Recovering affine curves over finite fields from $L$-functions

Jacobian conjecture as a problem on integral points on affine curves

It is shown that the $n$-dimensional Jacobian conjecture over algebraic number fields may be considered as an existence problem of integral points on affine curves. More specially, if the Jacobian conjecture over $\mathbb{C}$ is false, then for some $n\gg 1$ there exists a counterexample $F\in \mathbb{Z}[X]^n$ of the form $F_i(X)=X_i+ (a_{i1}X_1+\dots+a_{in}X_n)^{d_i}$, $a_{ij}\in \Z$, $d_i=2;3 $, $i,j=\overline{1,n},$ such that the affine curve $F_1(X)=F_2(X)=\dots=F_n(X)$ has no non-zero integer points.

Integral points of bounded degree on affine curves

We generalize Siegel's theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree d or less over some number field. Generalizing Picard's theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree d (or less) analytic cover of $\mathbb{C}$.

The Green-Tao theorem for affine curves over F_q

Green and Tao famously proved in a 2008 paper that there are arithmetic progressions of prime numbers of arbitrary lengths. Soon after, analogous statements were proved by Tao for the ring of Gaussian integers and by Lê for the polynomial rings over finite fields. In 2020 this was extented to orders of arbitrary number fields by Kai-Mimura-Munemasa-Seki-Yoshino. We settle the case of the coordinate rings of affine curves over finite fields. The main contribution of this paper is subtle choice of a polynomial subring of the given ring which plays the role of $\mathbb Z$ in the number field case…

从椭圆曲线的\(\ell\)-挠中恢复柯达依类型

Recovering Kodaira types from $\ell$-torsion on elliptic curves

研究如何通过赋予椭圆曲线\(E[\ell]\)距离函数，弥补惯性作用于\(E[\ell]\)及\(T_{\ell}(E)\)不足以区分潜在好约化和乘法约化、确定柯达依类型的问题，证明\(E[\ell]\)能确定椭圆曲线的柯达依类型。

仿射双曲曲线的线性与二次Chabauty方法

Linear and quadratic Chabauty for affine hyperbolic curves

该研究给出仿射双曲曲线线性与二次精细化Chabauty-Kim轨迹有限性的充分条件，通过构造基本群深度≤2商群实现，并结合Betts权滤波机制给出S-整点数量的无条件显式上界。

On abstract homomorphisms of Chevalley groups over the coordinate rings of affine curves

The goal of this paper is to establish a general rigidity statement for abstract representations of elementary subgroups of Chevalley groups of rank at least 2 over a class of commutative rings that includes the localizations of 1-generated rings and the coordinate rings of affine curves. This is achieved by developing the approach introduced in our previous work, and in particular by verifying condition (Z) over the class of rings at hand. Our main result implies, for example, that any finite-dimensional representation of SL_n(Z[X]) (for n at least 3) over an algebraically closed field of cha…

恢复超椭圆曲线约化类型的不变量

Invariants recovering the reduction type of a hyperelliptic curve

本文推广Tate算法至亏格g≥2的超椭圆曲线，定义绝对不变量以恢复潜在稳定模型及半稳定情形下的特殊纤维对偶图，并证明非半稳定时不存在此类不变量。

仿射查巴乌I

Affine Chabauty I

本文提出了一种基于广义雅可比簇和算术交理论的仿射曲线S-积分点求解方法，并给出了满足特定秩-亏格不等式的点数上界。

仿射曲线上整二次型的强逼近与 Hasse 原理

Strong Approximation and Hasse Principle for Integral Quadratic Forms over Affine Curves

本文利用 genus 理论研究仿射曲线坐标环上整二次型的表示问题，建立 spin 群强逼近与 Hasse 原理的联系，并给出强逼近失效的例子。