

We shall use later:

X(m) := characters that are trivial on (O_{L}/p^{m})^{×} under

(O_{L}/p^{m})^{×} ↪ 𝕋(W_{m}(𝔽)) = (O_{L} ⊗ W_{m}(𝔽))^{×} = ⊕_{i=1}^{g}W_{m}(𝔽)^{×},

where the map is given by

a ↦ (σ_{1}(a), …, σ_{g}(a)).

Let B be any W (𝔽)-algebra (e.g., B = W_{m}(𝔽)). We let

M(B, χ, μ_{N})

denote the modular forms of weight χ and level μ_{N} defined over B. This means the following:

• A level μ_{N} is an embedding

β_{N} : D_{L}^{−1} ⊗ μ_{N} ↪ A,

where \underline{A} is defined over a B-algebra. We remark that over ℂ this is the same as a Γ_{1}(N) level structure. We let

M(B, μ_{M})

denote the corresponding modular variety.

• A modular form f of weight χ is a rule

(\underline{A}, β_{N}, ω)_{/R} ↦ f(\underline{A}, β_{N}, ω) ∈ R,

where ω is a non-vanishing differential, such that f depends only on the isomorphism class of (\underline{A}, β_{N})_{/R}, commutes with base change and satisfies

f(\underline{A}, β_{N}, α^{−1}ω) = χ(α) · f(\underline{A}, β_{N}, ω), ∀α ∈ 𝕋(R).

We remark that over ℂ, if χ = χ_{1}^{r_{1}} … χ_{g}^{r_{g}} then this means f is a modular form in the previous sense of weight (r_{1}, …, r_{g}).

Fix m and N such that (N, p) = 1. Main Question: What can be said about

I_{m}◁ R_{m} := ⊕_{χ∈X}M(W_{m}(𝔽), χ, μ_{N}) ⟶^{q−exp.} W_{m}(𝔽)[[q^{α} : α ∈ D_{L}^{−1 +}]] ?

What was known is the case g = 1 and m = 1: Swinnerton-Dyer: I_{1} = (\overline{E_{p−1}^{∗}} − 1). Here we put in general E_{k} = 2^{g}ζ_{L}(1 − k)^{−1}E_{k}^{∗}

= 1 + 2^{g}ζ_{L}(1 − k)^{−1} ∑_{ν∈O_{L}^{+}} (∑_{𝔠|(ν)}Norm(𝔠)^{k−1}) e^{2πi·T r ντ}.

Deligne: \overline{E_{p−1}^{∗}} is the Hasse invariant. We shall generalize that:

• There exist partial Hasse invariants h_{1}, ⋯, h_{g} such that I_{1} = (h_{1}, …, h_{g}).

• We shall construct elements in I_{m} and say enough about it to derive congruences between values of the zeta function.

A bit on group schemes: Let G be a commutative finite group scheme of exponent p over a perfect field of characteristic p. To G one can associate its Dieudeonné module D which is a finite dimensional vector space over k endowed with two operators F, V satisfying:

F λ = λ^{p}F, λV = V λ^{p}, V F = F V = 0.

This association is a functor from such group schemes to such modules and in particular End(G) acts on D. It follows that Ker(F), Ker(V) and Ker(F) ∩ Ker(V) are characteristic sub-vector spaces.

This may applied now in the situation G = A[p], the p-torsion of a HBAV \underline{A}. It is know that

Ker(F) ≅ O_{L} ⊗ k = ⊕_{i=1}^{g}k,

and hence,

Ker(F) ∩ Ker(V) = ⊕_{i∈τ(\underline{A})}k,

for some subset τ(\underline{A}) of \{1, …, g\}. This subset is called the type of \underline{A}.

For every subset

τ ∈ {1, ⋯, g}

let W_{τ} to be the subset of the moduli space M(𝔽, μ_{N}) (of HBAV over 𝔽-algebras with μ_{N}-level) that is universal with respect to the property

τ(A) ⊇ τ.

G.- Oort: The subscheme W_{τ} is a closed regular subscheme of dimension g − |τ|, and

W_{τ} ∩ W_{σ} = W_{τ∪σ}.

Theorem 3. (G.) There exist HMFs

h_{1}, …, h_{g},

of weights

χ_{1}^{p}χ_{2}^{−1}, …, χ_{g}^{p}χ_{1}^{−1},

and q-expansion 1 at every cusp of M(𝔽, μ_{N}). We have

(h_{i}) = W_{{i}}.

Let h = h_{1} ⋯ h_{g}, then h is a modular form of weight Norm^{p−1} and q-expansion 1. Moreover, (h) is the complement of the ordinary locus.

Examples: g = 1, 2, 3. (g = 1 well known, g = 2 joint with Bachmat).

To study I_{m} (the kernel of ⊕_{χ∈X}M(W_{m}(𝔽), χ, μ_{N}) → W_{m}(𝔽)[[q^{α}]]) we shall use the technique of adding p to the level. Recall that p, m and N are fixed and (N, p) = 1. We want to add a μ_{p^{n}}-level to HBAV over W_{m}(𝔽)-algebras. That is,

β_{p^{n}} : D_{L}^{−1} ⊗ μ_{p^{n}} ↪ A.

Note that

D_{L}^{−1} ⊗ μ_{p^{n}} ≅ μ_{p^{n}}^{g},

hence the existence of such a level structure implies that A is ordinary. We define now a moduli scheme for every m ≥ 1 and n ≥ 0: T_{m,n} = ordinary (\underline{A}, β_{N}) over W_{m}(𝔽) algebras together with

β_{p^{n}} : D_{L}^{−1} ⊗ μ_{p^{n}} ↪ A. I.e., ordinary (\underline{A}, β_{Np^{n}}). Since we insisted on ordinary we are rewarded with the following:

T_{m,n} ⟶ T_{m,0},

is an étale Galois covering with Galois group (O_{L}/p^{n})^{×}. We also note that T_{m,n} ↪ T_{m+m′,n} is a closed immersion.

Note that T_{m,0}, being the complement of the ample divisor (h), is affine, hence so is T_{m,n} for every n. Let

V_{m,n} := regular functions on T_{m,n} (T_{m,n} = Spec(V_{m,n})). Let χ ∈ X be any character and consider it as χ : (O_{L}/p^{m})^{×} ⟶ W_{m}(𝔽). Let

V_{m,m}^{χ} = {f : [α]f = χ(α)f}.

Let

V_{m,m}^{K} = ∑_{χ∈X} V_{m,m}^{χ}.

It is the “Kummer part” of V_{m,m}. Theorem 4. (G.) 1. R_{m}/I_{m} ≅ V_{m,m}^{K}.

2. I_{m} ⊇< a(χ) − 1 : χ ∈ X(m)^{+} >. The map r : R_{m} ⟶ V_{m,m} has the following property: Let α ∈ (O_{L}/p^{m})^{×} act on a modular form f of weight χ by multiplying it by χ(α)and on a function h by [α]h(x) = h([α]x). Then the map r is equivariant for this action. One obtains the following Criterion: If r(∑ f_{χ}) = 0, then there exist scalars in a W_{m}(𝔽) algebra, such that a_{1} = f_{1} and

∑ a_{χ}χ(x) = 0, ∀x ∈ (O_{L}/p^{m})^{×}.

Application to congruences: Let p ≠ 2 and k ≥ 2 even. I. If k ≢ 0 (mod p − 1) then ζ_{L}(1 − k) is p-integral. Proof. Assume that ζ_{L}(1−k) is not p-integral. Consider E_{k} −1 where as before

E_{k} = 1 + 2^{g}ζ_{L}(1 − k)^{−1} ∑_{ν∈O_{L}^{+}} (∑_{𝔠|(ν)}Norm(𝔠)^{k−1}) e^{2πi·T r ντ}.

Then

E_{k} − 1 ≡ 0 (mod p).

Hence, the criterion gives that for some a we have

a · Norm^{k}(x) − 1 ≡ 0 (mod p), ∀x ∈ (O_{L}/p)^{×}.

Or, simply,

a · x^{k} − 1 ≡ 0 (mod p), ∀x ∈ (ℤ/p)^{×}.

This is easily seen to imply k ≡ 0 (mod p − 1).

II. Assume that k ≢ 0 (mod p − 1) and k ≡ k′ (mod p − 1). Then

ζ_{L}(1 − k) ≡ ζ_{L}(1 − k′) (mod p).

Proof. Consider E_{k}^{∗} − E_{k′}^{∗} − α, where

α = 2^{−g}(ζ_{L}(1 − k) − ζ_{L}(1 − k′)) ∈ ℤ_{p}.

Then

E_{k}^{∗} − E_{k′}^{∗} − α ≡ 0 (mod p).

This is true because of the nature of the coefficients of E_{k}: they are sums of k − 1 powers of integers. Applying the criterion we get that for some a, b we have

a · Norm^{k}(x) − b · Norm^{k′}(x) − α = 0, ∀x ∈ (O_{L}/p)^{×}.

Or simply that there exists a c such that

cx^{k} − α = 0 ∀x ∈ (ℤ/p)^{×}.

Taking x = 1 and then that k ≢ 0 (mod p − 1) one concludes that

α ≡ 0 (mod p).

On deriving the higher congruences:

In trying to follow through the same argument one faces the following obstacle: If k ≡ k′ (mod (p−1)p^{m}) it is not true that for every integer x

x^{k} ≡ x^{k′} (mod p^{m+1}).

Though, this does hold for x prime to p. One then attempts to remove the p-part of the σ-functions by considering

E_{k}^{†} := E_{k}^{∗}(τ) − p^{g(k−1)}E_{k}^{∗}(pτ).

This indeed works but for the fact that this difference is a modular form on Γ_{0}(p). However, one argues that in characteristic p it may considered as a (meromorphic) modular form of level 1. Why is that?

In terms of moduli problem, level Γ_{0}(p) means giving together with \underline{A} also an O_{L}-invariant subgroup H ⊂ A[p] of order p^{g}. If we work over W_{m}(𝔽) there exists a natural such choice over the open set T_{m,0} of the moduli space M(W_{m}(𝔽), μ_{N}). Indeed, T_{m,0} is the ordinary part and A[p]^{0}, the connected component of the identity, is such a group scheme. This allows us to pull-back E_{k}^{†} to a modular form on T_{m,0} and thus to a meromorphic modular form on M(W_{m}(𝔽), μ_{N}). One can then proceed and apply the same techniques for E_{K}^{†}, E_{k′}^{†}. Note that this explains where the Euler factor in the congruences comes from.

Back to the construction of modular forms:

• On T_{m,m} there exists a canonical modular form a(χ) of weight χ for every χ ∈ X.

• a(χ) descends to a modular form on T_{m,0} if and only if χ ∈ X(m).

• a(χ) descends to a holomorphic modular form on M(W_{m}(𝔽), μ_{N}) if and only if χ ∈ X(m) and writing χ = (χ_{1}^{p}χ_{2}^{−1})^{r_{1}} ⋯ (χ_{g}^{p}χ_{1}^{−1})^{r_{g}} we have each r_{i} ≥ 0.

• One then lets

h_{i} = a(χ_{i}^{p}χ_{i+1}^{−1}).

\underline{The definition of a(χ)}:

Using the fact that the given level

(D_{L}^{−1} ⊗ μ_{p^{m}})_{/R} ↪ A_{/R}

induces an isomorphism

𝔱_{D_{L}^{−1}⊗μ_{p^{m}}/R}^{∗} ⟶ 𝔱_{A/R}^{∗},

and that since

𝔱_{D_{L}^{−1}⊗μ_{p^{m}}}^{∗} ≅ O_{L} ⊗ R\frac{dq}{q},

it carries a canonical non-vanishing differential 1⊗\frac{dq}{q}, we find a canonical differential ω for 𝔱_{A/R}^{∗}.

Since O_{L} ⊗ R = ⊕_{i=1}^{g}R we get a canonical basis for 𝔱_{A/R}^{∗}, say ω_{1}, …, ω_{g}. One can check that these defines modular forms

a(χ_{1}), …, a_{(}χ_{g})

and we let

a(χ) = a(χ_{1})^{r_{1}} … a(χ_{g})^{r_{g}}, χ = χ_{1}^{r_{1}} ⋯ χ_{g}^{r_{g}}.

The map

r : R_{m} = ⊕M(W_{m}(𝔽), χ, μ_{N}) ⟶ V_{m,m}

is now defined simply by

f ∈ M(W_{m}(𝔽), χ, μ_{N}) ↦ f/a(χ).

The interpretation of the q-expansion as the expression of a modular form locally as a function in the local ring of a Tate object implies immediately that I_{m} = Ker(r). It is clear from the definition that

I_{m} ⊇< a(χ) − 1 : χ ∈ X(m)^{+} > .

How does one prove that

I_{1} = (h_{1} − 1, …, h_{g} − 1) ?

Given an element ∑ f_{χ} ∈ I_{1}, we first modify it by the h_{i}’s such that the following holds

r(∑_{n=1}^{t} f_{ψ_{n}}) = 0, i ≠ j ⇒ ψ_{i} ≠ ψ_{j} (mod X(1)).

Since the map r is equivariant for the (O_{L}/p)^{×} action and

V_{1,1} = ⊕_{χ∈X/X(1)}V_{1,1}^{χ},

(This is particular for the case m = 1) we conclude the

r(f_{ψ_{i}}) = 0, ∀i.

But for a fixed weight the q-expansion map is injective.

Lastly, I want to say something about the relation between modular forms, say modulo p, and in characteristic zero.

It need not be true that a modular form lifts to characteristic zero. For example, in characteristic zero every non-cusp form has parallel weight and we see that the h_{i}’s can not be lifted. Still, one can prove

Theorem 5. (G.) Let f be a modular form mod p, then there exist a modular form g over ℤ_{p} such that g has the same level as f and the reduction of the q-expansion of g modulo p is equal to the q-expansion of f.

Thus, “every q-expansion can be lifted (with the same level).”