The universal $p$-adic Gross-Zagier formula

TL;DR

This paper develops a universal $p$-adic Gross-Zagier formula for a specific automorphic group over a totally real field, linking $p$-adic heights, $L$-functions, and Heegner classes, with broad implications for number theory conjectures.

Contribution

It constructs an explicit universal Heegner class over a Hida family, interpolates classical Heegner classes, and proves a $p$-adic Gross-Zagier formula applicable to various conjectures.

Findings

Constructed a universal Heegner class interpolating classical classes.

Proved the $p$-adic height formula relates to cyclotomic derivatives of $p$-adic $L$-functions.

Implications for the Beilinson-Bloch-Kato conjecture and non-abelian Iwasawa theory.

Abstract

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