TY - JOUR

T1 - The universal p-adic Gross–Zagier formula

AU - Disegni, Daniel

N1 - Funding Information:
This work was supported by ISF grant 1963/20 and BSF grant 2018250. During the preparation of a first draft of this paper, the author was supported by a public grant of the Fondation Mathématique Jacques Hadamard.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - Let G be the group (GL 2× 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.

AB - Let G be the group (GL 2× 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.

UR - http://www.scopus.com/inward/record.url?scp=85134699380&partnerID=8YFLogxK

U2 - 10.1007/s00222-022-01133-w

DO - 10.1007/s00222-022-01133-w

M3 - Article

AN - SCOPUS:85134699380

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

ER -