Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
2,044
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Reconstruction of hyperbolic curves using the fundamental group
In the paper Curves and their Fundamental Groups written by Gerd Faltings, Mochizuki's proof of Grothendick's conjecture on anabelian curves is explained.
In the proof, he shows that for two ...
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1
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Locally analytic vectors of a quotient space
My question here is in connection with one of my previous question
"A definition of a (amalgamated) direct sum"
Following the notations there, my question is:
Why the locally analytic vectors of $B(...
- 685
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Oesterlé's unpublished bound on Uniform Boundedness
The bound in Merel's solution to the Uniform Boundedness conjecture is not explicit, as it relies on Falting's work on the Mordell conjecture. I think this still is the case.
But there are known ...
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Reduction of torsion points on Neron Model
Let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $R$ and residue field $k$. Let $A/K$ be an abelian variety with Neron model $\mathcal{A}/R$. We denote by $\tilde{\mathcal{A}}/k$ the ...
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Reference for map $\operatorname{Hom}^d(C,\mathbb{P}^1) \to \operatorname{Sym}^d(C)$
For a curve $C$ over a finite field, I am looking at the map $\phi: \operatorname{Hom}^d(C,\mathbb{P}^1) \to \operatorname{Sym}^d(C)$ where $\operatorname{Hom}^d(C,\mathbb{P}^1)$ are the functions of ...
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Supersingular isogenies of elliptic curves preserving divisibility of points
I hope my question is clear.
In summary, If $\phi:E\to E'$ is an isogeny and $P\in E$ is not divisible by $2$, under which conditions $\phi(P)\in E'$ is also not divisble by $2$.
Here is the detail ...
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Why does the Galois twist of this cover specialize to a certain field extension?
I didn't feel MO was the best place to ask this question, so apologies for this, but when I asked it at https://math.stackexchange.com/questions/2297837/why-is-this-cubic-polynomial-generic-for-cyclic-...
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exponential sum over variety
I am wondering where to find a good reference for bounds of the type
$$\sum_{x\in V(\mathbb{F}_p)} \chi(g(x))\psi(f(x))$$
where $V$ is a variety, $\chi$ is a multiplicative character over $\mathbb{...
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Recent progress toward Birch and Swinnerton-Dyer conjecture
Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after
The current status of the Birch & Swinnerton-Dyer Conjecture
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Reference on a result on local Galois representation associated to classic modular form in p-adic Hodge theory
At the end of Fontaine’s rings and p-adic L-functions, P. Colmez states a Theorem 8.4.8 (click here) of Faltings-Tusji-Saito without references.
So I am wondering is there any references for this ...
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Torsion points on twists of elliptic curves and products of fine modular curves over $\mathcal{M}_{1,1}$ vs over the $j$-line
Let $Y_1(n)$ (for $n\ge 4$) be the fine moduli scheme over $\mathbb{Q}$ parametrizing elliptic curves with a rational point of order $n$. Let $\mathbb{A}^1_j$ be the $j$-line over $\mathbb{Q}$, the ...
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The $\ell$- part of the class groups of the $p$-cyclotomic fields
Let $K_n = \Bbb Q(\mu_{p^{n+1}})$ and let $A_n$ be it's class group. Iwasawa theory tells us a lot about the $p$-part of $A_n$. For instance, we know quite a lot about how it varies with $n$.
I am ...
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A definition of a (amalgamated) direct sum
I am wondering about a definition of a direct sum in page $31$ of this paper by R. Liu.
I am following the notations in page $31$ of the above paper. Let $V$ be a crystalline irreducible ...
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Elliptic curves with the same mod $p$ representation
What is the largest prime number $p$ for which one knows examples of nonisogenous elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$ with isomorphic mod $p$ Galois representations: $E_1[p] \cong E_2[p]$...
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p-adic representations of $GL_2(\mathbb{Q}_p)$
Let $L$ be a finite extension of $\mathbb{Q}_p$. Colmez defines here
the trainguline representations which are extensions of Robba rings of dimension $1$. Then, in this paper he contructs the ...
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Is there any definition of $H^1$ in one dimensional Arakelov geometry
Consider a number field $K$ with ring of integers $O_K$. On the affine scheme $\overline X=\operatorname{Spec}(O_K)$ we have the well known one dimensional Arakelov geometry.
Let $\overline D=\sum_{\...
- 141
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Etale coverings of non-projective curves
For a smooth projective curve $Y$ over an algebraically closed field $k$ of characteristic 0, it is known that there exists a one-to-one correspondence between finite \'{e}tale morphisms $f:X\to Y$ of ...
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Why is $\mathbb{Q}_p(p^{1/p^\infty})$ a complete topological field?
In Matthias Wulkau's exposition of Scholze's thesis, the term perfectoid field is defined as follows:
Let $K$ be a field endowed with a non-archimedian absolute value $\lvert\cdot\rvert$, and let $\...
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Semi-Simplicity of Mod-$\ell$ Galois Representations
I am currently studying mod-$\ell$ Galois representations of $CM$ elliptic curves. More precisely, let $\ell$ be a prime, $K$ a number field and $E/K$ a $CM$ elliptic curve, where all endomorphisms of ...
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Gauss' Circle Problem at $\left ( \frac{1}{2}, \frac{1}{2} \right ) $
GCP (Gauss' Circle Problem) asks for a closed form for the number of square-lattice points inside a circle, centered at the origin, of radius $r$.
Let's denote by $N(r)$ the number of these points. ...
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G is p-divisible, about the affine rings of G[p]
Let $R$ be a ring and $G$ a $p$-divisible group over $R$. Since $G[p]$ is finite flat over $\text{Spec}(R)$, it is an affine (group) scheme, say $\text{Spec}(B)$. What can be said about $B$ beyond the ...
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Meaning of the determinant of cohomology
The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory ...
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Mumford-Tate conjecture cases with small $l$-adic monodromy groups
My question concerns the Mumford-Tate conjecture for abelian varieties over number fields.
Most proven cases (that I am familiar with) show that the l-adic monodromy group is as large as it can ...
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Deligne's Canonical Extension in Algebraic Varieties?
Suppose $C/\mathbb{Q}$ is an algebraic curve (not necessarily complete) defined over $\mathbb{Q}$, and $p$ is a $\mathbb{Q}$ valued point of $C$. Suppose there is a algebraic fibration
\begin{equation}...
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A surjective morphism of abelian varieties induces an epimorphism on the torsion subgroups
Let $f:A\to B$ be a surjective morphism of abelian varieties (over an algebraically closed field).
Why is it true that $f$ induces an epimorphism on the points of finite order $A_{\mathrm{tors}}\to ...
user107042
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$\mathbb Q_p$ étale local sytem in characteristic $p>0$
Let $k$ a field with $char(k)=p>0$, separable closure $k^{sep}$ and $f:X\rightarrow Y$ a smooth projective morphism of smooth variety over $k$.
1)Is it true that there exists a (EDIT) dense open ...
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Rank of elliptic surfaces
Is there any method to determine which elliptic curves over ${\mathbb Q}(t)$ have larger rank just from their equations—without knowing their exact rank—as with Mestres sums for elliptic curves?
For ...
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Dieudonne modules vs Dieudonne crystals reference/clarification
I've read a bit about Dieudonné modules, mainly from Fontaine's "Groupes p-divisibles sur les corps locaux" and Demazure's "Lectures on $p$-divisible groups". I am familiar with the main ...
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Does a generically injective morphism of schemes have a section?
Let $f \colon X \to Y$ be a generically injective, finite morphism between projective varieties over $\mathbb{C}$, with $Y$ non-singular. Does $f$ have a section i.e., a morphism $g:Y \to X$ such that ...
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Why it is difficult to define cohomology groups in Arakelov theory?
I have been puzzled by the following Faltings' remark in his paper Calculus on arithemetic surfaces (page 394) for a few months. He says:
If $D$ is a divisor on $X$, we would like to define a ...
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The role of Honda-Tate theory in (Scholze's refinement of) the Langlands-Kottwitz method?
I was wondering if somebody could give me a quick primer on Honda-Tate theory, specifically the number of isogeny classes of elliptic curves, and how specifically it's utilized in (Scholze's ...
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Endomorphism of a Jacobian
Let $C$ be a curve of genus $g$ defined over the finite field $k=\mathbb{F}_q$. Set $A=J(C)$ the jacobian of $C$, then $A$ is an abelian variety defined over $k$ of dimension $g$.
The algebra $End_k(...
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Relative position of elements in adic space
I'm trying to understand a certain function on a specific adic space I'm stuck on something silly and is probably due to my lack of understanding of the points in this case. This is Proposition 3.3.5 ...
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Splitting of polynomials over rational function fields
Let $K$ be a number field, and let $P(t,X)$ be a monic polynomial in $X$ with coefficients in $K(t)$.
I would like to understand the set $T$ consisting of those $t_0 \in K$ such that the polynomial $...
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Basis of homomorpshims of abelian varieties with minimal degree
Let $A, B$ be simple abelian varieties of dimension $g$ defined over a finite field $k$. We know that $Hom_k(A, B)$ is a free $\mathbb{Z}$-module of dimension $2g$.
Is it always possible to have a $\...
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morphism of abelian variety
Let $f: A \rightarrow B$ be a morphism of abelian varieties defined over a finite field $k$. Let $G$ be a finite group of $A$ and $\pi:A\rightarrow A/G$ the quotient morphism.
Looking at just the ...
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Endomorphism of Abelian variety over finite field
Let $A$ be an abelian variety defined over a finite field $k$ of characteristic $p$, such that $A/k$ is simple but not absolutely simple.
Let $f\in End_k(A)$ be an endomorphism defined over $k$, and ...
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Rational points on open subsets of affine space [closed]
Let $k$ be an infinite field. Assume that the index of the algebraic closure $\bar{k}$ over $k$ is strictly greater than $2$. Let $U$ be a non-empty open subset of some affine space over $k$. Is it ...
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Confusion regarding Riemann-Roch for vector bundles
Let $k$ be an infinite non-algebraically closed field, $X$ a smooth projective curve on $k$ and $E$ a locally-free sheaf on $X$ of rank at least $2$. Denote by $\bar{k}$ the algebraic closure of $k$, $...
- 1,949
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Poincaré duality and affine Lefschetz for flat cohomology
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p > 0$. Is there an affine Lefschetz theorem and Poincaré duality for sheaves represented by finite ...
user19475
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compute conjugacy classes of matrices over $\mathcal{Z}$
Given an irreducible polynomial $f(X)\in\mathbb{Z}[X]$, do you know an efficient algorithm to compute the number of conjugacy classes of matrices $A\in M_n(\mathbb{Z})$ with characteristic polynomial $...
- 389
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Weil pairing and Tate module for $p$-torsion in characteristic $p$
Let $A/\mathbf{F}_{p^n}$ be an Abelian variety with $\bar{A} = A \times_{\mathbf{F}_{p^n}} \bar{\mathbf{F}}_{p^n}$.
If $\ell \neq p$ is prime, there is a natural isomorphism $$\mathrm{H}^2_{\mathrm{...
user19475
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Uniqueness of finite flat models over bases of low ramification via Breuil-Kisin modules
Let $R$ be a complete DVR of mixed characteristic $(0, p)$, let $K$ be its fraction field, and assume that the absolute ramification index $e$ of $R$ satisfies $e < p - 1$ and that the residue ...
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Links between tight closure and deformation theory
I am looking for links between tight closure and deformation theory. As a sample question:
Question 1. Are there geometric interpretations in terms of deformation theory of
Frobenius rationality?
...
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Exponential diophantine equation system
I noticed a strange relation months ago :
$\begin{cases}3^5+10^2=7^3\\3+7=10\\2+3=5\end{cases}$
For the sake of math, I searched for positive integer non trivial (i.e. not containing any 0) ...
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votes
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502
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Lifting line bundles
Let $X$ be a smooth proper geometrically integral scheme over $\overline{\mathbb F_p}$. Assume $X$ is the specialization of a smooth proper scheme over $\mathbb Z_p^{nr}$. Let $L$ be an ample line ...
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3
votes
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About normalizers of infinite cyclic subgroups of Hilbert modular group
Consider $k$ a totally real finite extension of degree $n$ of $\mathbb{Q}$, i.e., all embeddings of $k$ in $\mathbb{C}$ have their image contained in the field of reals. Denote by $\mathcal{O}_k$ the ...
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independence of $\ell$ for $p$-adic cohomology of varieties over finite fields
Let $X/k$ be a smooth projective geometrically integral variety ($X = A$ an Abelian variety suffices) over $k = \mathbf{F}_q$ with absolute Galois group $\Gamma$, $\bar{X} = X \times_k \bar{k}$, $q = ...
user19475
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On nearby cycle sheaves and a 2-fibered product of topoi
In SGA7 Exposé XIII, Deligne introduces an algebraic theory of nearby cycle sheaves and vanishing cycle sheaves on schemes over the $S$, where $S$ is the spectrum of a Henselian discrete valuation ...
- 181
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Algebraization of Brauer classes in a paper of Lieblich
I am reading a paper of Lieblich on the unirationality of K3 surfaces and am trying to understand the result of Proposition 4.1:
Proposition 4.1: Let $k$ be an algebraically closed field of ...
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