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Since an algebraic number is zero if and only if any of its conjugates is zero, $I_f J_1$ is stable under $W_N$ and so indeed $W_N$ descends to an automorphism of $A_f$. Now, the important thing to re …

If by $f_1\equiv f_2$ modulo $p$, you mean that $a_n(f_1)\equiv a_{n}(f_2)$ modulo $p$ for all $n\in\mathbb N$ or maybe for all except finitely many, then this theorem cannot be true. Let's start with …

Given an eigencuspform $f$ of weight $k≥2$ (so in particular 2) and $p$ a prime of ordinary reduction (in particular good ordinary reduction), there is always a Hida family passing through $f$. This …

This semester, I teach a graduate course in epistemology of mathematics and as a case study, I assigned students a discussion on the epistemological status of Fermat's Last Theorem according to differ …

Write $L$ for the finite Galois extension of $\mathbb Q$ with Galois group $G_{\mathbb Q}/\operatorname{Ker}\rho_{F,v}^m$. Then $\rho_{F,v}^m(\operatorname{Frob}_p)$ is the identity in $\operatorname{ …

In fact much more than the equality of conductor is true: the local Galois representation $\rho_{F,\lambda}|G_{\mathbb Q_{p}}$ obtained by restricting $\rho_{F,\lambda}$ to the decomposition group at …

I think it helps to put things in a larger perspective. To an eigencuspform $f$ and a prime number $\ell$ is attached on the one hand an irreducible, admissible representation $\pi(f)_{\ell}$ of $\op …

The three articles referenced presented in logical order of exposition are respectively Faltings, Gerd Hodge-Tate structures and modular forms Math. Ann. 278 (1987) Tsuji, Takeshi $p$-adic étale coh …

This was getting too long for a comment I'll post it as an answer. Though the set of $K$-special points on the canonical model does not depend on the choice of an embedding of $K\hookrightarrow\opera …

Why not simply looking at the original source? Ramanujan made his famous conjectures in On certain arithmetical functions Transactions of the Cambridge Philosophical Society XXII (1916), a source whi …

The answer to the question in the title is yes, as explained in the last paragraph below. However, under a literal interpretation of "can" (implying actual feasibility), I believe the answer to the q …

In the ordinary case, the argument is simple so let me recall it here. The $p$-divisible group $J$ is an extension of an étale $p$-divisible group $J^{et}$ by a multiplicative $p$-divisible group $J^ …

This question is very simple. Would someone be so nice as to send me an electronic copy of M. M. Vishik, Non-Archimedean measures connected with Dirichlet series, Mat. Sb. (N.S.), 1976, Volume 99( …

First, a small clarification: level-lowering tells you that a modular representation in level $Np$ occurs in level $N$ only if by occurs you mean "is congruent to modulo $p$". That said, my answer to …

Examples of the phenomenon alluded to in Question 1 are actually plentiful. The first that came to my attention is described in M.Emerton $p$-adic families of modular forms (after Hida, Coleman, and …