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Questions tagged [arithmetic-geometry]

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

Arithmetic geometry is an explosively developing field of mathematics at the junction of modern algebraic geometry and number theory.

2042 questions
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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, especially for studying the fundamental groups of algebraic…
kiseki
  • 1,911
15
votes
3 answers

Integer points (very naive question)

Well, I don't have any notion of arithmetic geometry, but I would like to understand what arithmetic geometers mean when they say "integer point of a variety/scheme $X$" (like e.g. in "integer points of an elliptic curve"). Is an integer point just…
Qfwfq
  • 22,715
13
votes
2 answers

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 defines constructible sheaves modules over…
Jan Weidner
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10
votes
0 answers

Paths in $\mathrm{Spec} \, \mathbb{Z}$ and Kim's proof of Siegel's theorem for $\mathbb{P}^1 \setminus \{0,1,\infty\}$

This is motivated by a basic number theory question I asked the previous day: Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$? I noted there that the answer to the corresponding question is "no" in complex function…
Vesselin Dimitrov
  • 13,703
9
votes
2 answers

Picard number and torsion of Neron-Severi group of abelian varieties over a number field

Let $A$ be an abelian variety over a number field $k$ and let $NS_A$ denote its Neron-Severi scheme. Then the group of $k$-rational points of $NS_A$ is a finitely generated abelian group, i.e. $NS_A(k)=H^0(G_k,NS_A(\bar k)) = \mathbb{Z}^\rho \oplus…
Stefan Keil
  • 821
8
votes
0 answers

Frey-Mazur for abelian varieties

Let $K$ be a number field. The Frey-Mazur conjecture asserts the existence of a constant $N_K$ such that for all primes $p>N_K$, and all pairs of elliptic curves $E_1$, $E_2/K$, if $\overline{\rho}_{E_1,p} \sim \overline{\rho}_{E_2,p}$ then $E_1$ is…
Siksek
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5
votes
1 answer

connected component of the identity section in the special fiber of the Neron model under isogenies

Let $A$ and $B$ be two isogenous abelian varieties over a number field $K$ and let $\mathcal{A}$ and $\mathcal{B}$ denote their Neron models over $\mathcal{O}_K$. Let $v \in M_K^0$ denote a finite prime of $K$, $k_v$ its residue field,…
Stefan Keil
  • 821
4
votes
1 answer

Why is the kernel of reduction a pro-p group?

Let $A$ be an abelian variety over a $p$-adic field $K_v$, i.e., $K_v$ is a finite field extension of $\mathbb Q_p$, for $p$ a prime number. Denote by $k_v$ the residue field of $K_v$ and let $\mathcal{A}_v^{0}(k_v)$ be the smooth part of the…
Stefan Keil
  • 821
4
votes
0 answers

Weil cohomology theories "genuinely" of positive characteristic

One of the reasons why Weil cohomology theories are required to have coefficients in a field of characteristic 0 is that they are supposed to be robust enough to solve Weil conjectures, i.e. to count points (and if coefficients have…
user140971
3
votes
2 answers

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 $k$-point. Let $K$ be an algebraic extension of $k$ which…
James D. Taylor
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3
votes
0 answers

rational points of component group of the special fiber of the Neron model

Let $A$ be an abelian variety over a number field $K$ and let $\mathcal{A}$ denote its Neron model over $\mathcal{O}_K$. Let $v \in M_K^0$ denote a finite prime of $K$, $k_v$ its residue field, $\mathcal{A}_v = \mathcal{A} \times _{\mathcal{O}_K}…
Stefan Keil
  • 821
3
votes
0 answers

Wintenberger's mystery

Fontaine describes in §2 of this old survey work by Wintenberger and wonders (on p. 97) that the structures found by Wintenberger are a "complete mystery" and no-one knows a "reasonable geometric interpretation". Is that still a mystery?
Thomas Riepe
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3
votes
0 answers

Tate-Shafarevich group of non-principally polarized abelian variety

Let $A/k$ be an abelian variety over a number field $k$ with a polarization of minimal degree $d>1$. (Assume all Tate-Shafarevich groups to be finite.) What can one say about the order of $\mathrm{III}(A/k)$ in terms of being a multiple of a square?…
Stefan Keil
  • 821
3
votes
0 answers

points on non-hyperelliptic curves of genus 3

I have a non-hyperelliptic curve $C$ of genus 3 and I'm interested in finding the $K$-rational points on the curve with $K$ a fixed imaginary quadratic number field. As $C$ is a non-hyperelliptic genus 3 curve, the Jacobian $J_C(\mathbb{Q})$ of $C$…
Marcel
  • 31
3
votes
2 answers

Reference for Using Group Cohomology to calculate Etale Cohomology

I'm looking for a reference for the following statement: Let $X$ be a variety (over an algebraically closed field $k$), and let $F$ be a locally constant etale sheaf. Let $x \in X(k)$. Then $ \mathrm{H}^i(X, F) = \mathrm{H}^i(\pi_1(X,x), F_x)$. It…
user51764
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