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First of all, I am not sure I fully agree with the notion that Tamagawa numbers are harmless factors. What you wish for exists, and here is roughly why. The Birch and Swinnerton-Dyer conjecture is a …

In addition to François's answer, I'll address the second question. Are there any differences between the case over $\mathbb Q$ and a number fields $L$? The main difference - which can be dealt …

Jean-Marc Fontaine Groupes p-divisibles sur les corps locaux. Astérisque 47-48, Soc. Math. France, Paris (1977), i+262 pp (especially chapter V) This is probably the canonical answer to your question …

I am not sure what your criteria would be for a proof to be given a geometric interpretation, but the reason why weights "disappear" when we take the inverse limit on the level stems from the contract …

Going in the other direction, the Néron-Ogg-Shafarevich criterion and Weil pairing imply that the Tate module $T_{\ell}E$ is a Galois representation which is ramified at $p$. So if $n$ is large enough …

More explicitly, I would like to know if from these motives $M_{f}$ I can create an $\ell$-adic representation with values in some object of cohomological nature arising from $M_{f}$ (like motivic …

The residual representation of $G_{\mathbb Q_{p}}$ attached to an eigencuspform is markedly different depending on whether $p$ divides the coefficient $a_{p}$, the non-ordinary case, or not, the ordin …

EDIT: This is an answer to a different question, namely whether removing operators other than $U_p$ can result in a strict sub-algebra. In particular, the example given shows that $\mathbb T^{(2)}$ is …

I am not sure I really agree with the following quote (which is the opening paragraph of Modular forms and Galois cohomology by H.Hida) because I suspect that a mathematician valuing creativity and ve …

How did we end up with the such complicated constructions of $B$? To add to Laurent's answer remark that "these rings did not, however, come out of nowhere", I believe that in the early 80s, Fontain …

A Galois representation $\rho_\ell:\operatorname{Gal}(\bar{\mathbb Q}_{\ell}/\mathbb Q_{\ell})\longrightarrow\operatorname{GL}_2(\mathbb Q_{\ell})$ can be semistable (technically $B_{st}$-admissible i …

1) What generalizations of the Kummer congruences are known? This is somewhat imprecise as a question and in particular, I would dispute a little your assertion that This is probably the same …

I think that the answer to your questions depends in subtle ways on whether $r=0$ or $r=1$. In full generality, I believe you are right that none of the properties you state are known for all elliptic …

A natural generalization of the geometric modularity conjecture which is compatible with your formulation Do you expect some form of modularity to correspond to the existence of a map from some sp …

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 …