Questions tagged [arithmetic-geometry]

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

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4
votes
1answer
212 views

Torsion points on $E/\mathbb{Q}$ with large coordinates

Let $E/\mathbb{Q}$ be an elliptic curve with finitely many rational points. What are some examples where at least one rational point has large coordinates (compared to the height of $E$)?
2
votes
0answers
138 views

Schemes with common zeta function

If $S_\zeta$ is the set of all separated schemes of finite type over $\mathbb{Z}$ that have the same arithmetic zeta function $\zeta$, what more can we say about $S_\zeta$ assuming it is non-empty?
1
vote
1answer
107 views

Common prime of the finite number of order of imaginary quadratic field

This is from Silverman's 'the arithmetic of elliptic curves', exercise 5.5. Let $K$ be an imaginary quadratic field, and let $R_1...R_n$ be orders in $K$. I would like to prove that there are more ...
2
votes
0answers
175 views

Is there a smooth proper family whose fibers are not Mazur-Ogus?

Set $K$ to be a number field, denote by $\mathcal{O}_K$ the integer ring of $K$. My question is the following: Is there a smooth proper family $X \to \mathcal{O}_K$ whose fibers are not Mazur-Ogus?
12
votes
0answers
640 views

Seek for a algebro-geometric proof: the group homomorphism $\mathrm{SL}(2,\mathbb{Z}) \rightarrow \mathrm{SL}(2,\mathbb{Z}/N\mathbb{Z})$ is surjective

It is a well-known fact that the group homomorphism $\mathrm{SL}(2,\mathbb{Z}) \rightarrow \mathrm{SL}(2,\mathbb{Z}/N\mathbb{Z})$ is surjective. What I want is a proof by method of algebraic geometry. ...
8
votes
0answers
156 views

Zariski dense $K$-points for any non-trivial finite Galois extension $K/\mathbb{Q}$

Let $V$ be a smooth $\mathbb{Q}$-variety. Assume that for any non-trivial finite Galois extension $K/\mathbb{Q}$ the $K$-points are Zariski dense in $V$. Must $V$ have a $\mathbb{Q}$-point then?
4
votes
1answer
211 views

An explicit equation of the canonical morphism $X_1(N) \to X_0(N)$

I know there are some research about explicit equations for affine models in $\mathbb{A}^2$ of many modular curves over $\mathbb{Q}$, for example of $X_i(N), X(N)$ (where $i = 0, 1, 2$) for small $N$. ...
0
votes
0answers
56 views

An invariant subspace under $G_Q$ action , and BSD-rank

Let $E/Q$ be an elliptic curve. $E(\bar{Q})$ is a complicated abelian group, which equals to all closed points of $\bar{E}$, and also, a $G_Q$-Galois module. Its torsion part, $E(\bar{Q})_{tor}$ is a ...
4
votes
4answers
456 views

The integral closure $\overline{\mathbb{Z}}$ and the group $\overline{\mathbb{Z}}^{\times}$

Let $\mathbb{Q}$ be the field of rational numbers, and let $\overline{\mathbb{Q}}$ be its algebraic closure. Assume $\overline{\mathbb{Z}}$ is the integral closure of $\mathbb{Z}$ in $\overline{\...
5
votes
0answers
87 views

Are the nonnegative rationals diophantine with only two quantifiers?

Definition: A subset $D\subseteq \mathbb{Q}$ is diophantine if it is the projection of the zero set of a polynomial, i.e. there exists a polynomial $f\in\mathbb{Q}[X,Y_1,\dots,Y_n]$ for some $n$ such ...
5
votes
0answers
169 views

Explicit computations of the fundamental groups of perfectoid spaces

If $X$ is a perfectoid space then it has the same étale site as its tilt $X^\flat$. This means that the fundamental groups (suitably defined) of $X$ and $X^\flat$ are isomorphic. Can you give ...
2
votes
0answers
86 views

Cartier operator and logarithmic differentials

Let $k$ be an algebraically closed field of characteristic $p$, let $C$ be a curve over $k$ and let $\omega$ be a meromorphic differential form on $C$. If $\omega$ gets mapped to itself by the Cartier ...
3
votes
0answers
155 views

A question on the Bombieri-Lang conjecture

Let $X$ be a variety of general type, defined over a number field $K$. Then the Bombieri-Lang conjecture asserts that the set of rational points $X(K)$ (or $X(L)$ for any finite extension $L/K$) is ...
2
votes
2answers
157 views

Galois stable elements of the Picard group of a curve and the rational divisors

Let $C$ be a (smooth,proper) curve over a field $k$. Let $\operatorname{Div}_C(k)$ be the free abelian group generated by the closed points of $C/k$ and $k(C)^\times$ be the group of rational ...
3
votes
1answer
134 views

Generalization of torsion points on Jacobian of genus 2 over finite fields (with respect to the theta divisor)

Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \...
0
votes
0answers
113 views

Rational points on towers of surfaces

Take infinitely many 2-variable polynomials $p_k(X,Y)\in \mathbb{Q}[X]$ ($k\in \mathbb{N}$) and let $S_n$ be the surface given by $p_1(X,Y)=Z_1^2,\dots, p_n(X,Y)=Z_n^2$ Assume that no $p_k$ equals the ...
2
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0answers
176 views

Relation between stacky curves and “M-curves”

A tame stacky curve over a field $k$ is a geometrically connected proper smooth DM stack of dimension 1 which has a dense open substack which is a scheme, and whose automorphism group of each ...
1
vote
1answer
139 views

Complexity of a Diophantine equation having $\leq1$ solutions

We are provided a single Diophantine equation $$f(x_1,\dots,x_n)=0$$ having degree $\geq2$ and having the promise it has $\leq1$ solutions in the set $\{0,\dots,m-1\}^n$ and $t$ is the number of terms ...
8
votes
1answer
228 views

What's the average order of the reduction of a section of an elliptic curve

Suppose $E$ is an elliptic curve over $\mathbb Q$ and $x \in E(\mathbb Q)$ is not torsion. We can reduce $x \pmod p$ for a prime $p$ of good reduction and it will have some order $n_p$ in the group $E(...
2
votes
0answers
83 views

A computation of the rank of the Jacobian of a hyperelliptic curve over a number field using MAGMA

In this paper, the authors says that, in order to show the rank of a Jacobian over $\mathbb{Q}$ is 0, they use the L function. In the section 3.3, the authors compute the rank of the Jacobian of $X_1(...
3
votes
0answers
268 views

Why the curve $x^2+y^2+y+1=0$ has only one point over $\mathbb{F}_{3^7}$?

According to both sagemath and Magma the curve $x^2+y^2+y+1=0$ has only one point over $\mathbb{F}_{3^7}$. The projective closure has only one point too. Q1 What hypothesis are missing to not violate ...
4
votes
0answers
146 views

Ramification behavior of field given by adjoining $p$-torsion point on formal group of abelian variety

Setup. Let $p > 2$ be a prime, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and fix an algebraic closure $\overline{K}$ of $K$. Let $A/K$ be an abelian variety ...
3
votes
1answer
144 views

Integral models and adelic points

Let $k$ be a number field and denote by $\Omega _k$ the set of places of $k$, by $\Omega _\infty$ the set of archimedean places of $k$, and by $S$ a nonempty finite subset of $\Omega _k$ such that $\...
1
vote
1answer
114 views

The orders of $\mathbb{F}_{p^n}$- rational points of a fixed abelian variety and MAGMA computation

Let $A$ be an abelian variety over $\mathbb{F}_p$. Then of course for every natural number $i$, we have that $\# A(\mathbb{F}_{p^i})$ divides $\# A(\mathbb{F}_{p^{i+1}})$. But MAGMA says this is false:...
3
votes
1answer
192 views

Frobenius actions on de Rham cohomology of ordinary elliptic curves

In appendix 2 of Katz's "p-Adic properties of modular schemes and modular forms", he describes a certain "Frobenius" endomorphism on the de Rham cohomology of ordinary elliptic ...
3
votes
1answer
203 views

Mordell–Weil rank of some elliptic $K3$ surface

Consider a finite field $\mathbb{F}_q$ such that $q \equiv 1 \pmod3$ (i.e., $\omega \mathrel{:=} \sqrt[3]{1} \in \mathbb{F}_q$ for $\omega \neq 1$) and an element $b \in (\mathbb{F}_q^*)^2 \setminus (\...
8
votes
2answers
398 views

Chevalley-Warning-Ax for double covers

Let $f(x_1,\ldots,x_n)$ be a polynomial of degree $d$ with coefficients in the finite field $\mathbb F_q$ and let $V(f)\subseteq\mathbb F_q^n$ be its set of zeroes. Assume $d<n$. Then Chevalley ...
9
votes
1answer
465 views

Brauer-Manin obstruction on an open subset of an elliptic curve

First a disclaimer. This is an old question that I considered years ago and that I recently remembered. Since I am no longer in active research it may be considered as 'idle curiosity', although I ...
2
votes
0answers
145 views

Generalizations of Artin–Verdier duality?

Constructible étale abelian sheafs on $Spec\ O_\mathbb K$, for number fields $\mathbb K$, satisfy Artin-Verdier duality. Are there known any algebraic schemes or algebraic stacks, other than $Spec\ O_\...
4
votes
0answers
247 views

Formal integration (?) in Chabauty’s method

In Mccallum, Poonen’s paper “The method of Chabauty and Coleman”, the authors define, for the Jacobian $J$ of a geometrically connected smooth proper curve over the rational field and for $\omega \in ...
4
votes
1answer
409 views

The integral cohomology of the de Rham complex

Consider the usual de Rham CDGA $(\Omega^* Sym^*(V),d)$ for a free $\mathbb{Z}_{(p)}$-module $V$. What is known about its cohomology? It is easy to compute ranks of primary summands in $H^*(\Omega^* ...
1
vote
0answers
176 views

Shimura varieties which are not of abelian type but has a good modular description

Deligne's idea was that Shimura varieties should be understood as moduli space of motives(with extra structures). lot's of Shimura varieties of abelian type can be understood as moduli space of ...
9
votes
1answer
346 views

What is the indecomposable decomposition of holomorphic differentials of an Artin-Schreier curve C as a Z/p-representation?

I am attempting to decompose the holomorphic differentials of an Artin-Schreier-Witt curve as a $\mathbb{Z}/p^n$-representation. This is done in Theorem 1 of Madan-Valentini Automorphisms and ...
3
votes
0answers
127 views

2-fold linear cover of reductive group of type A

Let $F$ be a nonarchimedean local field of characteristic zero. Let $G=\operatorname{Res}_{E/F}\operatorname{GL}_n$ or $\operatorname{Res}_{E/F}\operatorname{U}_n$, where $\operatorname{U}_n$ is any ...
1
vote
0answers
71 views

A subgroup of the $n$-Selmer group

Let $p$ be an odd prime and for the purpose of this question let $n$ be an integer which is a power of $p$. Let $E$ be an elliptic curve over a number field $F$. The $n$-Selmer group, denoted by $S_n(...
41
votes
3answers
3k views

“Cute” applications of the étale fundamental group

When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the ...
3
votes
0answers
272 views

Uniformization of algebraic curves

Given an irreducible smooth complex-projective curve $X$, I will say that a subgroup $\Gamma< SL(2, {\mathbb R})$ weakly uniformizes $X$ if [corrected] there exists a nonconstant holomorphic map ...
5
votes
2answers
189 views

Formal models of rigid discs of any radii

sorry if this is a too vague. For $K$ some non-archimedean discretely valued field the rigid disc of radius 1 $\mathrm{Sp} \, K\langle T \rangle$ has a formal model $\mathrm{Spf} \, K^{\circ} \{T\}$. ...
24
votes
1answer
718 views

Relation between Schanuel's theorem and class number equation

(Crossposted on math stack exchange: https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation) It was recently brought to my attention that there ...
1
vote
0answers
74 views

Equidimensional Morphism

I am reading the paper "Relative Cycles and Chow Sheaves" due to Suslin and Voevodsky. Here we have the following definition: Definition 2.1.2. A morphism of schemes $p:X\rightarrow S$ is ...
2
votes
0answers
99 views

Moduli interpretation and Ogg's notation for the cusps on modular curves

In Ogg's paper "rational points on certain elliptic modular curves", the author says, using Ogg's notation for cusps, that for fiexed $d$, if $(y, N) = d$, then for any $x$ satisfying $(x, y,...
2
votes
1answer
176 views

Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases?

Help me please. Let $k$ be an algebraically closed field (I am mainly interested in $k = \overline{\mathbb{Q}}, \overline{\mathbb{F}_q}$). Consider a plane curve $C \subset \mathbb{A}^2$ of degree $d$ ...
8
votes
0answers
390 views

Elkies' theorem on supersingular primes and inertness

Suppose $E_{/\mathbb{Q}}$ is an elliptic curve over $\mathbb{Q}$ without CM. By Elkies' theorem, there exist infinitely many primes $p$ for which $E$ has supersingular reduction at $p$. Question. Is ...
2
votes
1answer
124 views

complement of “good reduction” points in p-adic shimura varieties

assume that $X$ is Siegel Shimura variety defined over $\mathbb{Z}_p$, you can take its p-adic formal completion $\mathfrak{X}$,and than take it's adic generic fiber $\mathcal{X}$ and get an adic ...
4
votes
0answers
218 views

Can arithmetic geometry accelerate the search for rational points in high dimensions?

There are several ideas in arithmetic geometry that can help in proving the absence of rational points as well exhibiting rational points on algebraic curves. I am aware there are some obstructions (e....
3
votes
1answer
145 views

Density of quadratic points on a hyperelliptic curve

We fix a binary form $F \in \mathbb{Z}[x,y]$ with non-zero discriminant and degree $d = 2g+2$, and consider the hyperelliptic curve $$C_F: \displaystyle z^2 = F(x,y).$$ We say that a point $(x,y,z)$ ...
2
votes
0answers
148 views

Finding rational points via birational map

Let $C$ be an affine curve given by $p_C(x,y)=0$ where $$ p_C=2x^3y + 2xy^3 +x^3 + y^3 + 5x^2y + 5xy^2 + 2x^2 + 2y^2 + 2x^2y^2 + 2xy $$ and let $\overline{C}$ denote the projective closure of $C$. For ...
7
votes
0answers
311 views

Status of the conjectured vanishing of Bloch-Kato H^2

There is a folklore conjecture that $\operatorname{Ext}^2$ vanishes in the category of geometric $p$-adic Galois representations (i.e. representations that are unramified almost everywhere and de Rham ...
2
votes
0answers
130 views

Abelian variety corresponding to a vector space

I would like to know what the following statement means: "Let $B_t$ be the Abelian subvariety in $J_t$ corresponding to the $\mathbb{Q}$-vector subspace $H^1(C_t,\mathbb{Q})_{van}$ in the space $...
-5
votes
1answer
705 views

Rational points on hypersurfaces [closed]

Let $\mathbb{Q} \subseteq K \subseteq \mathbb{R}$ be fields and let $I:=(f_1,..,f_l)$ where $f_i\in A:=K[x_1,..,x_n]$ let $ X:=\mathrm{Spec}(A/I)$. Let $F:=f_1^2+\cdots + f_l^2$ and let $Y:=\mathrm{...

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