Mathematical Sciences: Arithmetic on Elliptic Curves and Related Topics

本文研究的是数域上给定椭圆曲线的超奇异素数问题。直接目标包括证明超奇异素数在全虚情况下无穷的一个自然充分条件和这种素数密度的无条件显式下界。进一步的目标将包括非零固定迹问题（其中许多古老的经典问题是特殊情况）和椭圆曲线和高维阿贝尔变分算法中的各种其他开放问题。本研究的领域是数论的一个分支，研究某些三次方程及其有理解（椭圆曲线）的个数。这门学科在整个数学中都有联系，并在解决数论中涉及许多主题的许多问题方面发挥了重要作用。模素数方程的解的个数是本课题的中心焦点，也是本研究者的研究领域。

This research will be on problems related to the question of supersingular primes for a given elliptic curve over a number field. Immediate goals include a proof of a natural sufficient condition for the infinitude of supersingular primes in the totally imaginary case and unconditional explicit lower bounds on the density of such primes. Further goals will include the nonzero fixed trace problem (of which many old classical problems are special cases) and various other open problems in the arithmetic of elliptic curves and abelian varieties of higher dimension. The area of this research is in that branch of number theory which studies certain cubic equations and the number of rational solutions of them (elliptic curves). The subject has connections throughout mathematics and has been instrumental in solving many problems in number theory concerning many topics. The number of solutions of the equation modulo primes is a central focus in the subject and this is the area focused on by this investigator.