The universal $p$-adic Gross-Zagier formula

Abstract

Let $\mathrm{G}$ be the group $({\mathrm{GL}}_2\times \mathrm{GU}(1))/{\mathrm{GL}}_1$ over a totally real field $F$, and let $\mathcal{X}$ be a Hida family for $\mathrm{G}$. Revisiting a construction of Howard and Fouquet, we construct an explicit section $\mathcal{P}$ of a sheaf of Selmer groups over $\mathcal{X}$. We show, answering a question of Howard, that $\mathcal{P}$ is a universal Heegner class, in the sense that it interpolates geometrically defined Heegner classes at all the relevant classical points of $\mathcal{X}$. We also propose a `Bertolini-Darmon' conjecture for the leading term of $\mathcal{P}$ at classical points. We then prove that the $p$-adic height of $\mathcal{P}$ is given by the cyclotomic derivative of a $p$-adic $L$-function. This formula over $\mathcal{X}$ (which is an identity of functionals on some universal ordinary automorphic representations) specialises at classical points to all the Gross-Zagier formulas for $\mathrm{G}$ 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 $2[F:\mathbf{Q}]$ variables. Other applications include two different generic non-vanishing results for Heegner classes and $p$-adic heights.

arXiv comment89 pages, 1 figure. New in this version: a correction as Appendix B (to appear separately in Invent. math.)

References in corpus (24)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• Algebras of p-adic distributions and admissible representations
• Explicit Gross-Zagier and Waldspurger Formulae
• Hilbert modular forms and p-adic Hodge theory
• Syntomic cohomology and regulators for varieties over p-adic fields
• Variation of Heegner points in Hida families
• p-adic Cohomology and classicality of overconvergent Hilbert modular forms
• On the $p$-adic variation of Heegner points
• The p-adic Gross-Zagier formula on Shimura curves
• Darmon points on elliptic curves over number fields of arbitrary signature
• On $p$-adic Waldspurger formula
• CM cycles on Shimura curves, and p-adic L-functions
• $p$-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou
• On the $\mu$-invariant of the cyclotomic derivative of Katz p-adic L-function
• Heegner points in Coleman families
• On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions
• The $p$-adic Gross-Zagier formula on Shimura curves, II: nonsplit primes
• Local Langlands correspondence, local factors, and zeta integrals in analytic families
• Leading terms of anticyclotomic Stickelberger elements and p-adic periods
• Interpolation of Generalized Heegner Cycles in Coleman Families
• On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields
• p-adic Heights of Heegner points on Shimura curves
• Twisted triple product p-adic L-functions and Hirzebruch-Zagier cycles
• $p$-adic L-functions via local-global interpolation: the case of ${\mathrm{GL}}_2\times \mathrm{GU}(1)$

Cited by in corpus (6)

• Heegner points in Coleman families
• The $p$-adic Gross-Zagier formula on Shimura curves, II: nonsplit primes
• Local Langlands correspondence, local factors, and zeta integrals in analytic families
• Interpolation of Generalized Heegner Cycles in Coleman Families
• $p$-adic L-functions via local-global interpolation: the case of ${\mathrm{GL}}_2\times \mathrm{GU}(1)$
• The $p$-adic variation of the Gross-Kohnen-Zagier theorem