Exceptional zero formulae and a conjecture of Perrin-Riou

• 成果类型: Article
• 署名作者: Venerucci, Rodolfo
• 署名单位: University of Duisburg Essen
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0606-8
• 发表日期: 2016
• 页码: 923-972
• 关键词: adic l-functions; elliptic-curves; rational-points; cusp forms; Galois representations; iwasawa invariants; heegner points; modular-forms; hida families; VALUES

摘要

Let A/Q be an elliptic curve with split multiplicative reduction at a prime p. We prove (an analogue of) a conjecture of Perrin-Riou, relating p-adic Beilinson-Kato elements to Heegner points in A(Q), and a large part of the rank-one case of the Mazur-Tate-Teitelbaum exceptional zero conjecture for the cyclotomic p-adic L-function of A. More generally, let f be the weight-two newform associated with A, let f(infinity) be the Hida family of f, and let L p(f(infinity), k, s) be the Mazur-Kitagawa two-variable p-adic L-function attached to f(infinity). We prove a p-adic Gross-Zagier formula, expressing the quadratic term of the Taylor expansion of L p(f(infinity), k, s) at (k, s) = (2, 1) as a non-zero rational multiple of the extended height-weight of a Heegner point in A(Q).