The Analytic Theory of Division Fields and Spectral L-functions

杜克公爵的DMS-0355564奖和他的学生们将致力于两个广泛的问题：发展椭圆曲线和阿贝尔族的除域的解析理论，以及理解谱L函数及其导数的特定值的数论意义。第一部分可以被描述为‘非阿贝尔’解析数论，其动机是应用于椭圆曲线和阿贝尔族的mod p约化。到目前为止，假设广义黎曼假设(GRH)是可能的，而消除广义黎曼假设是许多挑战之一。第二部分的动机是Duke观察到的一些类比，即由单位生成数域作为L函数的导数的特定值(例如‘’斯塔克猜想‘’)和与黎曼曲面相关的L函数的导数的特定值。该提议的一个总体目标是扩大解析数论的范围，既包括在整个数论中，也包括在更广泛的数学中。在过去的几年里，很明显，解析数论的技术比人们以前认识到的具有更广泛的适用性。这种技术在实际应用中也具有潜在的重要性，特别是在密码学和相关领域。

Abstract for award DMS-0355564 of DukeDuke and his students will work on two broad sets of problems:to develop the analytic theory of division fields of elliptic curves and abelian varietiesand to understand the number-theoretic significance of special valuesof spectral L-functions and their derivatives.The first part may be described as ``nonabelian" analytic numbertheory and is motivated by applications to the mod p reductionsof elliptic curves and abelian varieties. So far progress has beenpossible assuming the Generalized Riemann Hypothesis (GRH), andits removal it one of the many challenges.The second part is motivated by some analogies Duke has observedbetween the generation of number fields by units occurring asspecial values of derivatives of L-functions (e.g. the ``Starkconjectures") and the special values of derivatives of Lfunctions associated to Riemann surfaces. A general goal of the proposal is to extend the reach of analytic number theory,both within number theory at large and also within mathematicsmore generally. It has become evident over the lastfew years that the techniques of analytic number theory have amuch broader applicability than was earlier appreciated.Such techniques also have potential importance in practical applications,especially in cryptography and related areas.