Math 726: L-functions and modular forms Fall 2011

Lecture 16 : Algebraic Modular forms cont’d

Instructor: Henri Darmon Notes written by: Maxime Turgeon

In the last lecture, we gave two important examples of marked elliptic curves over a ring:

1. The universal elliptic curve

(ℂ/⟨1, τ⟩)_{O_{H}} = (y^{2} = 4x^{3} − g_{4}(τ)x − g_{6}(τ), \frac{dx}{y})_{O_{H}};

2. The Tate elliptic curve

(ℂ^{×}/⟨q⟩, \frac{dt}{t})_{O_{D^{×}}} = (y^{2} = x^{3} − a(q)x + b(q), \frac{dx}{y})_{O_{D^{×}}},

where a(q) = \frac{1}{3} (1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n}) and b(q) = \frac{2}{27} (1 − 504 ∑_{n=1}^{∞} σ_{5}(n)q^{n}).

Note that the Tate elliptic curve can be viewed as an elliptic curve over the ring ℤ[\frac{1}{6}]((q)). But with more care (for example, by introducing new parameters), one can show that it can also be viewed as an elliptic curve over ℤ((q)):

(E_{T ate}, \frac{dt}{t})_{ℤ((q))} = (𝔾_{m}/⟨q⟩, \frac{dt}{t})_{ℤ((q))} .

We also defined a weakly holomorphic algebraic modular form of weight k and level 1 to be a rule

(E, ω)_{R} ⟶ f(E, ω)_{R} ∈ R,

subject to two conditions, namely, that it should be compatible with base change (see Lecture 15) and that it should satisfy the following homogeneity condition

f(E, λω) = λ^{−k}f(E, ω), for all λ ∈ R^{×}.

In particular, f (ℂ/⟨1, τ⟩, dz)_{O_{H}} ∈ O_{H} is the classical view of (weakly holomorphic) modular forms as holomorphic functions on H. Also, note that f (E_{T ate}, \frac{dt}{t})_{ℤ((q))} ∈ ℤ((q)) is the q-expansion of f.

Definition 1. An algebraic modular form over ℤ is a weakly holomorphic modular form over ℤ such that f (E_{T ate}, \frac{dt}{t})_{ℤ((q))} ∈ ℤ[[q]]. Moreover, if f (E_{T ate}, \frac{dt}{t})_{ℤ((q))} ∈ qℤ[[q]], we say that f is cusp form.

For an arbitrary ring S, f is a modular form if

f (E_{T ate}, \frac{dt}{t})_{S[[q]]} ∈ S [[q]] .

Thus, we say a modular form over S has Fourier coefficients in S (this is called the q-expansion principle).

Recall that we were able to give a good description of the ring of modular forms of level 1 in the classical setting, namely, we proved that the Eisenstein series E_{4} and E_{6} are generators. In a more general setting, we can give a good description of the ring of algebraic modular forms over ℤ[\frac{1}{6}].

Theorem 2. Let (E, ω)_{k} be a marked elliptic curve over a field k of characteristic different from 2 and 3. Then, there exist uniquely defined elements x, y ∈ H_{0}(E\{O}, O_{E}) (that is, functions regular outside the origin) satisfying

• The function x has a double pole at the origin, whereas the function y has a triple pole at the origin;

• x, y satisfy a cubic equation of the form y^{2} = x^{3} + ax + b;

• \frac{dx}{y} = ω.

Sketch of proof. First, we define two families of k-vector spaces:

L_{r} = {f ∈ H_{0}(E\{O}, O_{E}) | ord_{O}(f) < −r},
L_{r}^{∗} = {ω ∈ H_{0}(E, Ω^{1}) | ord_{O}(ω) > r},

where r ≥ 0 is an integer. The Riemann-Roch theorem gives the following relation between the respective dimensions of these two vector spaces:

dim_{k} L_{r} − dim_{k} L_{r}^{∗} = r + (1 − g),

where g is the genus; in our case g = 1. Using this relation, we can build the following table:

r dimL_{r} dimL_{r}^{∗}  basis for L_{r}
0  1  1  {1}
1  1  0  {1}
2  2  0  {1, x}
3  3  0  {1, x, y}
4  4  0  {1, x, y, x^{2}}
5  5  0  {1, x, y, x^{2}, xy}
6  6  0  \{1, x, y, x^{2}, xy, {x^{3} \atop y^{2}} \}

Table 1: Dimensions of the vector spaces

After eventualy rescaling x and y, we can assume x^{3} − y^{2} ∈ L_{5}, which gives us a dependence relation:

y^{2} = x^{3} + axy + bx^{2} + cy + dx + e, where a, b, c, d, e ∈ k.

When choosing the vector x, we could have chosen any vector of the form x + μ, with μ ∈ k. Similarly for y, we could have chosen any vector of the form y + μ′x + ρ′, where μ′, ρ′ ∈ k. Hence, by replacing y by y + μ′x + ρ′ (with appropriate μ′, ρ′), we can eliminate the xy and y terms, which gives a relation of the form

y^{2} = x^{3} + b′x^{2} + c′x + d′.

Note that at this stage, we have to assume that char k ≠ 2: in order to eliminate the xy term, we need μ′ = \frac{a}{2}, and to eliminate the y term, we need ρ′ = \frac{c}{2}, which only makes sense if 2 is invertible. Furthermore, by replacing x by x + μ (with appropriate μ), we can eliminate the x^{2} term, and this gives a relation of the form

y^{2} = x^{3} + a′′x + b′′.

At this stage, we have to assume that char k ≠ 3: to eliminate the x^{2} term, we need μ = \frac{−b′}{3} , which can only be done if 3 is invertible. With these choices, x and y are completely determined, up to the following change of variables:

(x, y) ⟼ (λ^{2}x, λ^{3}y),

for a nonzero λ ∈ k. The effect of this change of variables on the differential is \frac{dx}{y} ↦ \frac{1}{λ} \frac{dx}{y}. Thus, the condition ω = \frac{dx}{y} specifies the pair (x, y) uniquely.

The equation y^{2} = x^{3} + ax + b is called the canonical equation associated to the pair (E, ω). We thus get the following bijection:

\{ \begin{matrix}Marked elliptic \\ curves (E, ω)_{k} \end{matrix} \} ⟶ \{ \begin{matrix}(a, b) ∈ k^{2} satisfying \\ Δ(a, b) ≠ 0 \end{matrix} \} .

Note that we have two distinguished algebraic modular forms over ℤ[\frac{1}{6}]:

1. a(E, ω) = a, where a is the coefficient of x in the canonical equation of (E, ω). This a is a suitable multiple of the Eisenstein series of weight 4.

2. b(E, ω) = b, where b is the constant coefficient in the canonical equation of (E, ω). Note that b is a suitable multiple of the Eisenstein series of weight 6.

From the above discussion, we conclude that the ring of algebraic modular forms over ℤ[[\frac{1}{6}]] is generated by the modular forms a and b.