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

**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

600 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

331 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 ...

**7**

votes

**1**answer

350 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

437 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 ...

**2**

votes

**1**answer

302 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

173 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

334 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
$$
...

**4**

votes

**1**answer

315 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

233 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

517 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

350 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

222 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

117 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

194 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 ...

**2**

votes

**1**answer

274 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

207 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 ...

**2**

votes

**1**answer

274 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 ...

**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

294 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

313 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 ...

**11**

votes

**1**answer

580 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

727 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

155 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 ...

**1**

vote

**0**answers

143 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

290 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

239 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 ...

**10**

votes

**1**answer

396 views

### Pro-algebraic versus continuous Galois cohomology, and schematic homotopy types

I've been thinking about Bertrand Toen's approach to studying the homotopy theory of schemes, and I've come across an inconsistency in my understanding of the subject that I was hoping somebody might ...

**8**

votes

**3**answers

1k views

### How many integer points does my favorite ellipse go through?

The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is
$$
x^2+y^2 - ...

**3**

votes

**2**answers

142 views

### Lattice-point-free buffers around circles

Let $C(r)$ be the origin-centered circle of radius $r$,
and let $\beta(r)$ be the exterior buffer around $C(r)$:
the distance from $C(r)$ to the closest lattice point exterior to $C(r)$:
...

**4**

votes

**1**answer

156 views

### Comparison of cycle maps

Let $X$ be an algebraic variety over $\bar{\mathbb{Q}}$ of dimension $d$, then there is the l-adic cycle map $\mathrm{cl}_{et}:\mathrm{CH}^i(X)\rightarrow\mathrm{H}^{2i}(X,\mathbb{Q}_\ell(i))$ from ...

**7**

votes

**0**answers

177 views

### Lattice radial-step (ratchet) spirals

(30Oct13: Now solved; see Addendum.)
Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.
$S(r_0,\epsilon)$ begins with the arc ...

**6**

votes

**0**answers

228 views

### Kernels and cokernels for morphisms of abelian schemes up to isogenies

For $S$ a noetherian scheme, let $\mathcal{A}(S)$ be the additive category of abelian schemes over $S$ and $\mathcal{A}_{\mathbb{Q}}(S)$ be the category of abelian schemes up to isogenies, i.e. ...

**5**

votes

**1**answer

394 views

### How many combinations of intersections of n hyperplanes there are in which a hyperplane is not crossed more than k times?

Consider you have $n$ hyperplanes with dimension $k$ that are not coplanar. Each hyperplane intersects the others $n-1$ times. Any intersection of $k$ such hyperplanes produces a vector. The number of ...

**10**

votes

**0**answers

224 views

### Purity for abelian schemes up to $p$-isogenies

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the ...

**14**

votes

**2**answers

1k views

### Deligne Weil II

Deligne's Weil I has been published under the title "La conjecture de Weil: I" in 1974, and Weil II in 1980. So did Deligne know in 1974 that there would be a Weil II, and can one explain the period ...

**3**

votes

**1**answer

143 views

### Section on $\mathcal{X}(1)$ induced by a cusp of $X(1)$

I'm reading the article Generalized arithmetic intersection numbers of U. Kuehn. At the beginning of section 4.12 "Modular forms over $\mathbb{Z}$" we are in the following situation.
Let $\Gamma(1) ...

**0**

votes

**0**answers

664 views

### “Descending cohomology, geometrically” by Mazur:

(Exist texts of that talk or related texts: http://ttv.mit.edu/collections/harris60/videos/13881-problem-session-barry-mazur ?) Article: http://www.math.harvard.edu/~mazur/papers/page37.pdf

**2**

votes

**0**answers

315 views

### “Extended” Weil Cohomology Theories

According to Wikipedia, a Weil cohomology theory is a functor from the category of smooth projective varieties over a field $k$, to graded algebras over a field $K$ of characteristic zero, together ...

**9**

votes

**0**answers

207 views

### Relation between the arithmetic Frobenius and the Frobenius of the $\varphi$-module of an unramified representation

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field $k$. Suppose $V$ is an unramified representation with associated continuous homomorphism ...

**10**

votes

**1**answer

1k views

### Why A. Weil considered elimination theory to be eliminated?

It is well known that André Weil declared, in the 1940's, that elimination theory must be eliminated from algebraic geometry. I would like to understand his mathematical reasons to adopt such an ...

**1**

vote

**0**answers

107 views

### Rational solutions of equations of the form $y^2 x = f(x)$

Let $k$ be any number field, and suppose we want to study the $k$-rational points on
$$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 3. In other words, $y^2 x = f(x)$ is a ...

**4**

votes

**1**answer

375 views

### Cubic forms and Hasse Principle

It's well-known that quadratic forms over the rational numbers $\mathbf{Q}$ satisfy the Hasse-Minkowski theorem. This is to say that they are isotropic over $\mathbf{Q}$ if and only if they are ...

**4**

votes

**0**answers

150 views

### Shimura varieties and Maximal conditions

Working with Shimura varieties, I have been convinced to call them (or the families giving rise to them especially in $A_{g}$) somehow the "maximal" families. The motivation of this, has been for ...

**9**

votes

**0**answers

225 views

### congruences of level 1 and level p modular forms

I've been carrying out some experiments on the computer and I noticed the following congruence phenomenon: fixing a prime $p$, it seems that any modular form over $SL_2(\mathbb{Z})$ and of weight $k ...

**2**

votes

**1**answer

192 views

### Are Isom-schemes geometrically connected

This question is about properties of Isom-schemes that are well-known over algebraically closed fields.
Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically ...