The explicit Mordell Conjecture for families of curves (with an appendix by M. Stoll)

Note on the rank of quadratic twists of Mordell equations

14H52 : Elliptic curves Let E be the elliptic curve given by a Mordell equation y^2=x^3-A where A is an integer. For certain A, we use Stoll's formula to compute a lower bound for the proportion of square-free integers D up to X such that the Mordell-Weil rank of the quadratic twist by D is less than 2k, for given non-negative k. We also compute an upper bound for a certain average rank of quadratic twists of E.

利用秩0商策略计算某些亏格3曲线的有理点

Rational Points of some genus $3$ curves from the rank $0$ quotient strategy

本文提出一种算法，用于计算亏格3曲线中作为秩0亏格1曲线二次覆盖情形的全部有理点，并在Magma中实现，在约4万条曲线上验证，找到了达到Stoll上界的实例。

Uniform bounds for the number of rational points on curves of small Mordell--Weil rank

Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known by [CHM97] to follow from the Bombieri--Lang conjecture. A related conjecture posits the existence of a uniform bound $N_{{\rm tors},\dagger}(g,d)$ on the number of geometric torsion points of the Jacobian $J$ of $X$ which lie on the image of $X$ under an Abel--Jacobi map. For fixed $X$ this quantity was conjectured to be finite by Manin--Mumfor…

Some arithmetic properties on nonstandard rationals

For a given number field $K$, we show that the ranks of nonsingular elliptic curves over $K$ are uniformly finitely bounded if and only if weak Mordell-Weil property holds in all(some) ultrpowers $^*K$ of $K$. Also we introduce Nonstandard Mordell-Weil property for $^*K$ considering each Mordell-Weil group as $^*Z$-module, where $^*Z$ is an ultrapower of $Z$, and we show that Nonstandard Mordell-Weil property is equivalent to weak Mordell-Weil property in $^*K$. In Appendix, we showed that it is possible to consider definable abelian groups as $^*Z$-modules in a saturated nonstandard rational …

杰出范畴中的Hecke轨道与Mordell-Lang猜想

Hecke orbits and the Mordell-Lang conjecture in distinguished categories

本文在杰出范畴框架中引入$Σ$-轨道概念，统一了Shimura簇中的Hecke轨道与半阿贝尔簇中的有限秩子群，并借此证明Mordell-Lang型一般命题及Legendre族和$\mathcal{A}_g$中的无条件不太可能相交结果。

Finiteness Problems in Diophantine Geometry

This survey contains an exposition of ideas and results related to Faltings' proof of the conjectures of Shafarevich, Tate and Mordell. This paper originally appeared in 1986 as an Appendix to the Russian translation of Serge Lang, "Fundamentals of Diophantine Geometry" (Springer Verlag, 1983) published by "Mir", Moscow (MR0854670, 88a:11054). A history of the publication of the Appendix is briefly described by Lang in Section 4 of his paper "Mordell's review, Siegel's letter to Mordell, Diophantine geometry, and 20th century mathematics" that was published (in 1995) simultaneously in Notices …