Every elliptic curve $E/\mathbf Q$ is modular, in the sense that there exists a nonconstant morphism $X_0(N) \to E$ for some $N$. It is tempting to extend this definition in a na …

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$. An arithmetic subgroup is called congruence if it contains a subgroup of typ …

It is well-known that many modular forms can be expressed as infinite products. For instance, the most famous one is probably the expansion $$\Delta(q) = q \prod_{n=1}^\infty (1-q …

Let $f$ be a modular form on $\Gamma_0(4)$ of weight $k\in\tfrac 12\mathbb{Z}$ with trivial Nebentypus. Is it true that if you twist $f$ by $\tfrac 12$, i.e. look at the function …

Shimura lifts are correspondence between integer weight and half-integral weight automorphic forms. Half integral weight things are associated to representation of a double cover o …

$\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\PP}{\mathbb{P}}$ $\newcommand{\QQ}{\mathbb{Q}}$ $\newcommand{\hH}{\mathcal{H}}$ $\newcommand{\eE}{\mathc …

I've seen stated offhand in many sources that the cuspidal subgroup of the Jacobian of $X_0(N)$ is finite. Do they mean that the subgroup of the jacobian generated by $\mathbb{Q}$ …

Let $\sigma_m (n)=\sum_{d|n}d^m$, and $d(n)=\sigma_0(n)$ as stated in the title. My question is: Does the q-series $f(q)=\sum_{n\ge 1} d(n)q^n$ gives something like a modular form …

The paper can be found here: http://link.springer.com/content/pdf/10.1007%2FBF01078890.pdf EDIT: Russian original available at LINK In his proof of assertion 1, he "reduces to …

Suppose ell is prime and (N,ell)=1. Consider those power series over Z that are expansions at infinity of modular forms for gamma_0 (N) of weight a multiple of ell-1. I'll say that …

Actually I have a few related questions. Here, by $Y(1)$ I mean the affine $j$-line $\text{SL}_2(\mathbb{Z})\backslash\mathcal{H}$. I know $Y(1)$ is only a coarse moduli space, s …

Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from …

I have a question about the terminology for special values of L-functions. Is the following a correct description of standard usage: Suppose L(s) is an L-function which satisfies …

A Celebrated theorem of Coleman says that any overconvergent eigenform of weight $k$ and slope $< k-1$ is classical (here $k \geq 2$ and the level is $\Gamma_1(N) \cap \Gamma_0( …

This is more of a question about terminology than about math. The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly whic …