Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
2,042
questions
13
votes
1
answer
1k
views
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 ...
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.
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 ...
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 $\mathbf{Q}$. ...
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'...
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 ...
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 ...
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 ...
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 ...
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 $\overline{\mathbf{Q}}$. ...
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 )....
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 ...
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 ...
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 $k$-...
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 ...
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_{\...
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 ...
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 ...
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 ...
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 $S$?...
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, ...
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 ...
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{...
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{...
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 ...
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 ...
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 ...
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?
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?...
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....
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 $\...
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}...
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 $P$ ...
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 $\bigcup_nE(\mu_n)$, $\mu_n$...
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 $...
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, ...
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 ...
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. ...
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 $k$-...
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 ...
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 $X_d$ ...
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!
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 ...
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)...
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 ...
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 ...
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 ...
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 ...
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 ...