Answer Property [question]Definition [question]Example Problem Solution Counterexample Remark

[question]Theorem Lemma Proposition Corollary Conjecture Food for Thought

Math 726: L-functions and modular forms Fall 2011

Lecture 17 : Algebraic Interpretation of Modular Forms (Continued)

Instructor: Henri Darmon Notes written by: Dylan Attwell-Duval

Recall

We saw last time:

{Lattices in ℂ} ↔ { (E, ω)_{ℂ} : E = Elliptic curve over ℂ ω = generator of Ω^{1}(E) } .

Algebraic modular forms of level 1, over base ring S (\frac{1}{6} ∈ S)

We can generalize the RHS above to an arbitrary S-algebras R:

\{ (E, ω)_{R} : E = Elliptic curve over R ω = generator of Ω^{1}(E_{/R}) \} = \{ \begin{matrix}(y^{2} = x^{3} + ax + b, \frac{dx}{y}) \\ Δ(a, b) = 4a^{3} − 27b^{2} ∈ R^{×} \end{matrix} \} .

Notice that these pairs are in a clear bijection with the R-rational points of the affine space

Ỹ_{0}(1) := Spec(S[a, b][\frac{1}{Δ}]).

We thus define a weakly holomorphic modular form over S to be an element of the ring S[a, b][\frac{1}{Δ}].

The algebraic group 𝔾_{m} has an action on Ỹ_{0}(1) that is defined pointwise by

λ ∈ R^{×} → λ(a, b) = (λ^{4}a, λ^{6}b).

We define weakly holomorphic modular forms of weight k over S as the functions on Ỹ_{0}(1) satisfying the following homogeneity property with respect to this action:

f ∈ S[a, b][\frac{1}{Δ}] such that λ(f) = λ^{−k}.f.

Denoting this subring by (S[a, b][\frac{1}{Δ}])_{wt k}, we find that (S[a, b][\frac{1}{Δ}])_{wt 0}, or equivalently the set of functions on Ỹ_{0}(1) invariant under the action of 𝔾_{m}, is freely generated by the element j = \frac{4a^{3}}{Δ} .

Hence the scheme Y_{0}(1)_{/S} := Ỹ_{0}(1)/𝔾_{m} is equal to the affine line Spec(S[j]) and classifies all elliptic curves in the following sense:

If k is a field over S, then Y_{0}(1)_{/S}(k) = \{Elliptic curves over k\} /\{Isomorphisms over \overline{k}\}.

The curve Y_{0}(1) is the prototypical example of a modular curve.

Definition 1. A \underline{level N structure} on an elliptic curve E is a morphism α : μ_{N} → E of group schemes over S, where μ_{N} is the group scheme of N^{th}-roots of unity. If S is any field of characteristic 0 containing the N^{th}-roots of unity, α is equivalent to the datum of a distinguished point of order N in E(S).

Definition 2. A \underline{marked elliptic curve with level N structure} is a triple (E, α, ω)_{/S} where E is an elliptic curve over S, α a level N structure on E as above and ω a generator of Ω(E_{/S}).

1 Definition 3. An \underline{algebraic modular form} (resp. \underline{cusp form}) over S of weight k and level N is a rule f which for every R-algebra S associates to every triple (E, α, ω)_{/R} an element f((E, α, ω)_{/R}) ∈ R subject to

(a) f((E, α, ω)_{/R} ⊗_{φ} R′) = φ(f((E, α, ω)_{/R})) ∀φ : R → R′ (compatiblity with base change)
(b) f((E, α, λω)_{/R}) = λ^{−k}f((E, α, ω)_{/R}) ∀λ ∈ R^{×}
(c) f((𝔾_{m}/q^{ℤ}, (α : ζ_{N} → ζ_{N}^{a} · q^{\frac{b}{N}}), \frac{dt}{t})_{S(ζ_{N})((q^{\frac{1}{N}} ))}) belongs to S(ζ_{N})[[q^{\frac{1}{N}}]]
(If f is a cusp form, then this must lie in q^{\frac{1}{N}} S(ζ_{N})[[q^{\frac{1}{N}}]]).

Furthermore, if f satisfies the condition

(d) f((E, dα, ω)_{/R}) = ϵ(d) · f((E, α, ω)_{/R}) ∀d ∈ (ℤ/Nℤ)^{×}

where ϵ is a Dirichlet character, then we say f is of character ϵ.

We denote the corresponding S-modules by M_{k}(N; S) ⊃ S_{k}(N; S) and M_{k}(N, ϵ; S) ⊇ S_{k}(N, ϵ; S).

Theorem 4. Let N ≥ 1. Then there exists an affine curve Y_{1}(N) over ℚ with the property that for all fields k of characteristic not dividing N

Y_{1}(N)(k) = \{(E, α)_{k}\}/\{\overline{k} − isomorphism\}.

If N > 3 we even have

Y_{1}(N)(k) = {(E, α)_{k}}/{k − isomorphism}.

In that case, Y_{1}(N) has an integral model over ℤ which is smooth over ℤ[\frac{1}{N}] and satisfies

Y_{1}(N)(S) = \{(E, α)/S\}/\{Isomorphisms over S\} ∀ℤ[\frac{1}{N}] − algebras S.

\underline{Remarks:}

⋆ The construction of Y_{1}(N) follows exactly the same procedure as the construction of Y_{0}(1). See (Milne, Modular forms and modular curves) or (Diamond and Shurman) for full construction.

⋆ To see that Y_{1}(N)(k) = {(E, α)_{k}}/{k − isomorphism} when N > 3, observe that if φ : (E, α)_{k} ˜→(E′, α′)_{k} is defined over the algebraic closure \overline{k}, then in fact it is defined over k for the following reason:

If σ ∈ Gal(\overline{k}, k), then φ^{−1} ∘ φ^{σ} ∈ Aut\overline{k}(E, α) = \{1\}. Hence φ = φ^{σ} ∀σ ∈ Gal(\overline{k}, k) which means φ is defined over k.

The fact that Y_{1}(N) is not a fine moduli space for N = 2 or 3 is because there are elliptic curves with distinguished points of order 2 or 3 that have non-trivial automorphism groups.

Next Time

Eichler-Shimura Theory

We will see how an Eigenform f ∈ S_{2}(N, ϵ; S) corresponds to a compatible system of l-adic representations {V_{f,λ}}.