# Questions tagged [arithmetic-geometry]

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

1,629 questions
Filter by
Sorted by
Tagged with
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 $E$)?
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?
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 ...
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?
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. ...
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?
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 $N$. ...
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 ...
456 views

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 ...
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 ...
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 ...
228 views

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 ...
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 ...
144 views

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 ...
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 ...
145 views

409 views

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 ...
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 ...
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 $\mathrm{Spf} \, K^{\circ} \{T\}$. ...
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 ...
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 ...
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 $(x, y,... 1answer 176 views ### Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases? Help me please. Let$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$... 0answers 390 views ### Elkies' theorem on supersingular primes and inertness Suppose$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$. Question. Is ... 1answer 124 views ### complement of “good reduction” points in p-adic shimura varieties assume that$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}$and get an adic ... 0answers 218 views ### Can arithmetic geometry accelerate the search for rational points in high dimensions? There are several ideas in arithmetic geometry that can help in proving the absence of rational points as well exhibiting rational points on algebraic curves. I am aware there are some obstructions (e.... 1answer 145 views ### Density of quadratic points on a hyperelliptic curve We fix a binary form$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$(x,y,z)$... 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 ... 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 ... 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$...
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{...