Search Results

The set $\mathcal{L}_{\rho}$ is either empty, or a singleton finite. This follows from Faltings' isogeny theorem, which states that for any two elliptic curves (or, more generally, abelian varieties) …

Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has a …

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 $\Gamma_1(N)$ …

Serre proved in 1985 that weight 1 cusp forms are always lacunary; see Theorem 19 of Serre's article Serre, Jean-Pierre, Quelques applications du théorème de densité de Chebotarev, Publ. Math., Inst. …

Blasius and Rogawski's paper "Motives for Hilbert modular forms" (1993) proves a more general result for Hilbert modular forms over any totally-real field, which includes this as a special case.

The first half of the question has been answered in the comments, so let me address the second half of the question. We want to define Hecke operators on the complex $R\Gamma(X, \omega^k)$, because th …

The statement, as claimed, is false. Let $p = 2, N = 11$, and let $f_0$ be the unique normalised eigenform in $S_2(\Gamma_0(11))$; and set $f(\tau) = f_0(8\tau)$. Then $f \in M_2(\Gamma_0(Np^3))$, but …

This is a subtle issue (which has come up before on this site several times, see e.g. is the modular curve X(N) defined over Q? for a related question). Your $S(N)$ is naturally a scheme over $\mathbb …

The subjects of "p-adic L-functions" and "p-adic modular forms" are so closely intertwined that it's virtually impossible to talk about either one without immediately running into the other. Applicati …

I remember discussing this with Emmanuel Kowalski not long ago. The short answer is that generalising the result to $J_1(N)$ is an open problem, and seems to be very difficult.

You can compute these dimensions using modular symbols (an auxiliary space which has the same Hecke action as modular forms, but is easier to compute). Here's a Sage example for weight 4 cusp forms of …

The curve $X_1(Np^n)$ is connected, but the ordinary locus in this curve is not: if you remove the residue discs of the supersingular points, what's left "falls apart" into a disjoint union of several …

It is a widely believed conjecture that all elliptic curves, over any number field $K$, are modular (in the sense that there exists an automorphic representation [*] $\pi$ of $\operatorname{GL}_2 / K$ …

I'm wary of speaking about "the trivialisation of $\omega$", since the unit group $\mathcal{O}(Y(N))^\times$ (the group of modular units) is a pretty big group, and your trivialization will only be we …

Since this question has come alive again, let me point out that the Hecke operators cannot give a splitting of $h^1(E)$ into two pieces over $F$, since the Hecke correspondences on a modular curve are …