Modular Forms and Elliptic Curves Lecture Review

Flashcards covering the fundamentals of elliptic curves, modular forms, Eisenstein series, the modularity theorem, and Hecke operators.

For an elliptic curve $E$ and a prime $p$, how is the quantity $a_p(E)$ defined in the transcript?

$$a_p(E) = p - \# (\text{solutions in } \mathfrak{n}_p^2)$$

What does the Modularity theorem state regarding an elliptic curve $E/\mathbb{Q}$?

There exists an eigenform $f$ of weight $2$ such that $a_p(E) = a_p(f)$ for all primes $p$.

What are the three conditions for a function $f : \mathbb{H} \rightarrow \mathbb{C}$ to be a modular form of weight $k$ and level $\Gamma$?

1) $f$ is holomorphic, 2) $f(\gamma z) = (cz+d)^k f(z)$ for all $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma$, and 3) $f|_k \alpha$ is holomorphic at $\infty$ for all $\alpha \in SL_2(\mathbb{Z})$.

For an even integer $k \geq 4$, what is the value of the Eisenstein series $G_k$ at infinity?

$$2\zeta(k)$$

How is the modular discriminant $\Delta(z)$ defined in terms of the normalized Eisenstein series $E_4$ and $E_6$?

$$\Delta(z) = \frac{E_4(z)^3 - E_6(z)^2}{1728}$$

What is the valence formula (or $k/12$-formula) for a non-zero meromorphic modular form $f$ of weight $k$ for $\Gamma(1)$?

$$\sum_{z \in \Gamma(1)\backslash\mathbb{H}} \frac{1}{e(z)} \text{ord}_z(f) + \text{ord}_\infty(f) = \frac{k}{12}$$

The Ramanujan $\tau$-function represents which part of the modular discriminant $\Delta$?

The $m$-th Fourier coefficient in the expansion $\Delta = \sum_{m=1}^{\infty} \tau(m)q^m$.

According to the transcript, what is $\text{dim}_{\mathbb{C}}(M_2)$ for level $\Gamma(1)$?

$$\text{dim}_{\mathbb{C}}(M_2) = 0$$

For a modular form $f \in M_k(N, \chi)$, what is the general formula for the Fourier coefficients of $T_p f$ when $p$ is a prime?

$$a_n(T_p f) = a_{np} + \chi(p)p^{k-1} a_{\frac{n}{p}}$$

How is the hyperbolic measure $d\omega(z)$ on the upper half-plane $\mathbb{H}$ expressed in terms of the coordinates $z = w + ix$?

$$d\omega(z) = \frac{dw \wedge dx}{x^2}$$