A positive proportion of locally soluble quartic Thue equations are globally insoluble

Random Thue and Fermat equations

We consider Thue equations of the form $ax^k+by^k = 1$, and assuming the truth of the $abc$-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations $ax^k+by^k+cz^k = 0$ of degree at least six.

On the representation of integers by quadratic forms

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds for the least positive integer k such that the equation Q=k is insoluble in integers, despite being soluble modulo every prime power.

大多数奇次二元型不能本原表示平方数

Most Odd-Degree Binary Forms Fail to Primitively Represent a Square

本文研究固定首项系数的奇次超椭圆方程族，证明当次数充分大时超过74.9%的方程无本原整数解、超过71.8%处处局部可解但因Brauer–Manin阻碍违反Hasse原理，且比例可随首项系数素因子增多而提高至99.9%和96.7%。