Questions tagged [arithmetic-geometry]

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

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Number of homomorphisms from a group to $\mathrm{GL}_n(\mathbb{F}_q)$

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms ...
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Path spaces vs arc spaces

Let $X$ be a smooth projective variety over $\mathbf{C}$ and denote by $\mathcal{L}_m(X)$ the $m$-th jet space, a smooth $\mathbf{C}$-scheme representing the functor on $\mathbf{C}$-algebras $$A\...
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3 votes
1 answer
301 views

Rank of elliptic curves, parity, finiteness of Sha

$\newcommand{\Sha}{Ш}\newcommand{\alg}{\mathrm{alg}}\DeclareMathOperator\Sel{Sel}\DeclareMathOperator\rank{rank}$ Consider the elliptic curves $E$ of $j$-invariant zero that neither them nor their ...
EAg's user avatar
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1 answer
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Computation of the torsion of K-groups related to elliptic curves

Let $E$ be an elliptic curve over $\mathbb Q$. Let $F$ be the rational function field of $E$. The $K_2$ group of $F$ may be described by elements in $F^\times ⊗_\mathbb{Z} F^\times$ quotiented by the ...
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1 answer
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Galois invariants and tensor products

Consider a number field $K$ and a finite Galois field extension $L/K$. Let $E$ be an elliptic curve over $K$ and consider the abelian group $$E(L)\otimes L^{\times}.$$ Every element $g$ in $\text{Gal}(...
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2 votes
2 answers
369 views

Transition maps in trivial direct limit

If $\{X_i, i\in I\}$ is a directed system of abelian groups such that we have $$\varinjlim_{i\in I}X_i = 0$$ is it true that for every $i$ and large enough $j\ge i$ the transition map $f_{i,j} : X_i\...
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6 votes
1 answer
346 views

Adèlic points and algebraic closure

Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$. Let $F/K$ vary over all finite Galois number ...
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4 votes
1 answer
387 views

Tate-Shafarevich groups under finite Galois field extensions

Suppose $L/F$ is a finite Galois extension of number fields. Let $E$ be an elliptic curve over $F$ and $E_L$ its base change to $L$. Do we have that $\text{Sha}(E/F)$ is finite if and only if $\text{...
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3 votes
1 answer
402 views

Galois cohomology of abelian varieties

Suppose $A$ is an abelian variety over a number field $K$ and call $M$ the maximal torsion free quotient of $A(\overline{K})$ equipped with its Galois action. For the first Galois cohomology of $M$, ...
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2 answers
480 views

Smoothness of fibers over finite fields

Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties over a finite field of characteristic different from $2$. Is there any result on the existence of a point $y\in Y$ such that $X_y = ...
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Number of points of a quadric hypersurface over a finite field

Let $k = \mathbb{F}_q$ be a finite field with $q$ elements and $Q\subset\mathbb{P}^n_k$ a quadric hypersurface defined over $k$. By the Chevalley-Warning theorem if $n\geq 2$ then $Q$ has a point. Is ...
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Smooth surfaces in positive characteristic

Let $K = \mathbb{F}_p$ be a field of positive characteristic $p > 0$. Consider a surface in $\mathbb{A}^3_K$ of the following form $$ S = \{f_1(x_0)y_0^2+f_2(x_0)y_0y_1+f_3(x_0)y_0+f_4(x_0)y_1^2+...
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2 votes
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Confusion regarding Proposition 1.1 in Wiles's Fermat paper

This is from p. 459 of Wiles's Fermat paper. Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of ...
The Thin Whistler's user avatar
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Are tamely ramified representations $\widehat{\mathbb{Q}_p^\text{tr}}$-admissible?

Let $K$ be a finite field extension of $\mathbb{Q}_p$. Let $G_K$ denote the absolute Galois group of $K$, $I_K$ the inertia subgroup and $I_K^{(p)}$ the $p$-Sylow subgroup of $I_K$, i.e. the wild ...
Konstantin's user avatar
4 votes
1 answer
290 views

Del Pezzo surfaces of degree four and complete intersections of two quadrics

Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$. Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{...
Puzzled's user avatar
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3 votes
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How much results in Calabi-Yau manifolds and mirror symmetry depends on the existence of a ricci-flat metric?

An important result of CY manifold is the CY theorem, it talks about the existence of a ricci-flat metric. However, this theorem and its proof are highly analytic. There are many results about ...
Reflecting_Ordinal's user avatar
4 votes
0 answers
173 views

Rational points on ramified coverings of abelian varieties

Let $K$ be a number field $A$ an abelian variety over $K$ and $f: X \to A$ a possibly ramified covering of degree $d$ with $X$ a proper variety. My question is: Suppose that $f(X(K)) \neq A(K)$, can ...
Maarten Derickx's user avatar
8 votes
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Simultaneous rank jumping of elliptic curves over number fields

Here's something fun that I've been wondering about for a while out of curiosity. It has to do with the rank $\mathrm{rk}(E(K))$ of an elliptic curve $E$ over a number field $K$, and especially how it ...
Ariyan Javanpeykar's user avatar
7 votes
0 answers
213 views

Counting elliptic curves over finite fields with a prescribed number of points

Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$, and let $\mathcal{O}$ be an order in $K$ of discriminant $D$ and class number $h(\mathcal{O})$. Then the Hurwitz-...
Tristan Phillips's user avatar
4 votes
1 answer
222 views

Is Galois representation induced by semistable elliptic curve semistable?

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Aut{Aut}$Let $E$ be a semi stable elliptic curve. Let $\overline{\rho_\ell}: \Gal(\overline{\Bbb Q}/\Bbb Q)\to \Aut (E[l]) $ be mod $\ell$ ...
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1 answer
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Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $

Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ ...
Sky's user avatar
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15 votes
1 answer
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Is Mazur's analogy between arithmetic and topology formal, in any sense?

I preface my question by admitting I know no algebraic geometry nor algebraic number theory. I do know some algebraic topology. I'm a student. Recently I learned about sheaf cohomology. Then a little ...
Matthew Niemiro's user avatar
5 votes
1 answer
497 views

Lines on quadric surfaces

Consider a smooth quadric surface $Q\subset\mathbb{P}^3$ over a field $k$. Are there natural hypotheses one can put on $k$ in order to ensure the existence of a line defined over $k$ on $Q$?
Puzzled's user avatar
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1 vote
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About Definition 2 in Roĭtman's Paper

Let $A(X)$ be the Chow group of $0$-cycles on a nonsingular irreducible projective variety $X$ over an uncountable algebraically closed field of characteristic zero. In Definition 2 of Roĭtman's paper ...
Roxana's user avatar
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1 answer
229 views

integral points on elliptic curves in terms of discriminant

I am curios where in the literature was the first time written the following conjecture. Say we have we have an elliptic curve $E$ given by the Weierstrass equation $y^2=x^3+AX+B$ with $A,B\in \...
Vlad Matei's user avatar
1 vote
0 answers
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Drawing a 3D object in a 3D environment, and converting to math [closed]

So I have been granted a free time and I want to work on a project but first I had to research. As we know, lines have infinite points, and with lines, we can create infinite shapes. I want to let ...
Dead_Light's user avatar
5 votes
2 answers
548 views

Birational geometry over finite fields

I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ...
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5 votes
0 answers
259 views

Torsion points of an elliptic curve over number fields. Another proof of Silverman AEC theorem

I am studying the following theorem from Silverman's AEC: I am wondering whether there exists another proof that doesn't make use of formal groups and is still valid for a number field $K$. Could you ...
cartesio's user avatar
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7 votes
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Are unramified simple Rapoport-Zink spaces smooth?

I have read on different occasions that unramified simple Rapoport-Zink spaces are formally smooth, eg. stated in Remark 4.13 of Rapoport and Viehman's survey article. These spaces are formal schemes $...
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3 votes
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Smooth proper varieties over the integers that are not toric

Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric? By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by ...
Asvin's user avatar
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5 votes
2 answers
689 views

Embedding torsors of elliptic curves into projective space

Suppose I have a genus 1 curve $C$ over a field $k$. If $C$ has a point, then we can embed it into the projective plane by a Weierstrass equation. Now let us suppose that $C$ does not have a point (so ...
Asvin's user avatar
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3 votes
1 answer
485 views

Explicit defining equations for del Pezzo surfaces

Are there known explicit parametrizations of smooth del Pezzo surfaces, say of degree $5$ or higher, as surfaces cut out by equations in some projective space? The closest I've been able to find is on ...
Nicolas Banks's user avatar
7 votes
1 answer
871 views

L-functions and Galois representations: What’s the explicit relation?

It was mentioned in a lecture on Faltings’s proof of Mordell Conjecture that there’s some kind of correspondence between Galois representation (of cohomology, or some complex of constructible sheaves?)...
Wilhelm's user avatar
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7 votes
2 answers
670 views

Reference request: the geometry of vanishing cycle

I’m currently studying basics on étale cohomology, by Fu’s and Milne’s book. The formalization of vanishing cycle and nearby cycle particularly interests me. I realized it may relate with reduction ...
Wilhelm's user avatar
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2 votes
0 answers
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Relative homology in Fargues-Scholze paper

if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they ...
ali's user avatar
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3 votes
0 answers
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Does the construction of arithmetic toroidal compactification of $A_{g}$ depend on semistable reduction theorem?

If there is a good theory of arithmetic toroidal compactification over $\mathbb{Z}_{p}$ of the Siegel modular variety with deep enough level structure, then it seems like semistable reduction theorem ...
GTA's user avatar
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4 votes
0 answers
151 views

Constructing motivic representations through extensions of $\mathrm{SL}(2, \mathbb{Z})$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the Esquisse, Grothendieck explains how to find the representations of $G_{\mathbb{Q}}$ on Tate modules of Jacobians through its action on $\...
user149000's user avatar
4 votes
0 answers
201 views

Grothendieck group of admissible $p$-adic representations

Let $K$ be a $p$-adic local field; $G = \mathop{\mathrm{Gal}}(\overline K | K)$; $\tau \in \{\text{HT}, \text{dR}, \text{crys}\}$, $B_\tau$ the corresponding period ring; $\mathop{\mathrm{Rep}}_{\...
Aoi Koshigaya's user avatar
7 votes
0 answers
300 views

Number of rational points over finite fields mod $q$ is birational invariant

I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) \equiv \#(Y(\mathbf F_q)) \pmod q$, and I heard that the proof ...
Aoi Koshigaya's user avatar
6 votes
2 answers
664 views

Could the Weil zeroes of curves be evenly distributed?

If $X$ is a smooth, geometrically connected, projective curve of genus $g$ over $\mathbb{F}_q$, then the zeta function of $X$ is of the form $P(s)/(1 - s)(1 - qs)$, where $P(s)$ is a polynomial of ...
LeechLattice's user avatar
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2 votes
1 answer
396 views

Why geometric generic point (in abstract algebraic geometry) replace general points in the unit disk?

In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states: On $\mathbb{C}$ Lefshietz local results are as follows. Let $X$ be a ...
Roxana's user avatar
  • 519
4 votes
0 answers
134 views

Explicit toroidal compactification of Hilbert modular varieties

Hirzebruch's construction of toroidal compactification of Hilbert modular surfaces is explicit, namely one can explicitly choose rational polyhedral cone decomposition in a sort of optimal way using ...
GTA's user avatar
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3 votes
0 answers
344 views

Meaning of "the" general fiber in the paper "La conjecture de Weil : I"

In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states: Let $X$ be a non singular analytic space and purely of dimension $n+1$....
Roxana's user avatar
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1 vote
0 answers
239 views

To justify the intuition about #$E(\Bbb Q_p)$=$∞$

Let $E$ be an elliptic curve on $\Bbb Q_p$. $E_0(\Bbb Q_p)$ is points of $E(\Bbb Q_p)$ reduced to nonsingular points. How to prove #$E(\Bbb Q_p)$=$∞$ directly ? According to Silverman's book 'the ...
Duality's user avatar
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14 votes
2 answers
511 views

Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?

Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
Jonathan Love's user avatar
6 votes
0 answers
436 views

Proof of Lemma 6.5 in Scholze's Perfectoid Spaces

In the proof of Lemma 6.5(approximation lemma) in Scholze's Perfectoid Spaces, I have the following three questions about $h = f - g^\sharp_c$ and $g^\sharp_{c'}$. (Maybe it's something you'll find ...
user400965's user avatar
4 votes
1 answer
230 views

Relation between rational Tate module and universal cover of a p-divisible group

We can associate two $\mathbb Q_p$ vector spaces to a $p$-divisible group, and I'm a little confused about the relation between these two groups. First of all, I think part of my problem is that when ...
ali's user avatar
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5 votes
1 answer
348 views

diagonal cubic hypersurfaces

At the end of https://encyclopediaofmath.org/index.php?title=Cubic_hypersurface#References it is stated that the diagonal cubic hypersurface $$ \sum_{i=0}^{2m+1} a_i x_i^3 = 0, m\ge 2 $$ (and ...
W Sao's user avatar
  • 509
1 vote
0 answers
170 views

Does Lemma 5.4 in Deligne's Ramanujan paper generalize to Shimura varieties of PEL type?

It is generally not known if a smooth variety over a perfect field embeds into a smooth proper variety. Lemma 5.4 in Formes modulaires et représentations $\ell$-adiques provides such an embedding for ...
soft-drinks's user avatar
6 votes
0 answers
430 views

Cohomology theories for algebraic varieties over number fields

There is a standard line which is repeated by anyone writing/talking about motives and cohomology of algebraic varieties over number fields: namely, there are many such cohomologies and then the ...
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