# Questions tagged [arithmetic-geometry]

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

1,760
questions

13
votes

2
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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 ...

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 ...

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(...

1
vote

0
answers

80
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### 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 ...

3
votes

1
answer

407
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### 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 ...

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 ...

2
votes

1
answer

222
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### 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{...

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 ...

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 ...

-1
votes

0
answers

93
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### 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 ...

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}(...

3
votes

1
answer

155
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### 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 ...

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 ...

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\...

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 ...

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 ...

3
votes

1
answer

131
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### 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/...

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}(...

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 ...

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\...

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 ...

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 ...

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}(...

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\...

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 $K$.
...

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 ...

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{...

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 $M$, ...

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 = ...

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 ...

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+...

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 ...

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 ...

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 = \{...

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 ...

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 ...

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 ...

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-...

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 $\ell$ ...

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 $ ...

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 ...

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 $Q$?

1
vote

0
answers

96
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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 ...

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 \...

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 ...

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 ...

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 ...

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 $...

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 ...

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 ...