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The answer is on page 99 and in Technique de descente et théorèmes d’existence en Géométrie Algébrique, II (Section 3): the upshot is that the functor $F$ is not only pro-representable, it is strictly …

Let $A$ be a complete regular local noetherian ring of dimension $d>1$ and $B$ an $A$-algebra, finite and free as $A$-module. Assume moreover that there exists an open subset $U$ of $\textrm{Spec}\ A$ …

Am I missing something or is this the classical question of Serre? A class of examples is given in Exemples de variétés projectives en caractéristique p non relevables en caractéristique zéro. Proc. N …

Why did mathematicians begin to study CM abelian varieties? Because they could. The study of CM abelian varieties arguably starts with Fagnano's work on the length of the lemniscate. In 1799, Gauss l …

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 …

First of all, let us assume $p\neq 2$, otherwise I think $\bar{\rho}_n$ might be (absolutely) reducible even though $\bar{\rho}$ is not. Also, one has to make clear that the deformations parametrized …

I might be missing something (seeing that I don't know what a Chow motive is), but I think the answer is yes. It is a result of G.Laumon proved in Comparaison de caractéristiques d’Euler-Poincaré en c …

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 …

Franz Lemmermeyer's answer points out to a connection. Let me point out in the converse direction that believable heuristics suggest that if $E(\mathbb Q)=0$, then $\operatorname{Sel}_p(E)$ should be …

1) Iwasawa theory, as practiced by K.Iwasawa, is concerned with $\mathbb Z_{p}$-extensions. There is only one $\mathbb Z_{p}$-extension of $\mathbb Q$. Over more generally number fields, and in more g …

In the spirit of You Could Have Invented Spectral Sequences by T.Chow, I claim you could have invented $\operatorname{Ext}^{1}(\mathbb Q(0),M)$ as group of "rational points" of a motive. Here is how. …

This result (and much more) is in Exposé IX Modèles de Néron et monodromie by A.Grothendieck (in SGA7), and more precisely in section 11.1. In particular what you want is exactly Proposition 11.2 ther …

What I don't understand is why the exactly the same terms should appear in both sums. The Galois action on CM points is described in adelic terms via the fundamental theorem of complex multiplica …

First of all, your deformation problem is empty if $\det\bar{\rho}\neq\bar{\chi}_{cyc}$ so I am guessing that you are assuming this and likewise for the relevant restriction on $\bar{\rho}$ to $G_{\ma …

And so it turns out that I was in the audience of a seminar talk just today on this very subject. The opinion I expressed in comments is apparently not too far from the truth: V.Pilloni and B.Stroh no …