# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

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### Do many homogeneous polynomials help in faster integer root extraction?

Given $n$ homogeneous algebraically independent total degree $2$ polynomials with no $x_1^2,\dots,x_n^2$ variable in $\mathbb Z[x_1,\dots,x_n]$ with promise that it has non-zero integer roots with ...

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### Identifying elements in the kernel of an explicit endomorphism of a Jacobian variety

I hope this question fits here.
Let $H/k$ be a genus $2$ curve and $J$ its Jacobian variety. Since $J(k)\cong \text{Pic}^0(H)(k)$ we have that its generic point looks like $[(x_1,y_1)+(x_2,y_2)-2\...

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### Unique way to topologise finite algebra over Huber ring

Let me start with the following Lemma.
$\textbf{Lemma}$ Let $A$ be a Tate ring, and let $f\colon A\to B$ be a finite $A$-algebra. Then there is a unique way to topologise $B$ turning it into a ...

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### Current status of uniform boundness of rational points on higher genus curves

We know Lang conjecture can imply uniform boundless of rational points on higher genus (smooth projective) curves over a fixed number field by works of Mazur and others.
How is the conjecture of ...

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84 views

### Maximum number of bounded primitive integer points in a zero-dimensional system

Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...

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### What is the probability of 'yes' to this likely $coNP$ problem?

Pick a set of primitive (gcd of coordinates is $1$) integer points $\mathcal T$ in $\mathbb Z^n$.
Denote the set of $n$ many algebraically independent homogeneous system of polynomials (thus zero-...

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60 views

### Heights for rational points via Neron models

I only just started reading about heights in arithmetic geometry, so forgive me if this question is naive.
Suppose $E$ is an elliptic curve over a number field $K$ with ring of integers $R$ and let $\...

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### Connection Between Knot Theory and Number Theory

Is there any connection between knot theory and number theory in any aspects?
Does anybody know any book that is about knot theory and number theory?

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**1**answer

230 views

### Weil restriction of a projective variety to a finite extension over $\mathbb{Q}$

My question maybe a bit stupid, but I can't quite grasp what the Weil restriction actually does... In particular:
Given a projective variety $V$ defined over $L$ algebraically closed, of ...

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218 views

### Extending section of étale morphism of adic spaces

This question is related to Lifting points via étale morphism of adic spaces.
Fix a complete non-archimedean field $k$. Let $(A,A^+)$ be a complete strongly noetherian Huber pair over $(k,k^\...

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179 views

### On a refinement of Mordell's conjecture for curves

Let $C$ be an algebraic curve of genus $g \geq 2$, defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. It is then defined over a finite extension $K$ of $\mathbb{Q}$. We assume that $C(K) \ne \...

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### Down to earth, intuition behind a Anabelian group [closed]

An anabelian group is a group that is “far from being an abelian group” in a precise sense: It is a non-trivial group for which every finite index subgroup has trivial center.
I would like to know ...

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211 views

### The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve

Working over the complex numbers, consider a function $F\left(x,y\right)$ and a curve $C$ defined by $F\left(x,y\right)=0$.
I know that to construct the Jacobian variety associated to $C$, one ...

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69 views

### Igusa curve at infinite level

In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent....

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243 views

### Smooth proper variety over a number field with prescribed bad reductions

Given a number field $K$, and a finite set $S$ containing finite places of $K$. When can we find a smooth proper geomerically connected variety $X$ over $K$ such that $X$ has good reduction outside $S$...