Rational points on elliptic curves over totally real fields and p-adic L-functions

PI将推广到设置完全真实的领域的一些最近的工作由达蒙和Bertolini椭圆曲线的理性领域。 在这项工作中，Darmon和Bertolini推导出了椭圆曲线上Heegner点的p-adic解析公式，其中包含了曲线上二元p-adic L-函数的中心导数。这项工作与Birch和Swinnerton-Dyer的猜想直接相关，这是一个杰出的开放猜想，应该可以揭示有理数三次多项式方程的解集（整数的导数）。 这个猜想将解的集合与相关的L-函数的行为联系起来，L-函数是一个辅助函数，当它被看作是模变化素数的同余时，它被定义为多项式方程的解的个数。 至少从20世纪60年代以来，通过L-函数研究三次方程一直是富有成果的数学研究的中心。

The PIs will generalize to the setting of totally real fields some recent work by Darmon and Bertolini for elliptic curves over the rational field. In this work, Darmon and Bertolini derive a p-adic analytic formula for Heegner points on elliptic curves that involves the central derivative of the two-variable p-adic L-function attached to the curve.The proposed work is related directly to the conjecture of Birch and Swinnerton-Dyer, an outstanding open conjecture that should shed light on the set of solutions to cubic polynomial equations in rational numbers (quotients of whole numbers). This conjecture relates the set of solutions to the behavior of the associated L-function, an auxiliary function that is defined in terms of the numbers of solutions to the polynomial equation when it is viewed as a congruence modulo varying prime numbers. The study of cubic equations via L- functions has been the center of fruitful mathematical research at least since the 1960s.