Bounds for Completely Decomposable Jacobians

Integral Points on Hyperelliptic Curves

We give a completely explicit upper bound for integral points on (standard) affine models of hyperelliptic curves, provided we know at least one rational point and a Mordell-Weil basis of the Jacobian. We also explain a powerful refinement of the Mordell--Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on a two genus 2 hyperelliptic curves with Mordell--Weil Jacobian ranks of 3 and 6.

具有足够自同构的曲线的有效 Mordell 定理

Effective Mordell for curves with enough automorphisms

本文基于算术曲面的 Arakelov 理论，为具有足够自同构的亏格至少为 2 的曲线证明了完全显式且有效的有理点 Néron--Tate 高上界，并实际计算了一条雅可比矩阵 Mordell--Weil 秩为 2 的亏格 2 曲线的有理点。

Non-isogenous elliptic curves and hyperelliptic jacobians II

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n \ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other splits completely over $K$. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\bar{K}$ then there is …

雅可比簇与相对类群的指数

Exponents of Jacobians and relative class groups

针对有限域上曲线覆盖，证明相对类群指数下界，改进Stichtenoth结果并首次处理真正相对情形。

Chabauty-Coleman computations on rank 1 Picard curves

We provably compute the full set of rational points on 1403 Picard curves defined over $\mathbb{Q}$ with Jacobians of Mordell-Weil rank $1$ using the Chabauty-Coleman method. To carry out this computation, we extend Magma code of Balakrishnan and Tuitman for Coleman integration. The new code computes $p$-adic (Coleman) integrals on curves to points defined over number fields where the prime $p$ splits completely and implements effective Chabauty for curves whose Jacobians have infinite order points that are not the image of a rational point under the Abel-Jacobi map. We discuss several interes…

Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields

We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian surfaces, they are also moduli spaces for genus-2 curves covering elliptic curves via a map of fixed degree. We thereby extend classical work of Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska, Völklein, Magaard and others, producing explicit families of reducible Jacobians. In particular, we produce a birational model for the modul…

Principality of prime ideals of algebraic number fields

We discuss principality of prime ideals of finite algebraic number fields $L=K(θ)$ over an algebraic number field $K ([K:\mathbb{Q}]<\infty)$ defined by irreducible polynomials $f(x)\in \mathfrak{O}_{K}[x]$ and $f(θ)=0$. Our main Theorem says that if a principal prime ideal $(π)\subset \mathfrak{O}_{K}$ is relatively prime to conductor $\mathfrak{F} =\{α\in \mathfrak{O}_{L}|$ a principal ideal $(α)$ of $\mathfrak{O}_{L}\subset \mathfrak{O}_{K}[θ]\}$ and splits completely over $L$: $(π)\mathfrak{O}_{L}=\prod \mathfrak{p}_{i}$, then $\mathfrak{p}_{i}$ is a principal ideal of $\mathfrak{O}_{L}$ f…

通过特化显式地对特征为二的Kummer面进行去奇异性处理

Explicit desingularisation of Kummer surfaces in characteristic two via specialisation

本文通过显式构造和特化方法，研究了特征为二的Kummer面的去奇异性处理及良好减少的条件。

构造具有给定精细Humbert不变量的 genus 2 曲线

Constructing genus 2 curves with given refined Humbert invariants

本文提出算法，构造具有给定精细Humbert不变量的genus 2曲线

CM 理论、极大超椭圆曲线与 Chebyshev 多项式

CM theory, maximal hyperelliptic curves, and Chebyshev polynomials

本文研究由 Chebyshev 多项式定义的超椭圆曲线，结合 Frobenius 斜率、复乘 Abel 子簇与 Jacobian 的 $2$-下降技术，给出并排除若干 $\FF_{q^2}$ 上的极大性情形。