The universal p-adic Gross-Zagier formula

• 成果类型: Article
• 署名作者: Disegni, Daniel
• 署名单位: Ben-Gurion University of the Negev
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01133-w
• 发表日期: 2022
• 页码: 509-649
• 关键词: exceptional zero formulas; heegner points; elliptic-curves; hida families; chow groups; cycles; derivatives; heights; conjecture; characters

摘要

Let G be the group (GL(2) x GU(1))/GL(1) over a totally real field F, and let X be a Hida family for G. Revisiting a construction of Howard and Fouquet, we construct an explicit section P of a sheaf of Selmer groups over X. We show, answering a question of Howard, that P is a universal Heegner class, in the sense that it interpolates geometrically defined Heegner classes at all the relevant classical points of X. We also propose a 'Bertolini-Darmon' conjecture for the leading term of P at classical points. We then prove that the p-adic height of P is given by the cyclotomic derivative of a p-adic L-function. This formula over X (which is an identity of functionals on some universal ordinary automorphic representations) specialises at classical points to all the Gross-Zagier formulas for G that may be expected from representation-theoretic considerations. Combined with a result of Fouquet, the formula implies the p-adic analogue of the Beilinson-Bloch-Kato conjecture in analytic rank one, for the selfdual motives attached to Hilbert modular forms and their twists by CM Hecke characters. It also implies one half of the first example of a non-abelian Iwasawa main conjecture for derivatives, in 2[F : Q] variables. Other applications include two different generic non-vanishing results for Heegner classes and p-adic heights.