# Tagged Questions

**-2**

votes

**0**answers

75 views

### 2-torsion points on Frey-Hellegouarch curve. [closed]

Let us consider the so-called Frey-Hellegouarch curve
$E: Y^2 = X(X-a^l)(X+b^l)$.
Q: What are two independent ${\Bbb Q}$-rational points of order $2$ on $E$ ?

**5**

votes

**1**answer

268 views

### Connection of Galois representation and arithmetic geometry

This is might be a dumb question. There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that ...

**2**

votes

**0**answers

89 views

### Bloch Kato Exponential as formal lie group exponential

Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism
$$DR(V) ...

**2**

votes

**2**answers

335 views

### Elliptic curve E and Galois representation

Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then
Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$?
Next
...

**6**

votes

**1**answer

340 views

### Elliptic curve and Galois representation

For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by
$\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = ...

**9**

votes

**1**answer

281 views

### Is there a yoga of effectivity for motives and their realizations?

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...

**10**

votes

**0**answers

94 views

### Is there literature on approaching semisimplicity of l-adic cohomology using vanishing cycles

Let $K$ be a finitely generated field, and $X/K$ a smooth proper variety. Let $G$ denote the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$. Let $\ell$ be a prime different from $\mathrm{char}(K)$. A ...

**3**

votes

**1**answer

150 views

### Galois representation attached to $3$-torsion points of an elliptic curve

Let
$ E $ - Elliptic curve defined over $ {\mathbb{Q}} $.
$G_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) $ of $\mathbb{Q}$.
$ E[3] $ - $3$-torsion points ...

**8**

votes

**1**answer

197 views

### Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology

This question is inspired by the question: Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
Let $K$ be a number field (or finitely generated field of ...

**2**

votes

**1**answer

162 views

### When does the filtration in the limit of the Leray spectral sequence split?

Let $\ell$ be a prime, and $k$ a field of characteristic $\ne \ell$. Let $f \colon X \to Y$ be a proper map of smooth projective $k$-varieties. The Leray spectral sequence says
$$
E_{2}^{pq} = ...

**5**

votes

**0**answers

144 views

### When are Galois representations with open image attached to elliptic curves?

Let $K$ be a number field with absolute Galois group $G_K$.
Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in ...

**6**

votes

**0**answers

239 views

### Galois action on $E_n$-operads

Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for ...

**1**

vote

**1**answer

200 views

### Finite Flat Group Schemes for Modular Forms of Higher Weight

Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from an abelian variety, ...

**20**

votes

**1**answer

642 views

### What are the possible motivic Galois groups over $\mathbb Q$?

Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...

**0**

votes

**0**answers

143 views

### Uniqueness of decomposition of completely reducible representations

Let $X$ be a smooth, separated scheme of finite type over $\mathbb{F}_q$ where $q=p^r$ for some $r>0$. Let $gcd(l,p)=1, \rho:W(X) \to GL_r(\mathbb{Q}_l)$ be a Weil representation which is ...

**8**

votes

**2**answers

541 views

### What is a(n algebro-geometric) family of modular forms?

We know that a family of elliptic curves is a morphism of schemes $f:X \to Y$ such that the fiber of every point of $Y$ is an elliptic curve (and we usually require the morphism to be smooth, proper, ...

**4**

votes

**1**answer

489 views

### Should the etale cohomology of a smooth projective variety (over rationals) be semi-simple; why?

$\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$
Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $K=\mathbb{Q}$ ...

**2**

votes

**3**answers

333 views

### Useful notion of unramified Galois representation

Let $\mathbf C(t)$ be the field of rational functions and let $\overline{\mathbf C(t)}$ be an algebraic closure. Let $G$ be the Galois group of $\overline {\mathbf C(t)}$ over $\mathbf C(t)$.
Let ...

**8**

votes

**2**answers

359 views

### Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples?

The Fontaine-Mazur conjecture predicts that an $l$-adic Galois representation of a number field is 'geometric' if it is unramified outside a finite set of primes and is De Rham for primes lying over ...

**4**

votes

**1**answer

1k views

### Why are Galois Representations so important in Number theory ?

Dear everyone,
Motivation :
From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of elliptic curves have ...

**3**

votes

**2**answers

365 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 ...

**6**

votes

**0**answers

372 views

### dimensions of Galois representations appearing in the cohomology

Let's take $D$ to be $n^2$-dimensional central algebra with an involution $*$ over a CM field $E=FE_0$ (where F is totally real, and $E_0$ is imaginary quadratic). Define
$$G(R) = \{ x \in D \otimes ...

**4**

votes

**1**answer

701 views

### Eichler-Shimura for Shimura curves

Hi,
What is the statement of the Eichler-Shimura relation for Shimura curves? And where
can one find a proof?
Thanks

**1**

vote

**1**answer

768 views

### Rapoport-Zink proof of purity of monodromy

Hi,
Does anyone know if the article
"Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindene Zyklen in ungleicher Charakteristik", INvent. Math, 68 (1980)
by ...

**22**

votes

**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 ...

**8**

votes

**3**answers

2k views

### When is the Galois representation on the étale cohomology unramified/Hodge-Tate/de Rham/crystalline/semistable?

Let $X/K$ be a variety over a global field $K$. When (and why) is the Galois representation $H^i_{et}(X \times_K \bar{K}, \mathbf{Q}_\ell)$ unramified at a place $v$ of $K$?
I guess this is true if ...

**12**

votes

**2**answers

810 views

### Obstructions to descend Galois invariant cycles

Let $X$ be a smooth projective variety over $F$, and $E/F$ - finite Galois extension.
There is an extension of scalars map $CH^\*(X) \to CH^\*(X_E)$. The image lands in the Galois invariant part of ...

**5**

votes

**1**answer

885 views

### Learning about Galois representations

My goal was to learn about l-adic representations on some example — I'm a newbie in these topics.
Thus take pt = Spec F_q, ...