The p-adic Gross-Zagier formula for elliptic curves at supersingular primes
The p-adic Gross-Zagier formula for elliptic curves at supersingular primes
复制标题
DOI：
10.1007/s00222-012-0400-9
复制
发表时间：
2013
影响因子：
3.1
通讯作者：
Shin-ichi Kobayashi
中科院分区：
文献类型：
--
作者：
Shin-ichi Kobayashi
设为素数，并设为定义在导体N的ℚ上的椭圆曲线。设K是一个虚二次域，具有判别素数TOP，使得N的所有素因数都是墨迹。B.Perrin-Riou建立了p-进Gross-Zagier公式，该公式将EoverK的p-adicL-函数的一阶导数与K的Heegner点的p-进高联系起来，当K具有良好的正常还原ATP时。在这篇文章中，我们证明了在好的超素数下，割圆ℤp-扩张的p-进Gross-Zagier公式。我们的结果适用于完全的Birch和Swinnerton-Dyer猜想。假设Eoverℚ的解析秩为1，并假设岩泽猜想对所有好素数成立，而p-进高对对所有好的普通素数不等于零，则我们的结果暗示了直到坏素数的全Birch和Swinnerton-Dyer猜想。特别地，如果有复杂的乘法和解析秩为1，则完整的Birch和Swinnerton-Dyer猜想是真的，直到坏素数和2的幂。
Letpbe a prime number and letEbe an elliptic curve defined over ℚ of conductorN. LetKbe an imaginary quadratic field with discriminant prime topNsuch that all prime factors ofNsplit inK. B. Perrin-Riou established thep-adic Gross-Zagier formula that relates the first derivative of thep-adicL-function ofEoverKto thep-adic height of the Heegner point forKwhenEhas goodordinaryreduction atp. In this article, we prove thep-adic Gross-Zagier formula ofEfor the cyclotomic ℤp-extension at goodsupersingularprimep. Our result has an application for thefullBirch and Swinnerton-Dyer conjecture. Suppose that the analytic rank ofEover ℚ is 1 and assume that the Iwasawa main conjecture is true for all good primes and thep-adic height pairing is not identically equal to zero for all good ordinary primes, then our result implies the full Birch and Swinnerton-Dyer conjecture up to bad primes. In particular, ifEhas complex multiplication and of analytic rank 1, the full Birch and Swinnerton-Dyer conjecture is true up to a power of bad primes and 2.