Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, ...

**6**

votes

**3**answers

384 views

### Torsion group of the following elliptic curve

Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i $ be the elliptic curve defined by $y^2 = x^3 + n_i$.
Can one compute the torsion group $...

**2**

votes

**1**answer

112 views

### Does the isotropic group of local galois representation have finite index?

Let $\rho : G\to \mathrm{GL}_n(\mathbb Z_p)$ be a crystalline representation of $G=\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$. For any non zero element $a\in \mathbb{Z}_p^n$ , it spans a rank one ...

**8**

votes

**1**answer

327 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 ...

**4**

votes

**1**answer

211 views

### log smooth curve vs pointed node curve

F.Kato has a statement said that a log smooth curve $f:(X, M,\alpha)\to (k,N,\beta)$ where $k$ is an algebraic closed field and $N$ is some fine log structure on $k$, is equvalent to pointed node ...

**3**

votes

**0**answers

158 views

### log structure on Witt ring and Frobenius map

I am a little confused about the log structure of Witt ring and its Frobenius map.
Let $k=\bar{\mathbb{F}}_p$, $W:=W(k)$ the Witt ring.
We know that to deal with the semistable reduction, i.e. scheme ...

**3**

votes

**1**answer

132 views

### Deforming curves to other curves over the field of rational numbers

Let $X$ and $Y$ be smooth projective geometrically connected curves over $k$ of genus $g$ at least two.
If $k$ is an algebraically closed field of characteristic zero, there exists a connected ...

**7**

votes

**2**answers

444 views

### Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is

Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space.
If $\mathcal X$ can be defined over $\overline{...

**0**

votes

**0**answers

114 views

### Question about the “middle” intermediate Jacobian

Suppose $Y$ is a smooth projective variety of dimension $2p-1$ over $\mathbb{C}$. I have a few questions about the $p^{~\text{ th}}$ intermediate Jacobian $J^p(Y)$ of $Y$.
Does it come from (i.e. is ...

**2**

votes

**1**answer

265 views

### representation of algebraic fundamental group of projective line minus three point

everyone, I want to ask is there any result in the literature
similar to the following:
Let $ X=\mathbb{P}^1\backslash \{0,1,\infty\}$, then $X$ is defined over $\mathbb{Z}$. Let $X_{\mathbb{Q}}$ ...

**1**

vote

**0**answers

92 views

### degree of isogenies between Jacobians and Abelian Varieties

Let $K$ be a local field of characteristic zero and positive residual characteristic. Let $A$ be a simple abelian variety and assume we have an isogeny $f:Jac_C\rightarrow A$ with $C$ a smooth curve ...

**11**

votes

**3**answers

733 views

### What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(...

**12**

votes

**2**answers

364 views

### Existence of local sections

I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.
Take a number field $K$, and let ...

**2**

votes

**0**answers

176 views

### integral hard Lefschetz

I am looking for examples $(X,\eta)$ where the integral hard Lefschetz is an isomorphism:
$X/k$ is a smooth projective variety of dimension $d$ over an algebraic closure of a finite field and $\eta \...

**1**

vote

**1**answer

209 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**

votes

**1**answer

310 views

### What is the probability that a randomly chosen number from set of c.e.number is period(number)?

What is the probability that a randomly chosen number from the set of c.e.numbers is period(number)?
What is the probability that a randomly chosen number from the set of computable numbers is ...

**3**

votes

**1**answer

295 views

### What is the relation between KC and height of rational number?

Roughly speaking,Kolmogorov Complexity of a bits string or a description is the minimal length of programs outputing a bits string,and height of rational number is logarithm of the largest numerator ...

**2**

votes

**1**answer

127 views

### Determining existence of p-adic point on a plane curve

Is there an implemented algorithm that will take a polynomial $f(x,y)\in\mathbb Z[x,y]$ and a prime $p$, and determine whether the equation $f(x,y)=0$ has a solution over $\mathbb Q_p$?

**1**

vote

**1**answer

194 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 ...

**7**

votes

**0**answers

111 views

### Can the failure of the multiplicativity of archimedean L-factors be corrected?

My question is parallel to J. Borger' question:
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
As emphasized by Scholbach in his paper on special values of L-...

**2**

votes

**0**answers

87 views

### Bounds for the Tamagawa number of the Jacobian of a hyperelliptic curve

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ and let $C$ be a hyperelliptic curve of genus $g$ defined over $K$ with Jacobian $J$.
Suppose that $C$ is given by ...

**11**

votes

**1**answer

618 views

### Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...

**10**

votes

**0**answers

162 views

### Totally real points on curves

Let $X$ be a smooth, projective (geometrically integral) curve defined over $\mathbb{Q}$ with genus $g \geq 3$. Suppose that $X(\mathbb{R}) \neq \emptyset$. Does $X$ have a point defined over a ...

**9**

votes

**1**answer

341 views

### Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)

Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in Goldman-...

**7**

votes

**1**answer

368 views

### Finite morphisms to projective space

Let $X$ be a projective variety of dimension n. Then there exists a finite surjective morphism $X \to \mathbf P^n$. Let $d$ be the minimal degree of such a finite surjective morphism.
Let $d^\prime ...

**10**

votes

**1**answer

446 views

### Degeneration of the Hodge spectral sequence for scheme over truncated Witt ring?

Let $f:X\to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb{Q}$-scheme), then Deligne has shown:
The Hodge-De Rham spectral sequence $E^{a,b}_1=...

**3**

votes

**1**answer

304 views

### Number field of degree 5

I am interested in field extensions of the rationals. About degree 3 extensions there are many refrences including the famous paper of Shanks "The Simplest Cubic Fields". In particular he gave ...

**2**

votes

**1**answer

175 views

### Potential good reduction of abelian varieties

In Corollary 3 on page 498 of the article "Good reduction of abelian varieties" it says that, under some specified conditions, the minimal subextension $L/K$ of $\overline{K}/K$ over which an abelian ...

**3**

votes

**1**answer

344 views

### Base change in crystalline cohomology?

Does one have a base change theorem in crystalline cohomology like in étale cohomology?
Suppose one has the following cartesian diagram
$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}...

**4**

votes

**1**answer

323 views

### What is “special” maximal compact subgroup of algebraig group over local field?

Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field.
Here, I think the word "compact" is used ...

**1**

vote

**1**answer

256 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 ...

**7**

votes

**1**answer

530 views

### Explicit calculation of Weil Deligne representations

According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations.
However, given a galois representation, it is usually difficult ...

**7**

votes

**2**answers

356 views

### Is there a largest prime p such that J_0(p) completely splits into elliptic curves

The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...

**6**

votes

**0**answers

225 views

### The Rappoport-Zink spectral sequence vs. the one of the complement of a normal crossing divisor

As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale ...

**1**

vote

**0**answers

118 views

### Covers of modular curves

I'm interested in covers of modular curves (especially cyclic covers) and I'm sure there's a lot of information out there available on this topic. However, I'm unable to locate any literature (on ...

**2**

votes

**1**answer

196 views

### A problem in intersection theorem

I'm reading the paper:
SGA 7 II, Intersections sur les surfaces regulieres.
In Papge 6 , I cannot understand why there is sign $-1$ in the formula (1.10.4):
Let $S$ be a trait, for any $\mathcal O_S$...

**2**

votes

**1**answer

276 views

### Pencil with desired Jet in Algebraic geometry(new!)

Let $k$ be an algebraic closed field.
Let $n$ be a positive integer.
Let $X$ be an irreducible, proper and smooth scheme over $k$ with an immersion $i:X\hookrightarrow E:=\mathbb P^N_k$ with $N$ ...

**3**

votes

**1**answer

215 views

### A problem on Jets in algebraic geometry

Let $k$ be a perfect field, let $n$ and $m$ be two positive integers.
Consider $X=\mathbb P_k^n\times \mathbb P_k^m$. Let $x_0=(1,0,\cdots,0;1,0,\cdots,0)\in X$ be fixed.
For any pair of integers $(...

**5**

votes

**0**answers

174 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 $GL_2(\hat{\mathbb{...

**2**

votes

**1**answer

276 views

### Explicit basis for the space of global sections of a twisted arithmetic ideal sheaf

Assume $x\in X=\mathbb{P}^1_{\mathbb{Z}}$ is a closed point with $f(x)=p\in Y$ where $f:X\rightarrow Y$, here $Y=Spec(\mathbb{Z})$. Assume $k(x)=\mathbb{F}_p$ and denote by $I_x$ the ideal sheaf of $x$...

**1**

vote

**0**answers

97 views

### Benchmark problems for computing rational points on varieties

Are there standard benchmark problem sets used for empirically evaluating algorithms designed for computing rational points on (various classes of) algebraic varieties?
If so, could you please point ...

**4**

votes

**1**answer

296 views

### Are elliptic Kummer extensions big?

Loosely speaking, are elliptic Kummer extensions big? More concretely:
Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime, and
let $F$ be a subfield of $\overline{\mathbb{Q}}$ ...

**3**

votes

**1**answer

320 views

### Elliptic units and Euler system

Maybe this question is quite obscure and ambiguous. I am really sorry for such ambiguity.
My question is, what is the good thing we get from defining elliptic units and Euler system? There are lots ...

**10**

votes

**1**answer

590 views

### The Sato-Tate conjecture for hypersurfaces?

The Sato-Tate conjecture for elliptic curves $E$ predicts the distribution of the eigenvalues of Frobenius at $p$ on the Tate module of $E$ as $p$ varies in terms of the distribution of the ...

**5**

votes

**2**answers

746 views

### Faltings height of a CM abelian variety

Let A be a CM abelian variety, say simple of dimension g, with $End(A) = O_K$, where $K$ is a CM field
of degree $2g$.
Is there an upper bound for the Faltings height $h(A)$ in terms of the ...

**3**

votes

**1**answer

160 views

### How do ideal sheaves behave on the special fibers of the projective line over the integers?

Let $X=\mathbb{P}^1_{\mathbb{Z}}$ and $Y\subset X$ be a local complete intersection of codimension two with Ideal sheaf $I_Y$.
(I'm mostly interested in the case where $Y$ is a single point $x$ ...

**2**

votes

**0**answers

92 views

### Vanishing theorems that work in positive characteristic

Let $X$ be a smooth projective variety over a field of characteristic $p>0$ of dimension at least $2$. I am looking for some examples when $H^2(\mathcal{O}_X)$ vanishes. Is there any standard way ...

**2**

votes

**0**answers

154 views

### algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields

(transcendence of canonical heights)
Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...

**3**

votes

**2**answers

293 views

### isogeny and congruence subgroup

Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$.
By ...

**3**

votes

**1**answer

188 views

### Etale covers of products of curves

Is a finite etale cover of a product of curves again a product of curves?
The answer is no in general. Here's one way to construct an example. Take the product $A$ of two elliptic curves and an ...

**4**

votes

**1**answer

241 views

### Is there a formula for a hyperelliptic curve over QQ, such that its Jacobian contains a rational torsion point of extact order n, for any given n?

Given a positive integer $n$, is there an algorithm (or even better a closed formula) that provides me with a hyperelliptic curve $C/\mathbb Q$, such that its Jacobian $J:=Jac(C)$ possesses a $\mathbb ...