# Tagged Questions

**1**

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**2**answers

146 views

### On the conductor of the Groessencharacter of a CM elliptic curve

Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that ...

**1**

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**1**answer

191 views

### Arithmetic property of a surface of general type

In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...

**1**

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**1**answer

163 views

### Etale cohomology and restricted direct product

[migrated from math.stackexchange: http://math.stackexchange.com/questions/727896/etale-cohomology-and-restricted-direct-product]
$\newcommand{\h}{\operatorname{H}}$
Let $k$ be a global field, $A$ an ...

**1**

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**1**answer

143 views

### Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...

**10**

votes

**1**answer

991 views

### Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?

From a naive outsider's viewpoint, just watching the MO postings
in those three fields scroll by, and hearing of breakthroughs in the news,
it appears there might be increasing overlap among the ...

**16**

votes

**1**answer

449 views

### Is there a known example of a curve X of genus > 1 over Q such that we know the number of points of X over the n-th cyclotomic field, for every n?

By Falting's theorem, these numbers are of course finite. Is there an example where we can explicitly compute them for every $n$?
Thank you!

**5**

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**1**answer

257 views

### Inertia subgroup in the ordinary reduction case when $p=2$

Dear MO,
Let $K/\mathbb{Q}_2$ be a finite extension, and let $E/K$ be an elliptic curve with good ordinary reduction, and such that $\mathbb{Q}_2(j(E))=K$. Let ...

**6**

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**2**answers

405 views

### Jacobians defined over smaller fields

Let $L/K$ be an extension of number fields.
Let $X$ be a curve over $L$ which can not be defined over $K$. Let $J(X)$ be the Jacobian of $X$ over $L$.
In general, the Jacobian $J(X)$ probably ...

**5**

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**1**answer

543 views

### Special value of $L$-function

Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the
algebraic part ...

**4**

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**0**answers

255 views

### On Stickelberger's Theorem over function fields

Here is the setup to Stickelberger's theorem over number fields (following Washington's book Intro. to cyclotomic fields).
Let $M/\mathbb{Q}$ be a finite abelian extension with galois group $G$. ...

**2**

votes

**1**answer

434 views

### Conductor of an elliptic curve

Given any elliptic curve over $\mathbb{Q}$ of conductor $N$, by modularity of elliptic curves,
there exists a surjective morphism from $X_0(N)$ $\rightarrow$ $E$.There may be several such 'N' and ...

**7**

votes

**1**answer

2k views

### Order of Ш (Sha)

To prove the BSD conjecture, one has to know about 'the finiteness of the Shafarevich Tate group'. But, an example of an elliptic curve of rank 2 (whose Sha group $Ш(E/\mathbb{Q})$ is finite) is not ...

**3**

votes

**1**answer

1k views

### Trace of Frobenius over $F_q$

Let $q_0$ be a prime and $q$ = $q_0^n$.
Let $a(F_q/F_{q_0})$ denote any integer which is trace of Frobenius over the field $F_q$ for some elliptic curve which can be defined over $F_{q_0}$.
It is ...

**4**

votes

**1**answer

943 views

### Bound for the number of rational points on the modular curve

By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977),
we know that the only rational points of X_0(N) for N any prime > 163 are
the two cusps (o) and (oo) (|X_0(N)(Q)| = 2 ...

**4**

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**0**answers

115 views

### Detecting linear dependence on multiplicative groups

Let G = $\mathbb{G}_m^2/\mathbb{Q}$ and let $\Gamma \subseteq G(\mathbb{Q})$ be a free abelian group of rank 2. Assume that the set of primes $p$ for which $\Gamma \mod p$ is cyclic has positive ...

**6**

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**1**answer

644 views

### $\ell$-adic Weil cohomology theory

I have a reference or counterexample request. Suppose $k$ is a field and $\ell\neq char(k)$. There are several common references that show that $H^i_{et}(-, \mathbb{Q}_\ell )$ is a Weil cohomology ...

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**0**answers

266 views

### What is the shape of the zeta function of a singular hypersurface?

So let $X$ be a projective hypersurface inside $\mathbb{P}_{\mathbb{Z}}^n$ of degree $d$.
Assume that
(a) $X(\mathbb{C})$ and $X(\overline{\mathbb{F}}_p)$ are irreducible,
(b) and that ...

**10**

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**0**answers

509 views

### Points of bounded height in a number field

Let $K$ be a number field of absolute degree $d$, let $B$ be a positive real number, and write $S(K, B) = \{x \in K : H(x) \leq B\}$. Here $H$ is the absolute multiplicative height of an algebraic ...

**6**

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**2**answers

756 views

### “Bad” reduction of Shimura curves via dual graphs

I have the following naive (and inexpert) question about the
reduction of Shimura curves at primes dividing the discriminant
of the underlying quaternion algebra. It requires some background
to ...

**3**

votes

**1**answer

685 views

### How do I visualize finite covers of curves over non-algebraically closed fields?

If $L$ is algebraically closed, fields of transcendence degree one over $L$ correspond to algebraic curves over $L$ up to birational equivalence, and finite extensions correspond to finite Galois ...

**2**

votes

**2**answers

489 views

### When is the period of elliptic curve over the rationals transcendental?

Given an elliptic curve $E/\mathbf{Q}$, when is its period transcendental/algebraic?

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**1**answer

391 views

### Hasse principle for a group

In the paper "Hasse principle" for $PSL_2 (\mathbb Z)$ and $PSL_2(\mathbb F)$ there's a definition of a Hasse principle for a group $G$, but I don't completely get it. Is there a more motivated ...