Let $f = \sum_n a_n q^n$ be a cuspidal newform of weight $k$ on $\Gamma_0(N)$ for some $N$. Let $K_f$ be the number field generated by the $a_q$ as $q$ runs over all primes. My q …

Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Arti …

If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the Galois r …

Let $F_v$ be a local field. Let $\sigma_v$ be a two-dimensional representation of $Gal(\overline{F}_v:F_v) \rtimes W_{F_v}$. Now, there exists an infinite-dimensional representatio …

Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conject …

Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from …

Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic represe …

Bogomolov's paper "Sur l'algébricité des représentations l-adiques" proves that the image of the $\ell$-adic Galois representation associated to an abelian variety over a number fi …

$\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$ Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $ …

Hi, Suppose that $F$ is a local field, $G_F$ its Galois group, $I$ the inertia subgroup, $k$ its residue field. Let $X$ be a finite type scheme over $k$. Let $C$ be a constructibl …

We know that a family of elliptic curves is a morphism of schemes $f:X \to Y$ such that the fiber of every point of $Y$ is an elliptic curve (and we usually require the morphism to …

Hi all, I am sorry to ask a stupid question but I am really confused right now. Kitagawa-Mazur constructed a $2$-variable p-adic L-function attached to Hida families of modular …

Let $f : G' \subset G$ be an injection between profinite groups such that $G'$ is normal in $G$ (typical situation which I deal with : $G$ the absolute Galois group of a local fiel …

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations …

Let $X$ be a smooth, separated scheme of finite type over $\mathbb{F}_q$ where $q=p^r$ for some $r>0$. Let $gcd(l,p)=1, \rho:W(X) \to GL_r(\mathbb{Q}_l)$ be a Weil representation w …