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Here's some piece of the bigger picture. Maass forms and holomorphic modular forms are both automorphic representations for $GL(2)$ over the rationals. An automorphic representation is a typically hug …

No there are not any mistakes in these papers of any interest. In the 1990s there were a bazillion study groups and seminars across the world devoted to these papers; I personally read all three of th …

Let me highlight some issues that Emerton doesn't: 1) you seem to hint that you don't know that modular forms can be viewed as a product of a bunch of local terms. So there is an adelic story, where …

Say $E$ is an elliptic curve over the rationals, of conductor $N$. There's a covering of $E$ by the modular curve $X_0(N)$, and if you rig it right then you can define this map over $\mathbf{Q}$: ther …

I could have made this question very brief but instead I've maximally gone the other way and explained a huge amount of background. I don't know whether I put off readers or attract them this way. The …

The fact that Hecke operators (double coset stuff coming from $SL_2(\mathbf{Z})$ acting on modular forms) and Hecke algebras (locally constant functions on $GL_2(\mathbf{Q}_p)$) are related has nothin …

If $f\in\mathbf{C}[[q]]$ is non-constant, and algebraic over $\mathbf{C}[q]$ (in the sense that it is a root of a polynomial with coefficients in in $\mathbf{C}[q]$) then can $f$ be the $q$-expansion …

A weight 2 modular form which happens to be a normalised cuspidal eigenform with rational coefficients has a natural geometric avatar---namely an elliptic curve over the rationals. It seems to be a su …

The main point is that the basic definitions work fine but the link with arithmetic is much more "vague". Look at early papers of Tony Scholl. There are Galois representations attached to certain non- …

There has been a lot written already about this question. but here is a simple answer. The Hodge--Tate weights of the Tate module of an elliptic curve are 0 and 1. The Hodge--Tate weights of the Galoi …

By "level $\ell$" I assume you mean "level $\Gamma_1(\ell)$". Here's a proof. By the Eichler-Shimura theorem, the system of eigenvalues associated to the modular form shows up in $H^1(SL(2,\mathbf{Z} …

The right way to do this sort of question is to apply Saito's local-global theorem, which says that the (semisimplification of the) Weil-Deligne representation built from $D_{pst}(\rho_{f,p})$ by forg …

[I took the time to chase this up so may as well post it as an answer.] There is a (cuspidal) modular (eigen)form of level $\Gamma_0(512)$ and weight 2, which if I remember correctly was shown to me …

It is. I want to argue the following way: if the polynomial is non-constant then after scaling it has integral coefficients and so the reduction of the p-adic form mod p^n will be a classical form who …

Oh, I know how I would try and build examples. First I would write down a random $A_5$ extension $K$ of $\mathbf{Q}$, ramified at some primes (in fact I would look in a table, e.g. in Buhler's thesis …