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The universal $p$-adic Gross-Zagier formula

Daniel Disegni

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.

The universal $p$-adic Gross-Zagier formula

Abstract

Let be the group over a totally real field , and let be a Hida family for . Revisiting a construction of Howard and Fouquet, we construct an explicit section of a sheaf of Selmer groups over . We show, answering a question of Howard, that is a universal Heegner class, in the sense that it interpolates geometrically defined Heegner classes at all the relevant classical points of . We also propose a `Bertolini-Darmon' conjecture for the leading term of at classical points. We then prove that the -adic height of is given by the cyclotomic derivative of a -adic -function. This formula over (which is an identity of functionals on some universal ordinary automorphic representations) specialises at classical points to all the Gross-Zagier formulas for that may be expected from representation-theoretic considerations. Combined with a result of Fouquet, the formula implies the -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 variables. Other applications include two different generic non-vanishing results for Heegner classes and -adic heights.

Paper Structure

This paper contains 170 sections, 9 theorems, 423 equations, 1 figure.

Table of Contents

  1. Introduction and statements of the main results
  2. The $p$-adic Beı linson--Bloch--Kato conjecture in analytic rank $1$
  3. Selmer groups according to Bloch--Kato and Nekovář
  4. The $p$-adic Beı linson--Bloch--Kato conjecture for Hilbert modular forms
  5. Notation
  6. A variant for selfdual Hilbert modular forms
  7. Addendum to the historical overview: higher-rank cases
  8. The $p$-adic Gross--Zagier formula for arbitrary weight
  9. Algebraic groups and Shimura varieties
  10. $p$-adic automorphic representations
  11. Automorphic and Galois representations
  12. Heegner cycles
  13. Multiplicity one
  14. Local toric periods
  15. Global pairings and $p$-adic heights
  16. ...and 155 more sections

Key Result

Theorem 1

Let $\pi_{0}$ be a Hilbert modular form over $L$ of weight $\underline{w}$, and let $\chi$ be a Hecke character of $E$ over $L$ of weight $\ul$. Let $V:=V_{\pi_{0}|G_{E}}\otimes V_{\chi}$ . Suppose that:

Figures (1)

  • Figure 1: An illustration of the proof of Theorems \ref{['GZ thm']} and \ref{['ugz thm']}. Each of the (infinitely many) floors corresponds to a representation $\Pi$ as in Theorem \ref{['GZ thm']}, each apartment to a quadruple ${\bf f}=(f_{1}, f_{2}, f_{3}, f_{4})$, and the building to a Hida family. A light being on indicates that the corresponding Gross--Zagier formula is proven. On 'most' floors corresponding to a $\Pi$ of trivial weight, all lights are on by dd-pyzznonsplit. In this paper, we construct the lift corresponding to the formula of Theorem \ref{['ugz thm']}, with doors (interpolation statements) onto special apartments in each floor (the formulas of Theorem \ref{["GZ thm'"]} in § \ref{['B ord']}, equivalent to Theorem \ref{['GZ thm']} for certain special quadruples ${\bf f}$). As soon as the lights in a dense set of floors in the building are on, the light in the lift is on; this allows to turn on the light in all the special apartments. Finally, the multiplicity-one principle allows to propagate the electricity among different apartments on the same floor.

Theorems & Definitions (74)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • proof
  • proof
  • proof
  • ...and 64 more