# Tag Info

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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 ...

### 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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### 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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### Cube root of discriminant of elliptic curve

One way to think about this is via the modular curves parametrizing elliptic curves $E$ with either $E$ or $\Delta^{1/3}$ rational. Note that $\Delta^{1/3}$ is rational iff $j^{1/3}$ is rational, ...
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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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### 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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### 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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### 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,...

### 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 ...

### 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.

### 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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### 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 ...

### 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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### 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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### 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 ...

### 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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### 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 ...