# Tagged Questions

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### How does one see level structure in the ell-adic Galois representation

This is probably a very easy question for those familiar with the arithmetic theory of abelian varieties because it is either possible or impossible for "trivial reasons".
Let $A$ be an abelian ...

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**3**answers

424 views

### Another question related to the isogeny theorem for elliptic curves

I was reading the following question: About isogeny theorem for elliptic curves and was interested in the following statement at the end of Torsten Ekedahl's answer:
"Note also that the situation is ...

**18**

votes

**1**answer

1k views

### Modern proof of Serre's open image theorem?

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic ...

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**0**answers

133 views

### Construction of RM abelian variety from eigenform

Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...

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192 views

### Where can I find a copy of Serre's Cours au college de France 1985-1986?

Hi,
I was wondering: where might I be able to find a copy of this work online?
And are there any other resources for the proof of the open image theorem for abelian varieties with endomorphism ring ...

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votes

**2**answers

360 views

### Quotients of Tate modules

Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let
$\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over
$K$ and denote by $T_{p}A$ its ...

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votes

**1**answer

483 views

### Mumford-Tate group and Galois representations

Could someone please point me towards a proof of why the image of a Galois representation on the Tate-module of an abelian variety is limited by its Mumford-Tate group?

**2**

votes

**2**answers

312 views

### CM abelian varieties and potential good reduction

Let $F$ be a number field and $A$ an abelian variety over $F$. It is known that if $A$ has complex multiplication, then it has potentially good reduction everywhere, namely there exists a finite ...

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**1**answer

482 views

### Serre's open image theorem for products of elliptic curves over function fields via specialization

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6
′′, p. 325):
Let $K$ be a number field and let
...

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**0**answers

198 views

### Mod m versions of the toric part of Tate modules

Let $A$ be a polarized abelian variety over a local field $K$ with residue characteristic $p$. In the course of proving that a polarized abelian variety $A/K$ has semi-stable reduction iff for all ...

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**1**answer

537 views

### Galois action on one-dimensional quotients of l-adic cohomology

Let $A$ be an abelian variety of dimension $g$ over a number field $K$, and $\ell$ be a rational prime. Suppose that the Galois action on the $\ell$-adic cohomology $H^k(A, \mathbb{Z}_\ell) ...

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**1**answer

3k views

### Fontaine-Mazur for GL_1

For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified ...

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**0**answers

309 views

### Ext of Tate-modules of abelian varieties

Let $K$ be a local field (in fact, finite extension of $\mathbb{Q}_p$) and let $A$ and $B$ be abelian varieties over $K$.
Associated to $A$ and $B$ are the Tate-modules $T_p(A)$ and $T_p(B)$.
Both ...