A converse to a theorem of Gross, Zagier, and Kolyvagin

• 成果类型: Article
• 署名作者: Skinner, Christopher
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.191.2.1
• 发表日期: 2020
• 页码: 329-354
• 关键词: rational-points; elliptic-curves; zeta-functions

摘要

Let E be a semistable elliptic curve over Q. We prove that if E has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes, then rank(z)E(Q) = 1 and #III(E) < infinity double right arrow ords(s=1)L(E, s) = 1. We also prove the corresponding result for the abelian variety associated with a weight 2 newform f of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for f and H-f(1)(Q, V), where V is the p-adic Galois representation associated with f, that ensure that ord(s=1)L(f, s) = 1. The main theorem is proved using the Iwasawa theory of V over an imaginary quadratic field to show that the p-adic logarithm of a suitable Heegner point is non-zero.