Questions tagged [arithmetic-geometry]

Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

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A curve with bad reduction for which the jacobian has good reduction

Let $K$ be a number field. If $X$ is a curve over $K$ with good reduction at a place $v$ of $K$, then the Jacobian of $X$ also has good reduction at $v$. This follows from the functoriality of the ...
Edgar's user avatar
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2 votes
0 answers
424 views

Are torsors over unipotent groups trivial

I might have misunderstood something I heard somewhere. Are torsors over unipotent groups trivial? I couldn't find this in some standard references.
Harry's user avatar
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33 votes
3 answers
3k views

Arithmetic geometry examples

(This is inspired by Algebraic geometry examples.) I want to collect here (counter)examples in arithmetic geometry. Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...
4 votes
1 answer
4k 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 ...
Srilakshmi's user avatar
3 votes
1 answer
927 views

Construction of Kummer map for abelian variety

Let $A$ be an abelian variety over the rational numbers $\mathbf{Q}$. Let $V=T_p A \otimes \mathbf{Q}_p$ be the $\mathbf{Q}_p$-Tate module of $A$. Let $G$ be the absolute Galois group of $\mathbf{Q}$. ...
Harry's user avatar
  • 1,203
1 vote
0 answers
443 views

Why do twists of an algebraic group over k correspond to k-torsors over G

Let $G$ be an algebraic group over a field $k$. Let $k^s$ be the separable closure of $k$. I can't seem to figure out why isomorphism classes of twists of $G$ correspond to $k$-torsors over $G$. It'...
Harry's user avatar
  • 1,203
5 votes
1 answer
801 views

Does the Mordell conjecture imply the Shafarevich conjecture

The base field is a number field. It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin). Is the converse also true? Note that both conjectures are now ...
Bobby's user avatar
  • 51
4 votes
0 answers
401 views

Every curve is a Hurwitz space in infinitely many ways

Diaz, Donagi and Harbater proved that every curve over $\overline{\mathbf{Q}}$ is a Hurwitz space. A Hurwitz space is a connected component of the curve $H_n$. The curve $H_n$ is (the ...
Harized's user avatar
  • 163
2 votes
0 answers
313 views

CM abelian variety from an algebraic Hecke character?

Hi, Given an algebraic Hecke character $\chi$ of a number field $k$ there should be a "rank 1 CM-motive" $M$ with $\overline Q$-coefficients such that $L(s,M) = L(s,\chi)$. This follows from the ...
Nicolás's user avatar
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2 votes
0 answers
206 views

algebraic de Rham cohomology of hypersufaces

For a smooth hypersurface $X\subset\mathbb{P}^n_k$, where $k$ is an algebraic closed field of charactersitc $p>0$. How to compute its algebraic de Rham cohomology explicitly? or equivalently its ...
henckcn's user avatar
  • 41
4 votes
2 answers
337 views

Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group

A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. ...
Harized's user avatar
  • 163
4 votes
1 answer
1k 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 )....
Srilakshmi's user avatar
7 votes
2 answers
508 views

Zograf's bound on the index of a modular curve for Shimura curves

I've been reading Voight's paper on Shimura curves and it prompted the following question; see http://www.cems.uvm.edu/~voight/articles/shimura-clay-proceedings-071707.pdf for which notes I'm talking ...
Ariyan Javanpeykar's user avatar
0 votes
0 answers
82 views

Extending functions on curves to functions on abelian varieties

Say I have a function $f_g$ on the moduli space of curves $M_g$ for all $g\geq 1$. Is there some way of extending this to the moduli space of abelian varieties $A_g$ in a nice way? What if I have ...
Harized's user avatar
  • 163
2 votes
1 answer
493 views

Etale group schemes over a local ring

Let $p$ be a prime number and $R$ be a Noetherian local ring of characteristic $p$ with residue field $k$. Let $G$ be a finite etale subgroup scheme over $R$ of order $p$. Suppose that the etale $k$-...
A.E.'s user avatar
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7 votes
1 answer
738 views

$p$-adic uniformization not from the Drinfel'd spaces?

It seems that when we talk about the $p$-adic uniformization, we typically mean those uniformized by either the Drinfel'd upper spaces (for which we think of the examples of Mumford curves and some ...
genshin's user avatar
  • 1,305
5 votes
4 answers
517 views

What is the obstruction for a local set of models of a curve to come from a global model?

If $X_{\mathbb{Q}}$ is a curve over $\mathbb{Q}$, we get a curve $X_{\mathbb{Q}_p}$ over $\mathbb{Q}_p$ for every prime $p$. My question is about the reverse process. Say we are given curves $X_{\...
James D. Taylor's user avatar
4 votes
0 answers
124 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 ...
Tzanko Matev's user avatar
6 votes
4 answers
1k views

Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry

I hear that the nonstandard methods are applied to many problems in various fields of mathematics such as functional analysis, topology, probability theory and so on. So, I have become interested in ...
Hiro's user avatar
  • 945
5 votes
3 answers
949 views

The historical development of automorphic geometry

Background: Today the notion of automorphic geometry is often framed in the context of the Langlands program, in particular what is sometimes called the Langlands reciprocity conjecture. This is ...
Laie's user avatar
  • 1,694
10 votes
0 answers
398 views

Is there an algorithm which determines if a curve has good reduction outside a given set of primes

Fix a number field $K/\mathbf{Q}$, a finite set of places $S$ in $K$, an integer $g$ and a curve $X$ over $K$ of genus $g$. Is there an algorithm which tells you if $X$ has good reduction outside $S$?...
Ali's user avatar
  • 153
16 votes
2 answers
5k views

why we need rigid geometry?

Hello, everyone. I want to ask some questions about rigid geometry. 1.what is the motivation of rigid geometry? 2.what is the applications of rigid geometry for solving arithmetic problems, ...
kiseki's user avatar
  • 1,911
10 votes
1 answer
1k 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 ...
Matt's user avatar
  • 970
3 votes
1 answer
466 views

Brauer-Manin obstruction and Hasse principle

I am looking for varieties without $\mathbf{Q}$-rational points where the absence can be explained by the Brauer-Manin obstruction, but not by the absence of adelic points varieties without $\mathbf{...
user avatar
2 votes
2 answers
524 views

branch points of modular parametrization of an elliptic curve

Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X_0(n) \to E$. Consider composition $f:X_0(n)\to \mathbf{P}^1_K$, where we compose with degree 2 cover $E\to \mathbf{...
Maxim's user avatar
  • 23
13 votes
2 answers
2k views

Why is the definition of l-adic sheaves so complicated?

I find the definition of constructible $\bar{\mathbb Q}_l$-sheaves (or their derived category) on a variety of positive characteristic quite involved and ad Hoc. Roughly it goes as follows: First one ...
Jan Weidner's user avatar
  • 12.8k
1 vote
1 answer
358 views

Manin-Drinfeld and constructing a finite morphism with two given ramification points

Fix a smooth projective connected curve $X$ over $\overline{\mathbf{Q}}$ of genus $g\geq 1$ and distinct points $x,y \in X$ such that $x-y$ has infinite order in the Jacobian. Can we always find a ...
Ariyan Javanpeykar's user avatar
3 votes
0 answers
308 views

Invertible Hasse-Witt for non-ordinary curves

Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the ...
Cyrus's user avatar
  • 395
2 votes
0 answers
353 views

modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes"

Is there a modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes" http://www.jstor.org/pss/1970644 i.e. using Grothendieck's language of schemes and in English?
user avatar
1 vote
0 answers
115 views

singularities $\mathcal{A}_{g,d}$ in positive characteristic

Hi, I would like to understand the singularities of the moduli spaces of abelian varieties with polarization of degree $d>1$ in characteristic $p>0$ when $p|d$. Do you know some good references?...
uuk's user avatar
  • 11
6 votes
0 answers
2k views

project proposal: English translation of Deligne's "La conjecture de Weil : II" [closed]

First of all, I hope this "question" is appropriate here. If not, please delete it. I would like to propose a translation project of Deligne's "La conjecture de Weil : II" 52_137_0">http://www.numdam....
user avatar
2 votes
1 answer
401 views

Does each finite morphism of curves have a model whose minimal resolution is semi-stable

Let $\pi:Y\to X$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$. Question. Does there exist a finite field extension $L/K$ and a regular model $\...
Ariyan Javanpeykar's user avatar
5 votes
0 answers
388 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 $X(\mathbb{C}...
Hugo Chapdelaine's user avatar
0 votes
0 answers
278 views

well known facts on openness condition

Hi, I would like to understand and prove the following two "well-known" facts: 1) If $B$ is a scheme and $P$ a property for which I know: i) if $B=Spec(V)$ where $V$ is a complete DVR, if $P$ ...
uuuk's user avatar
  • 1
9 votes
1 answer
413 views

finiteness of torsion points of an abelian variety over a totally real field?

Ribet has shown the following result: if $A$ is an abelian variety over a number field $E$, then the torsion subgroup of $A(E^{cyc})$ is finite. Here $E^{cyc}$ is the union $\bigcup_nE(\mu_n)$, $\mu_n$...
genshin's user avatar
  • 1,305
4 votes
1 answer
380 views

references for theta characteristic

Hi, I am looking for references on theta characteristics. In particular I am interesting in understanding the isomorphism $\Omega_A^g\cong\mathcal{O}_A(\Theta)^2$ where $A$ is an abelian variety and $...
uujk's user avatar
  • 43
3 votes
1 answer
700 views

Is every Weil divisor on an arithmetic surface Q-Cartier

This question is about a technical issue I ran into. Let $S$ be a connected 1-dimensional Dedekind scheme, and let $X\to S$ be a flat projective integral normal 2-dimensional scheme. (For simplicity, ...
Ariyan Javanpeykar's user avatar
5 votes
0 answers
827 views

Motivic Galois group and Shimura varieties

Hi, Suppose that one has a Shimura variety $Sh(G,X)$ where $(G,X)$ is the corresponding Shimura datum and suppose that it can be interpreted as a moduli space of motives (e.g. PEL type Shimura ...
unknown's user avatar
  • 647
6 votes
0 answers
294 views

Does a lower bound for models of finite group schemes exist?

Let $R$ be a discrete valuation ring (as beautiful as you like) and set $K:=Frac(R)$. Let $G_K$ be a finite $K$-group scheme, $G_1$ and $G_2$ two affine and flat models of $G_K$ of finite type, i.e. ...
Federigo's user avatar
3 votes
2 answers
418 views

Does a curve have infinitely many $K$-rational points under these hypotheses?

The question was a bit long for the title, so let me explain what I mean here. Let $k$ be some field (ideally, a number field). Let $X$ be a curve that is defined over $k$, such that it has a $k$-...
James D. Taylor's user avatar
3 votes
2 answers
731 views

Almost Northcott properties for heights of abelian varieties

Let $h$ be a function on the moduli space of abelian varieties of dimension $g$ over $\overline{\mathbf{Q}}$. Let $K$ be a number field and let $g\geq 2$ be an integer. Fix a real number $C$. Does ...
Hafez's user avatar
  • 31
2 votes
1 answer
244 views

Is there an easier argument to prove that almost all of these curves have no semi-stable reduction

Fix a number field $K$ and a polynomial $F(x)\in K[x]$ of degree at least $4$. For a squarefree integer $d$, define the curve $X_d$ over $K$ by the equation $dy^2 = F(x)$. Note that the curves $X_d$ ...
Ariyan Javanpeykar's user avatar
7 votes
1 answer
1k views

References for bad reduction of Jacobians of modular curves?

Hi, Where can I learn about the reduction of the Jacobians of modular curves such as X_0(N) and X_1(N) at primes p dividing N? Thanks!
Nicolás's user avatar
  • 2,802
9 votes
1 answer
653 views

Can we always find a curve which doesn't have semi-stable reduction

Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction ...
Shaye's user avatar
  • 145
1 vote
0 answers
230 views

Lower bound for intersection number

The base scheme is an algebraically closed field. Let $X\to \mathbf{P}^1$ be an arithmetic surface over $\mathbf{P}^1$ and let $P$ be a section of $X\to \mathbf{P}^1$. Let $D$ be an effective (edited)...
Taicho's user avatar
  • 225
24 votes
2 answers
4k views

A route towards understanding Shimura varieties?

I'm in the embarrassing situation that I want to ask a question that was already asked, but (for complicated reasons) never answered. I'd like to try with a blank slate. Shimura varieties show ...
user9509's user avatar
  • 415
3 votes
0 answers
202 views

Computing Elliptic Curves of Conductor Divisible by a Large Prime Factor

A little while ago, I came across a paper (or slides from a talk or something) that seemed to suggest that the modular symbol method for computing elliptic curves over $\mathbf{Q}$ of prescribed ...
NPC's user avatar
  • 299
1 vote
0 answers
204 views

Which rational subfields are corresponding to the two dimensional subspaces of holomorphic differentials

I implemented the algorithm that Felipe Voloch's suggested in his reply to the question: Subfields of a function field the algorithm is here: Subfields of a function field I considered the ...
Syed's user avatar
  • 601
2 votes
1 answer
264 views

Comparing heights of rational points on curves through covers

Let $a$ be a closed point in $\mathbf{P}^1_{\overline{\mathbf{Q}}}$. Let $Y \cong \mathbf{P}^1_{\overline{\mathbf{Q}}} $ and let $\pi:Y\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$ be a finite morphism ...
Showmie's user avatar
  • 23
15 votes
1 answer
751 views

Crystalline realization of mixed Tate motives

Deligne and Goncharov, in their article of 2005, mention that the crystalline realization functor has yet to be worked out. What's the current state of the literature on this? And how big of an issue ...
Ishaidc's user avatar
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