Search Results

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- …

I think the answer to your second question is "no". For example if $k=2$ and $f$ and $g$ correspond to elliptic curves over $Q$ with positive rank, then the only critical point is $s=1$ and (at least …

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 …

It's deeper than theta cycles, I think. I am going to assume that $N$ is prime to $p$ -- you don't say this in your question but most of my answer assumes this in a very serious way. If $f$ is ordin …

If you think about this question in terms of automorphic representations then it sort of becomes trivial. The space $Sk(\Gamma(N))$ can be re-interpreted as the direct sum of $\pi^{U(N)}$, where $\pi$ …

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} …

If I've understood your question correctly, you're right that $C(q,f)\not=0$ always and there is a natural representation-theoretic proof of this result (before I start let me say that I don't know ho …

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 …

No they're not in general affinoid. The problem is that the zero locus of the power series is computed within a space which is almost never affinoid -- for example in the modular curve case the ambien …

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 …

Presumably you want the form (let me call it g) of level Np^3 to be new at p, otherwise it's trivial. Let me also assume ell isn't p. If the form g is new at p, and has level Gamma0(p^3) at p, then …

Joel -- it's difficult to work out what you're asking. Of course both possibilities can occur, as Wanax said. Furthermore both possibilities can occur even for the same modular form. For example, if y …

I can give you a "formula" in the sense that I can give you an algorithm to compute the number in any given case. If $\ell\not=p$ is prime then an old result of Carayol and Livn\'e says that the condu …

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 …