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Unfortunately there isn't a simple direct relationship between OC mod forms and OC cohomology! The miracle is that they contain the same finite-slope Hecke data. As you've guessed, one proves this by …

Suppose $f$ is a weight $k$ cuspidal Hecke eigenform on the full modular group, with its L-function $L(s,f)$ normalized such that $s=1/2$ is the central point. Is a bound of the form $\int_{-1}^{1}| …

One of the canonical references for questions like this is Serre's "Quelques applications du theoreme de densite de Chebotarev", Publ. Math. IHES 54. He proves, for example, that the number of primes …

Write $f=\sum c_i f_i$ as a sum over new eigenforms. Your condition is thus equivalent to $\sum c_i \lambda_i(p)=0$ for all $p$. Taking the absolute value squared of this and summing over $p\leq X$ …

Suppose $F/\mathbb{Q}$ is a totally real field of degree $d$ and class number one. Fix an ordering $\sigma_1, \dots, \sigma_d$ on the embeddings of $F$. Is $\sum_{\alpha \in \mathcal{O}_F}e^{2\pi i …

Yes. Bruinier and Ono have shown in their paper "Heegner divisors, L-functions and harmonic weak Maass forms" that the vanishing or nonvanishing of central derivatives of twisted L-functions like thi …

Let $X_0(N)$ be the usual modular curve. For $\chi$ a quadratic Dirichlet character of conductor $D$, define the homology class $c(\chi)=\sum_{i=0}^{|D|-1}\chi(i)\{\frac{i}{|D|}{\infty} \}$; here $\{ …

The infinite-dimensional unitary representations of $SL_{2}\left(\mathbb{R}\right)$ appearing in the right-regular representation on $L^{2}\left(H\right)$ are precisely the unitary representations of …

I think the right approach would be to observe that $R^n$ is the leading term in $j(n\tau)$ as above. Then there is a modular polynomial $\Phi_n$ which satisfies $\Phi_n(j(x),j(nx))=0$. For small $n …

Let $S$ be the (countable) set of holomorphic cuspidal new eigenforms of weight $\geq 2$. Any $f\in S$ has a level $N_f$ and a canonically normalized Fourier expansion $f(z)=\sum_{n=1}^{\infty}a_f(n)e …

Suppose $f$ is a weight $2$ level $N$ cusp form. When can we realize the mod-$\ell$ representation of $f$ in a form of weight $2$ and level $Np^3$, where $p$ is some prime not dividing $N$? I assume …

First of all, let me apologize in advance for the terseness of this question. It seems that by now there are well-developed techniques (the "Taylor-Wiles-Kisin" method) for proving modularity lifting …

Let me start with a simple observation. Suppose $f$ is a weight two newform of level $p^3$. Write $d$ for the size of the Galois orbit $f^\sigma, \sigma \in \mathrm{Gal}(\overline{\mathbf{Q}}/\mathb …

Let $\mathscr{C}$ be the Coleman-Mazur-Buzzard eigencurve of some fixed tame level $N$. Are there any known examples of a singular point $x\in \mathscr{C}$ which lies in a unique irreducible componen …

I think this may be a silly question, but here goes. Let $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{F}_p})$ be a representation; say $\rho$ is of S-type i …