26 votes
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In what sense is $p$-adic Hodge theory related to ordinary (complex) Hodge theory?

So, the problem as I see it is as follows. The term '$p$-adic Hodge theory' can mean one of two things: The study of $p$-adic representations of $p$-adic fields. Comparison theorems. What you're ...
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11 votes

Galois theory, had it solved any major problems beside its original applications to classical problems?

This is a very reasonable question. Indeed, not all "generalizations" are anything more than keeping-busy, or technical improvements. But, in fact, "Galois theory" is constantly invoked in algebraic ...
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8 votes
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Galois representations and isogenies of elliptic curves

The Galois representation is reducible iff there is a one-dimensional Galois-stable $\mathbb{F}_{\ell}$-subspace, say $C$. Then $E \rightarrow E/C$ is a $\mathbb{Q}$-rational isogeny. Conversely, if ...
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8 votes
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Cube root of discriminant of elliptic curve

One way to think about this is via the modular curves parametrizing elliptic curves $E$ with either $E[3]$ or $\Delta^{1/3}$ rational. Note that $\Delta^{1/3}$ is rational iff $j^{1/3}$ is rational, ...
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7 votes
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The mod $p$ Galois representation of the Frey curve is unramified away from $2, p$

Well, it was easy for Serre to check ;D This is not obvious, but once you know the right tools to use, it does become an easy exercise. A great resource for learning about the proof of FLT is this ...
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7 votes
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How does Galois group acts on etale cohomology?

Please correct me if I am wrong, but I really don't see where we need the proper assumption to get a Galois action. So let $X/k$ be any scheme, $\overline{k}$ a separable (or algebraic) closure of $k$...
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7 votes
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What is a semistable representation?

An E-rep of $G_K$ (E either $Q_p$ or $\bar{Q}_\ell$, K an extension of $Q_p$) being semistable basically means that it looks like a representation coming from the etale cohomology of a variety with ...
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  • 1,806
7 votes
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The splitting of Galois representations

This is false for simple group theoretical reasons. Suppose that $V$ is an absolutely irreducible representation of a group $G$ which has odd dimension $d$ and which is self-dual up to twist, say $V \...
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7 votes
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Why does having an integral model make étale cohomology unramified?

This is really a local question: you may as well assume $X$ is defined over a local field $L$, it doesn't matter whether it comes from a number field. The point is that if $S = \operatorname{Spec} O_L$...
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6 votes

the corresponding modular form of an elliptic curve

Yes. (a) You compute the conductor of your elliptic curve (say using Tate's algorithm, or some variant). (b) You then compute all the weight two Hecke eigenforms at that level (using modular ...
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6 votes
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What is the Weil group of a global field $K$?

In general, the concept of a Weil group arises via the theory of class formations, with the standard introductory reference being Tate's Number theoretic background. In the case that the field is (...
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6 votes
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Mod $p$-representation ($p$-torsion points ) of elliptic curve (over number fields) with CM can be irreducible?

Certainly. It will (typically) be irreducible for primes where the curve has supersingular reduction. Indeed, for such primes, the p-torsion as a module over the endomorphism ring $O$ is isomorphic to ...
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6 votes
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Why $\mathbb{Q}(x_1,y_1,...,x_n,y_n)/\mathbb{Q}$ is a Galois extension? where $E[m]=\{(x_1,y_1),...,(x_n,y_n)\}$ is the m-torsion group.

An algebraic field extension $L/K$ is Galois if it is normal and separable. In our case, we only need to show that $L/K$ is normal. However, normality is equivalent, that for every field homomorphism $...
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5 votes

What is the intuition behind the Fontaine-Mazur Conjecture?

Suppose first that $K$ is a finite extension of some $\mathbb Q_p$, with abs. Galois gp. $G_K$. A $p$-adic rep. of $G_K$ coming from geometry satisfies some basic conditions: it is pot. semi-stable,...
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  • 1,527
5 votes

Can one check by hand whether the Tate module of an elliptic curve is semi-simple

Have you tried looking at Serre's "Abelian $\ell$-adic representations and elliptic curves"? That might shed some light. If you just want some individual examples, you can look at elliptic curves ...
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5 votes

Galois theory, had it solved any major problems beside its original applications to classical problems?

Wiles' proof of Fermat's Last Theorem makes extensive use of Galois theory.
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  • 1,216
5 votes

The mod $p$ Galois representation of the Frey curve is unramified away from $2, p$

To elaborate a little more on the existing answer: There is a criterion (Neron--Ogg--Shafarevic) whichs says that $E$ has good reduction at $\ell \neq p$ iff the Galois action on the $p$-adic Tate ...
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5 votes
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Reference to proof of Weil-Langlands theorem

As noted by Mathmo123 in comments, this is a form of the converse theorem. There is the original paper of Weil in which he proved this (but I'm not sure if he treated the weight 1 case which is ...
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topology on the ring of Witt vectors in the theory of period rings of Fontaine

There are (at least) two different topologies on all those rings, and that is the source of your confusion. Both on $W(R)$ and $W(fof(R))$ one has the $p$-adic topology, in which e.g. a basis of ...
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5 votes

What are Robba rings and why are they important?

One sentence of buzzwords to help your literature search: it connects the field of norms (or, more fashionably, tilting) with Fontaine's functors of $p$-adic Hodge theory. The original work in this ...
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5 votes
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Property of a Galois representation associated to an elliptic curve

For an elliptic curve over $\Bbb{Q}$ or any field of characteristic $0$ $$x^3+ax+b = (x-e_1)(x-e_2)(x-e_3)$$ then since $-(x,y) = (x,-y)$ we see $(\infty,\infty),(e_1,0),(e_2,0),(e_3,0)$ is in the 2-...
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5 votes
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Fixed subfield of symmetric rational functions $K(s_1,\ldots,s_n)$ under $A_n$

You have the field $L$ with a faithful action of $S_n$. You have $F$, the fixed field of $S_n$ and $E$, the fixed field of $A_n$. Recall the theorem of Galois theory that states that if $G$ is a ...
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4 votes

The Frobenius Trace for an elliptic curve

The key fact you need is this: if $E/K$ is an elliptic curve which has complex multiplication over $K$, then the associated $\ell$-adic Galois representations $$\rho_{E,\ell}:G_K\to\mathrm{GL}_2(\...
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4 votes
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References about moduli space of abelian varieties with level structure

I've made this community wiki for obvious reasons. Arithmetic Moduli of Elliptic Curves by Katz and Mazur. This is the ideal comprehensive reference if you want to work with just elliptic curves. It ...
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Does the supposed to exist functor considered in Langlands program bear a peculiar name?

The (classical) Langlands correspondence is still being pieced together and is not so precise that it can be thought of as a specific functor between well-defined categories.
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4 votes
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Definition of Galois representation unramified at a prime.

$\require{AMScd}$ If $\rho$ factors through some $\overline{\rho}\colon \mathrm{Gal}(L/K) \to \mathrm{GL}_n(F)$ with $L/K$ unramified at $\mathfrak{p}$, then the image of $I_{\mathfrak{p}}$ under $\...
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4 votes
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irreduciblity of $\ell$-adic representation attach to the elliptic curve over $\mathbf{Q}$ with complex multiplication

An elliptic curve over $\mathbf Q$ cannot have complex multiplication (defined over $\mathbf Q$). It's possible for a rational elliptic curve to have extra endomorphisms, but these will only be ...
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4 votes
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Determinant representation of Tate module and cyclotomic character

I'm going to approach your questions in reverse order. For an abelian variety of dimension $g$, it is well known that the determinant representation of the Tate module is $\chi^g$, i.e. the $g$-th ...
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4 votes
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Integral models of $p$-divisible groups

Just to get this off the unanswered list. This is false even if you replace $\mathbb{C}_p$ by a finite extension of $\mathbb{Q}_p$. When I want to think of concrete examples of $p$-divisible groups ...
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