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

这些pi将把Darmon和Bertolini最近关于有理域上椭圆曲线的一些工作推广到全实域的设置。在这项工作中，Darmon和Bertolini推导了椭圆曲线上Heegner点的p进解析公式，该公式涉及曲线上的两变量p进l函数的中心导数。所提出的工作与Birch和Swinnerton-Dyer的猜想直接相关，这是一个杰出的开放猜想，它应该揭示有理数（整数商）的三次多项式方程的解集。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，当它被视为变化素数的同余模时，它被定义为多项式方程的解的数量。通过L函数来研究三次方程，至少从20世纪60年代以来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.