Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
2,042
questions
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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 = \...
5
votes
0
answers
428
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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_{...
12
votes
0
answers
285
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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(\...
11
votes
1
answer
683
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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 ...
8
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253
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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 $\...
3
votes
1
answer
254
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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 ...
0
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1
answer
589
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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 ...
31
votes
2
answers
1k
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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 ...
15
votes
2
answers
792
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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, ...
13
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3
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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 ...
3
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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}...
4
votes
1
answer
593
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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{...
5
votes
1
answer
691
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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}^...
6
votes
2
answers
686
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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 ...
31
votes
3
answers
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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 $\...
10
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1
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413
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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,...
3
votes
1
answer
398
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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 ...
40
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4
answers
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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 $\...
3
votes
1
answer
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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 ...
8
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380
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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=...
11
votes
1
answer
358
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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 ...
6
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1
answer
254
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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 ...
1
vote
1
answer
140
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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 ...
3
votes
1
answer
300
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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^...
5
votes
0
answers
659
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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{...
6
votes
1
answer
509
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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 ...
18
votes
1
answer
1k
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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(\...
6
votes
1
answer
346
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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 ...
3
votes
0
answers
127
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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 ...
2
votes
0
answers
214
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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 ...
4
votes
1
answer
311
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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\...
27
votes
1
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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. ...
11
votes
1
answer
881
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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 ...
6
votes
1
answer
804
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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 $J_p$ ...
4
votes
1
answer
303
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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 $\mathbb{Q}$). ...
6
votes
2
answers
2k
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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 ...
7
votes
1
answer
390
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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 $\rho: \pi_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ ...
6
votes
1
answer
179
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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 ...
6
votes
0
answers
367
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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 $\...
2
votes
1
answer
226
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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(...
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 ...
3
votes
0
answers
131
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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 ...
3
votes
1
answer
306
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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 ...
5
votes
1
answer
1k
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"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 ...
11
votes
3
answers
550
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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 ...
8
votes
0
answers
592
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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 $\ell$-...
10
votes
1
answer
871
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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, ...
24
votes
1
answer
2k
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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' ...
1
vote
1
answer
162
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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 ...
5
votes
1
answer
615
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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 ...