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 = Elliptic curve over C
ω = generator of Ω1 (E)
(E, ω)C :
.
Algebraic modular forms of level 1, over base ring S ( 16 ∈ S)
We can generalize the RHS above to an arbitrary S-algebras R:
)
( 2
)
(y = x3 + ax + b, dx
E = Elliptic curve over R
y
.
(E, ω)R :
=
ω = generator of Ω1 (E/R )
∆(a, b) = 4a3 − 27b2 ∈ R×
Notice that these pairs are in a clear bijection with the R-rational points of the affine space
1
Y˜0 (1) := Spec(S[a, b][ ]).
∆
We thus define a weakly holomorphic modular form over S to be an element of the ring
S[a, b][ ∆1 ].
The algebraic group Gm has an action on Y˜0 (1) that is defined pointwise by
λ ∈ R× → λ(a, b) = (λ4 a, λ6 b).
We define weakly holomorphic modular forms of weight k over S as the functions on Y˜0 (1)
satisfying the following homogeneity property with respect to this action:
f ∈ S[a, b][
1
] such that λ(f ) = λ−k .f.
∆
Denoting this subring by (S[a, b][ ∆1 ])wt k , we find that (S[a, b][ ∆1 ])wt 0 , or equivalently the set
of functions on Y˜0 (1) invariant under the action of Gm , is freely generated by the element
3
j = 4a∆ .
1
Hence the scheme Y0 (1)/S := Y˜0 (1)/Gm 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 Y0 (1)/S (k) = {Elliptic curves over k} /{Isomorphisms over k}.
The curve Y0 (1) is the prototypical example of a modular curve.
Definition 1. A level N structure on an elliptic curve E is a morphism α : µN → E of
group schemes over S, where µN 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 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 ).
Definition 3. An algebraic modular form (resp. 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 R0 ) = ϕ(f ((E, α, ω)/R )) ∀ϕ : R → R0 (compatiblity with base change)
(b) f ((E, α, λω)/R ) = λ−k f ((E, α, ω)/R ) ∀λ ∈ R×
b
1
(c) f ((Gm /q Z , (α : ζN → ζNa · q N ), dtt )S(ζ )((q N1 )) ) belongs to S(ζN )[[q N ]]
N
1
1
(If f is a cusp form, then this must lie in q N S(ζN )[[q N ]]).
Furthermore, if f satisfies the condition
(d) f ((E, dα, ω)/R ) = (d) · f ((E, α, ω)/R ) ∀d ∈ (Z/N Z)×
where is a Dirichlet character, then we say f is of character .
We denote the corresponding S-modules by Mk (N ; S) ⊃ Sk (N ; S) and Mk (N, ; S) ⊇
Sk (N, ; S).
Theorem 4. Let N ≥ 1. Then there exists an affine curve Y1 (N ) over Q with the property
that for all fields k of characteristic not dividing N
Y1 (N )(k) = {(E, α)k }/{k − isomorphism}.
If N > 3 we even have
Y1 (N )(k) = {(E, α)k }/{k − isomorphism}.
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In that case, Y1 (N ) has an integral model over Z which is smooth over Z[ N1 ] and satisfies
Y1 (N )(S) = {(E, α)/S}/{Isomorphisms over S}
∀Z[
1
] − algebras S.
N
Remarks:
? The construction of Y1 (N ) follows exactly the same procedure as the construction of
Y0 (1). See (Milne, Modular forms and modular curves) or (Diamond and Shurman) for full
construction.
? To see that Y1 (N )(k) = {(E, α)k }/{k − isomorphism} when N > 3, observe that if
ϕ : (E, α)k →(E
˜ 0 , α0 )k is defined over the algebraic closure k, then in fact it is defined over k
for the following reason:
If σ ∈ Gal(k, k), then ϕ−1 ◦ ϕσ ∈ Autk (E, α) = {1}. Hence ϕ = ϕσ ∀σ ∈ Gal(k, k) which
means ϕ is defined over k.
The fact that Y1 (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 ∈ S2 (N, ; S) corresponds to a compatible system of
l-adic representations {Vf,λ }.
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