Questions tagged [arithmetic-geometry]

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

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Good cohomological setting for binary operations on arithmetical functions

Is there currently a good abstract theory (derived from algebraic geometry and cohomological theories) to study binary operations on arithmetical functions like the Dirichlet convolution $$f\star g = \...
C. Dubussy's user avatar
5 votes
0 answers
428 views

Torsionfree crystalline cohomology implies torsionfree etale cohomology?

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$. Let $W=W(k)$ be the ring of Witt vectors of $k$. Assume that the crystalline cohomology $H^2_{...
Monsie's user avatar
  • 81
12 votes
0 answers
285 views

Modularity of endomorphism algebras

This question is about comparing Hecke algebras and endomorphism algebras. Let $\mathbf{A}_f$ be the ring of finite adèles of $\mathbf{Q}$ and let $K$ be a compact open subgroup of $\mathrm{GL}_2(\...
François Brunault's user avatar
11 votes
1 answer
683 views

Galois Representations and Rational Points

Suppose $X$ and $Y$ are two connected smooth projective varieties over $\mathbb{Q}$ (of the same dimension) that have the same $\ell$-adic Galois representations (up to semisimplification). What is ...
user103716's user avatar
8 votes
0 answers
253 views

Ramification for subgroups of Lubin-Tate formal group

Let $K/\mathbb{Q}_p$ be a finite field extension and $E/\mathcal{O}_K$ be the local N\'eron model of a CM elliptic curve with CM by $\mathcal{O}_F$ and let $G\subseteq E[p^n]$ be a subgroup over $\...
Vincent's user avatar
  • 443
3 votes
1 answer
254 views

Arithmetic projective duality

Projective duality is a duality that associates to a (smooth) subvariety X of $\mathbb{P}^n$ the dual variety $X^*\subset\mathbb{P}^{n*}$ of tangent hyperplanes. What makes the duality interesting ...
Bear's user avatar
  • 845
0 votes
1 answer
589 views

Regularity of schemes under base change

Let $K$ and $K'$ be number fields $K \subset K'$, and let $R$ and $R'$ be the corresponding ring of integers. Let $S = Spec\ R$ and $S' = Spec\ R'$. Suppose $X \to S$ be an arithmetic surface that is ...
Chitrabhanu's user avatar
31 votes
2 answers
1k views

The Sylvester-Gallai theorem over $p$-adic fields

The famous Sylvester-Gallai theorem states that for any finite set $X$ of points in the plane $\mathbf{R}^2$, not all on a line, there is a line passing through exactly two points of $X$. What ...
François Brunault's user avatar
15 votes
2 answers
792 views

Multizeta function values

Francis Brown Theorem says that $\zeta(a_{1},\dots ,a_{r})$ the multi-zeta value of weight $N=a_{1}+\dots +a_{r}$ is a $\mathbb{Q}$-linear combination of elements of the set $S=\{\zeta(a_{1},\dots, ...
Nguyen lan Lee's user avatar
13 votes
3 answers
1k views

Is the map on étale fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?

$\DeclareMathOperator\Spec{Spec}$Let $k \subset L$ be two algebraically closed fields of characteristic $0$. Let $U \subset \mathbb P^n_k$ be a smooth quasi-projective variety and let $U_L$ denote the ...
Aaron Landesman's user avatar
3 votes
0 answers
95 views

CM abelian surfaces (computed locally)

Let $K$ be a CM field such that $[K:\mathbb{Q}] = 4$ and let $K^+\subseteq K$ be the totally real subfield of $K$. For simplicity, assume that $K/\mathbb{Q}$ is Galois and suppose that $p\in \mathbb{Z}...
Vincent's user avatar
  • 443
4 votes
1 answer
593 views

finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme

Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme. Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{...
user avatar
5 votes
1 answer
691 views

Understanding Siegel's Theorem on integral points

Siegel's theorem states the following: Let $C$ be a smooth projective curve over a number field $K$. Let $\tilde C\subset C$ be an open affine subvariety, and $i:\tilde C\hookrightarrow \mathbb{A}^...
Andrew NC's user avatar
  • 2,011
6 votes
2 answers
686 views

Integer roots of a symmetric polynomial

The question is very simple and I apologize for that, but I am not an expert of this kind of problem. Given the polynomial $$ P(x_1,\ldots,x_{2n})=x_1^2+\ldots+x_n^2-x_{n+1}^2-\ldots-x_{2n}^2,$$ I ...
Felice Iandoli's user avatar
31 votes
3 answers
1k views

Consequences of Shafarevich conjecture

The Shafarevich conjecture states that the Galois group $\mathrm{Gal}({\overline{\mathbf{Q}}/\mathbf{Q}^{ab}})$ is a free profinite group, where $\mathbf{Q}^{ab}$ is the maximal abelian extension of $\...
Muhammed Ali's user avatar
10 votes
1 answer
413 views

Is there a notion of hyperbolicity for number rings?

For algebraic curves over a nice enough field $k$, we have a notion of what it means to be hyperbolic: If $\overline{C}$ is a smooth projective curve of genus $g$ and $P_1,\dots,P_n$ are closed points,...
Matthias Wendt's user avatar
3 votes
1 answer
398 views

Crystalline extension the $p$-adic cyclotomic character

Let $\epsilon_p$ be the $p$-adic cyclotomic character, $F$ be a real quadratic extension of $\mathbb{Q}$ in which $p$ splits, $\psi$ be an odd character of $G_\mathbb{Q}$ of finite image and with ...
Adel BETINA's user avatar
  • 1,046
40 votes
4 answers
3k views

Why are Green functions involved in intersection theory?

I've been learning Arakelov geometry on surfaces for a while. Formally I've understood how things work, but I'm still missing a big picture. Summary: Let $X$ be an arithmetic surface over $\...
Dubious's user avatar
  • 1,237
3 votes
1 answer
1k views

Is the Tate-Shafarevich group of a rational elliptic curve finite?

It seems that Lan Nguyen proved in a preprint on arxiv of 2013 that the Tate-Shafarevich group of a rational elliptic curve is finite. However, I couldn't find any published version thereof. So is it ...
Sylvain JULIEN's user avatar
8 votes
0 answers
380 views

Voronoi summation and functional equation, from the point of view of distributions

Consider the Voronoi summation formula for the sum of squares function $r_2$, in terms of Bessel function $J_0$: $$\sum_{n=0}^\infty r_2(n) \int_0^\infty \pi J_0(2\pi\sqrt{nx}) f(x) \, dx = \sum_{n=...
Serendipity's user avatar
11 votes
1 answer
358 views

Is there a presentation to the kernel of the prime-to-$p$ fundamental short exact sequence of curves over finite fields?

Let $X$ be $\mathbb{P}^1_{\mathbb{F}_q}\smallsetminus \{a_1,...,a_r\}$, where $a_1,...,a_r$ are some $\mathbb{F}_q$-rational points. Let $\bar X :=X_{\bar{\mathbb{F}}_q}$. There is a short exact ...
Andrew NC's user avatar
  • 2,011
6 votes
1 answer
254 views

Does exist a "product formula" for arithmetic surfaces?

Let $K$ be a number field, then the Arakelov geometry of $\operatorname{Spec }O_K$ can be interpreted by means of the adelic ring $\mathbb A_K$. In particular, a key ingredient is the product formula ...
user100660's user avatar
1 vote
1 answer
140 views

Transformation of height on projective varieties

I am trying to find a reference for the following statement: let $X$ and $Y$ be projective varieties defined over $ \mathbb{Q}$ and $\phi: X \to Y$ be a rational map defined over $ \mathbb{Q}$. Denote ...
Keivan Karai's user avatar
  • 6,064
3 votes
1 answer
300 views

hard Lefschetz isomorphism for rational Tate module

Let $k$ be a finite field, $\ell \neq \mathrm{char} k$ be prime, $X/k$ be a smooth projective geometrically integral variety of dimension $d$, and $\mathcal{A}/X$ be an Abelian scheme. Let $\eta \in H^...
user avatar
5 votes
0 answers
659 views

Basic question on p-adic Hodge theory

I am starting to study the rudiments of p-adic Hodge theory and I have the following basic question. Let $\chi$ be the unramified quadratic character of $G_p = \mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{...
Michael's user avatar
  • 111
6 votes
1 answer
509 views

Endomorphisms of elliptic curves with CM; can we have an order?

Let $E$ be an elliptic curve over $\mathbb C$ with CM by ring of integers $O_K$ of an imaginary quadratic number field $K$. Let $O$ be an order of $O_K$. Is there a number field $L$ such that $E$ has ...
Ciro's user avatar
  • 119
18 votes
1 answer
1k views

On Tate's "Endomorphisms of Abelian Varieties over Finite Fields", sketch of proof of main result?

Let $k$ be a field, $\overline{k}$ its algebraic closure, and $A$ an abelian variety defined over $k$, of dimension $g$. For each integer $m \ge 1$, let $A_m$ denote the group of elements $a \in A(\...
user avatar
6 votes
1 answer
346 views

Infinitely many primes $p \in \mathbb{Z}$ where reduced curve $E/\mathbb{F}_p$ has Hasse invariant $1$?

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and fix a Weierstrass equation for $E$ having coefficients in $\mathbb{Z}$. Are there infinitely many primes $p \in \mathbb{Z}$ such that the ...
user388407's user avatar
3 votes
0 answers
127 views

On Abhyankar's results cited in a paper of Manin titled "Correspondences, Motifs and Monoidal Transformations"

Consider the following from this paper "Correspondences, Motifs and Monoidal Transformations" of Manin here. Theorem. Nonsingular three-dimensional projective unirational varieties $V$ over ...
user100749's user avatar
2 votes
0 answers
214 views

p-divisible groups over a p-adic field

p-div gps over a finite field (e.g., $\mathbb F_p$) are classified by Dieudonne modules. There are also results for p-div gps over integers of local fields (e.g., over $\mathbb Z_p$). However, are ...
guestguest's user avatar
4 votes
1 answer
311 views

Counting integral points on a surface

Let $f$ be a homogeneous polynomial with integral coefficients of 4 variables $a$, $b$, $c$ and $d$. Suppose $f$ is invariant under the rotation that rotates $(a,b)\in\mathbb{R}^2$ and $(c,d)\in\...
Fan Zheng's user avatar
  • 5,119
27 votes
1 answer
1k views

Analogies between classical geometry on complex surfaces and Arakelov geometry

This is my first question on this wonderful site. The following question about Arakelov geometry is gonna be quite long and wide; to be clear one of that kind of questions that are usually ignored. ...
user100660's user avatar
11 votes
1 answer
881 views

Inverse Limits in the category of Perfectoid Spaces

First I apologize as the following is likely littered with misunderstanding. My understanding at least from the discussion in Scholze-Weinstein is that inverse limits in the category of adic spaces ...
Alexander's user avatar
  • 861
6 votes
1 answer
804 views

A Jacobian with a good reduction, which is simple : how is the reduction of the curve?

Let $C$ be a smooth genus $g>1$ curve defined over a number field (say over $\mathbb Q$). Let $J(C)$ be its jacobian. Suppose that $J(C)$ has good reduction $J_p$ at a prime $p$ and moreover $J_p$ ...
Xavier Roulleau's user avatar
4 votes
1 answer
303 views

Albanese of Siegel modular variety $\mathcal{A}_2$

Jacobian varieties of Shimura curves are very interesting objects. For one thing they provide a geometric relation between elliptic curves and modular forms of weight 2 (say we are over $\mathbb{Q}$). ...
Bear's user avatar
  • 845
6 votes
2 answers
2k views

Sketch of Weil's proof of the Riemann hypothesis for curves

I was wondering if anybody could provide a sketch of Weil's proof of the Riemann hypothesis for curves that uses the Jacobian $\text{Pic}^0(X)$ and a bit of the intersection theory on $X \times X$ and ...
user avatar
7 votes
1 answer
390 views

Reference result: proof of theorem of Kazhdan-Margulis on monodromy group of a Lefschetz pencil of odd fiber dimenion is "as big as possible"

In Deligne's paper on his first proof of the Weil conjectures, we have the following result. Theorem 5.10 (Kazhdan-Margulis). L'image de $\rho: \pi_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ ...
user avatar
6 votes
1 answer
179 views

If $X$ is a genus $g\geq 2$ curve over a number field $K$, then is there a bound on $X_L(L)$ for $L/K$ that depends only on $X$ and $[L:K]$?

If $X$ is a genus $g\geq 2$ curve over a number field $K$, then $X(K)$ is finite by Falting's Theorem. My question is how does $X_L(L)$ behave for finite field extensions $L/K$? In particular, is ...
Andrew NC's user avatar
  • 2,011
6 votes
0 answers
367 views

Conjecture on classification of $p$-divisible over the ring of integers of $\widehat{\bar{\mathbb{Q}}_p}$

I am reading the paper of Fargues Quelques résultats et conjectures concernant la courbe. In the end of this paper, there is a conjecture on the classification of $p$-divisible groups over $\...
Mayday's user avatar
  • 193
2 votes
1 answer
226 views

Pencil of polynomials mostly irreducible?

Let $p,q \in \mathbb{Q}[x]$ two relatively prime polynomials. Let $h\in \mathbb{R}$ any number and let $F_h(x) = p(x) + h \cdot q(x)$. What can be said about the irreducibility of the polynomial $F_h(...
Gabor Lippner's user avatar
6 votes
1 answer
294 views

Abelian varieties with p-rank zero

Let $X$ be an abelian variety over a finite field of characteristic $p$ such that the $X[p]=0$. In other words, none of the Newton slopes are $0,1$. QUESTIONS. (a) Is it possible for the ...
Student88's user avatar
  • 337
3 votes
0 answers
131 views

Arithmetic version of "Attaching maps" for moduli of curves

I am looking for a reference for attaching maps of moduli of curves with marked points. Especially I would like to know whether they descend over $\mathbb{Z}$. On one hand this seems very hard to ...
Bear's user avatar
  • 845
3 votes
1 answer
306 views

Vojta's conjecture on the bounded degree algebraic points over projective line?

I want to know the accurate version of the Vojta's conjecture on the bounded degree algebraic points over projective line, which is known as an extension of the Roth's theorem for bounded degree ...
Sajad Salami's user avatar
5 votes
1 answer
1k views

"Role" of cohomology of coherent sheaves in SGA 4.5, étale cohomology

As the question title suggests, what is the role cohomology of coherent sheaves plays for SGA 4.5, étale cohomology? Why are they so important for the construction and establishing properties of étale ...
user avatar
11 votes
3 answers
550 views

When $k = \mathbb{F}_q$ finite field, $X$ always has $k$-rational point, and so $A \simeq X$?

Let $k$ be an arbitrary field. Let $(A, e)$ be an abelian variety over $k$, and let $X$ be a torsor for $A$, i.e. $X$ is a proper smooth $k$-variety, and there is an $A$-action acting $:A \times X \to ...
user avatar
8 votes
0 answers
592 views

A Generalization of the Tate-Shafarevich/Tate/Fontaine-Mazur Conjectures

Let $A$ be an abelian variety over a number field $k$. The Tate-Shafarevich conjecture says that the Tate-Shafarevich group of $A$ is finite. A weakening of this conjecture states that the $\ell$-...
David Corwin's user avatar
  • 15.1k
10 votes
1 answer
871 views

Dwork's proof of rationality of zeta function, crux of his generalization of a result of Borel along the way

In this article by Katz and Tate here, there's a nice account of Dwork's argument for showing the rationality of the zeta function part of the Weil conjectures. Here is an excerpt. To recapitulate, ...
user avatar
24 votes
1 answer
2k views

Clarifying the connection between 'etale locally' and 'formally locally'

The following type of statements is often bandied about around the mathematical watering hole: "etale close is closer than formally close". Namely, if one imagines some sort of 'absolute topology' ...
SomeGuy's user avatar
  • 833
1 vote
1 answer
162 views

Another fix field of a certain galois group action

Let $E=\mathbb{F}_p(\!(u)\!)$, the Laurent series field over $\mathbb{F}_p$. Let $K/E$ be a finite normal separable extension. Consider the field $L=K(x \mid x^p-x-a=0 \text{ for some } a \in K)$. Let ...
Louis's user avatar
  • 189
5 votes
1 answer
615 views

Brauer group of a product of curves

By a famous theorem of Tate, we know that the Tate conjecture holds for a product of curves over a finite field. But this implies that the Brauer group of a product of curves (over finite field) is ...
Sylvain Lefuste's user avatar

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