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One expects that the majority of algebraic curves over number fields having genus $> 1$ should not be modular in this sense. For instance, take a sufficiently general genus 2 curve $C$ over $\mathbf{Q …

The correct setting for this construction turns out to be projective varieties, so let me suppose we have a smooth variety $X$ inside $\mathbf{P}^N$, for some $N \ge 1$, defined by the vanishing of so …

It's "well known" that, for any weight $k$ and level $N$, the space $S_k(\Gamma_1(N))$ of cusp forms of that weight and level has a basis in which all the Hecke operators act by matrices with entries …

You can do this in Sage but "in disguise". The idea is that if $f(z)$ is a cusp form for $\Gamma(N)$, then $g(z) := f(Nz)$ is a cusp form for a certain subgroup intermediate between $\Gamma_0(N^2)$ an …

Let $\Gamma$ be a finite-index subgroup of $\operatorname{SL}_2(\mathbb{Z})$. I've seen it stated (in a comment in the code of a computer program) that the graded ring $$ M(\Gamma, \mathbb{C}) = \bigo …

Let $k \ge 4$ be an even integer, and let $d$ be the dimension of the space $M_k(\operatorname{SL}_2(\mathbb{Z}))$ of modular forms of level 1 and weight $k$. Then the space of Hecke operators acting …

This is more subtle than it looks. I asked exactly the same question some years back (see here); but I'm not going to flag this question as duplicate, because the answer that was given to my question …

This question already has multiple nice answers, but I am going to add one more thing which isn't quite covered by the existing posts. One distinctive advantage of the $\Gamma_0(N)$ and $\Gamma_1(N)$ …

Yes, this is a very active area -- one of the major themes of current research in number theory. Much of the recent work has focussed on proving something slightly weaker, but easier to get at, than …

I would quibble with your definition of "arithmetic". There are discrete subgroups of $SL(2, \mathbf{R})$ coming from quaternion algebras, which are not commensurable with $SL(2, \mathbf{Z})$, but are …

This is a generalisation of my earlier question about generators for the level 1 Hecke algebra. Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$, and $k \ge 1$ an integer. T …

The Manin-Drinfeld theorem has various equivalent statements. Let $\Gamma$ be a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$. Then: for any congruence subgroup $\Gamma$ of $\mathrm{SL}_2(\math …

It's worth distinguishing between the prime coefficients $a_p$, and the coefficients $a_n$ for general $n$. Let's look at $a_p$ first. Firstly: for elliptic curves, it is fairly easy and elementary t …

You can certainly attach $L$-functions to half-integer weight eigenforms, but you don't get anything really new by doing so: they turn out be versions of $L$-functions of integer weight modular forms. …

I did the computation in Sage, and there is no such form $f$. There are 21 Galois orbits of newforms of level $\Gamma_0(1776)$ and trivial character, of which the largest has size 3, and none of the r …