Questions tagged [arithmetic-geometry]

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

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Polynomial values are powers of two

The initial question comes from Komal in 1999. Namely it asks to show that for infinitely many $n$ there is a polynomial $f\in\mathbb{Q}[X]$ of degree $n$ such that $f(0),f(1),\dotsc,f(n+1)$ are ...
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1 vote
0 answers
92 views

Singularities of arithmetic surface

I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves. The setting is the following: We have a DVR with uniformiser $t$, characteristic of ...
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3 votes
0 answers
70 views

Rationality of plane curves with a certain property

Let $C\subset\mathbb{C}^2$ be an irreducible algebraic curve defined over a number field $F.$ Suppose that for any $(z, w)\in C, z\in \mathbb{\overline{Q}}, w\in\mathbb{\overline{Q}},$ either $z\in F(...
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  • 31
1 vote
0 answers
80 views

Image of the Kummer map for abelian varieties over $p$-adic local fields

The following statement might be well-known to the community: let $K$ be a finite extension of $\mathbb{Q}_p$ for some prime $p$. Let $A$ be an abelian variety over $K$. Then the image of the Kummer ...
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  • 1,180
3 votes
1 answer
407 views

If we have a nice formula for number of points on a curve over finite fields, can we get some geometric information of the curve from the formula?

Let $p$ be a prime number and let $q = p^2$. Let $C$ be a separated scheme of finite type over $\mathbb F_q$ of dimension $1$. If we know that for every $\alpha \in \mathbb Z_{>0}$, "the ...
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4 votes
0 answers
76 views

Newton stratification in the Shimura variety for $\mathrm{GU}(1,n-1)$ over a ramified prime

Consider the PEL Shimura variety $\mathrm{Sh}_{K}$ for $\mathrm{GU}(1,n-1)$, where $K\subset G(\mathbb A_f)$ is an open compact subgroup which is small enough, and $G$ is the group of unitary ...
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  • 395
2 votes
1 answer
222 views

Rational points on a special class of surfaces

Consider a smooth surface of the following form $$ S = \{f(x,y,t) = p_0(t)x^2+p_1(t)xy+p_2(t)x+p_3(t)y^2+p_4(t)y+p_5(t) = 0\}\subset\mathbb{A}^3 $$ over $\mathbb{Q}$, and set $$ U_S = \{t' \in \mathbb{...
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  • 71
7 votes
1 answer
405 views

Weil height vs Moriwaki height

Let $X$ be a projective veriety over a number field. After fixing an embedding into $\mathbb P^n$ (i.e. a very ample line bundle $L$), one can define the Weil height $\hat h_{L}$ by restriction of the ...
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  • 1,217
4 votes
0 answers
176 views

Torsionness of the kernel of the pullback map of Picard groups of a normalization map

Let $X$ be a (irreducible) projective variety over a number field $k$, $\pi: \tilde X \to X$ be its normalization, and $\pi^{*}: \mathrm{Pic}(X) \to \mathrm{Pic}(\tilde X)$ be the corresponding map of ...
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-1 votes
0 answers
93 views

Is there an superpolynomial integral point degree $2$ family satisfying Coppersmith's bounds?

Is there an irreducible degree $2$ bivariate curve (so of genus $0$) which satisfies Coppersmith's bounds but has superpolynomial number of integral points satisfying the bounds (allowed by Falting's ...
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8 votes
1 answer
375 views

Why is the category of motives generated by varieties?

I'm reading Ayoub's paper Motifs des varietes analytiques rigides, but I'm not quite familiar with motives. In this paper, he defines the category of motives to be $\mathbf{RigDM}^{\rm eff}_{\rm Nis}(...
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3 votes
1 answer
155 views

Stabilizers in abelian varieties are also abelian? reference request

Let $K$ be a field of characteristic $0$ (number fields is a sufficient generality), $A/K$ an abelian variety, and $X\subseteq A$ a closed reduced subscheme. I am looking for a reference for the ...
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3 votes
0 answers
203 views

Grothendieck trace formula for arbitrary morphisms

The Grothendieck trace formula can be viewed as a generalization of the Lefschetz trace formula in étale cohomology from constant sheaves to constructible $l$-adic sheaves, but restricting to the ...
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4 votes
1 answer
240 views

Rational points of bounded height on a variety

I would like to ask for some clarification on the following argument which I can not quite understand. There is a variety $X$ of dimension $n$ over a number field with a degree two map $f:X\...
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  • 351
5 votes
1 answer
328 views

Does p-adic etale cohomology know the variety has ordinary reduction or not?

For a smooth proper variety $X$ over discrete valuation ring $\mathcal{O}$ of mixed characteristic $(0,p)$, let $X_K$ be the generic fibre over a generic point $\textbf{Spec} K$ and let $X_k$ be the ...
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  • 159
5 votes
1 answer
243 views

About closed points in symmetric product schemes over a finite field

Let $k=\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation): Let $N$ be a positive ...
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  • 1,614
3 votes
1 answer
131 views

On the exactness of some completed tensor products

Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper https://arxiv.org/abs/...
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4 votes
1 answer
201 views

Finding the $K=\mathbb{Q}(\sqrt{6})$-rational points on the twist of $X_{0}(26)$

Let $K=\mathbb{Q}(\sqrt{6})$. I am looking to determine all $K$-rational points on the curve $$C: y^{2}=3x^6-24x^5+24x^4-54x^3+24x^2-24x+3.$$ More precisely, $C$ is a twist of the modular curve $X_{0}(...
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  • 43
4 votes
0 answers
215 views

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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  • 2,041
2 votes
0 answers
141 views

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
215 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 ...
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  • 41
5 votes
1 answer
163 views

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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  • 7,982
2 votes
1 answer
384 views

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
215 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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1 vote
0 answers
120 views

On closed subsets in spaces of adèlic points

Consider as in Adèlic points and algebraic closure $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$. ...
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6 votes
1 answer
326 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
276 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
266 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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4 votes
2 answers
437 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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4 votes
1 answer
241 views

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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  • 387
3 votes
1 answer
244 views

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
0 answers
455 views

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 ...
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2 votes
0 answers
67 views

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 ...
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4 votes
1 answer
237 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 = \{...
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  • 387
3 votes
0 answers
154 views

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 ...
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4 votes
0 answers
135 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 ...
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  • 1,775
8 votes
0 answers
201 views

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 ...
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7 votes
0 answers
169 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-...
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4 votes
1 answer
149 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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  • 333
0 votes
1 answer
142 views

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 $ ...
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  • 693
12 votes
1 answer
1k views

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 ...
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5 votes
1 answer
186 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 $Q$?
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  • 387
1 vote
0 answers
96 views

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 ...
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  • 407
4 votes
1 answer
204 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 \...
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1 vote
0 answers
72 views

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 ...
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5 votes
2 answers
494 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
210 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 ...
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  • 233
7 votes
0 answers
181 views

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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  • 395
3 votes
0 answers
118 views

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 ...
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  • 6,782
5 votes
2 answers
621 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 ...
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