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The $p$-adic Gross-Zagier formula on Shimura curves, II: nonsplit primes

Daniel Disegni

Abstract

The formula of the title relates $p$-adic heights of Heegner points and derivatives of $p$-adic $L$-functions. It was originally proved by Perrin-Riou for $p$-ordinary elliptic curves over the rationals, under the assumption that $p$ splits in the relevant quadratic extension. We remove this assumption, in the more general setting of Hilbert-modular abelian varieties.

The $p$-adic Gross-Zagier formula on Shimura curves, II: nonsplit primes

Abstract

The formula of the title relates -adic heights of Heegner points and derivatives of -adic -functions. It was originally proved by Perrin-Riou for -ordinary elliptic curves over the rationals, under the assumption that splits in the relevant quadratic extension. We remove this assumption, in the more general setting of Hilbert-modular abelian varieties.

Paper Structure

This paper contains 66 sections, 2 theorems, 184 equations, 1 figure.

Table of Contents

  1. Introduction and statement of the main result
  2. The main ideas in a classical context
  3. Review of Perrin-Riou's proof
  4. The nonsplit case
  5. Subtleties
  6. Statement
  7. Abelian varieties parametrised by Shimura curves
  8. Heegner points
  9. A local unit of measure for invariant functionals
  10. $p$-adic heights
  11. The $p$-adic $L$-function
  12. The $p$-adic Gross--Zagier formula
  13. A new proof of a result of Greenberg--Stevens
  14. Exceptional cases and non-vanishing results
  15. Organisation of the paper
  16. ...and 51 more sections

Key Result

Theorem 1

There is a function characterised by the following property. For each complex geometric point $s=\chi_{F}\in \mathscr{Y}_{F}^{\rm l.c.}(\mathbf{C})$, with underlying embedding $\iota\colon L(\chi_{F})\hookrightarrow \mathbf{C}$, where ${e}_{p}(V_{(A, \chi')}):= \prod_{v\vert p} e_{v}(V_{(A, \chi')})$.

Figures (1)

  • Figure 1: A road sign in Croatia.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['A']}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 19 more