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Gan--Gross--Prasad cycles and derivatives of $p$-adic $L$-functions

Daniel Disegni, Wei Zhang

Abstract

We study the p-adic analogue of the arithmetic Gan-Gross-Prasad (GGP) conjectures for unitary groups. Let $Π$ be a conjugate-selfdual cuspidal automorphic representation of GL_{n} x GL_{n+1} over a CM field, which is algebraic of minimal regular weight at infinity. We first show the rationality of twists of the ratio of L-values of $Π$ appearing in the GGP conjectures. Then, when $Π$ is p-ordinary at a prime p, we construct a cyclotomic p-adic L-function $L_p(M_Π)$ interpolating those twists. Finally, under some local assumptions, we prove a precise formula relating the first derivative of $L_p(M_Π)$ to the p-adic heights of Selmer classes arising from arithmetic diagonal cycles on unitary Shimura varieties. We deduce applications to the p-adic Beilinson-Bloch-Kato conjecture for the motive attached to $Π$. All proofs are based on some relative-trace formulas in p-adic coefficients.

Gan--Gross--Prasad cycles and derivatives of $p$-adic $L$-functions

Abstract

We study the p-adic analogue of the arithmetic Gan-Gross-Prasad (GGP) conjectures for unitary groups. Let be a conjugate-selfdual cuspidal automorphic representation of GL_{n} x GL_{n+1} over a CM field, which is algebraic of minimal regular weight at infinity. We first show the rationality of twists of the ratio of L-values of appearing in the GGP conjectures. Then, when is p-ordinary at a prime p, we construct a cyclotomic p-adic L-function interpolating those twists. Finally, under some local assumptions, we prove a precise formula relating the first derivative of to the p-adic heights of Selmer classes arising from arithmetic diagonal cycles on unitary Shimura varieties. We deduce applications to the p-adic Beilinson-Bloch-Kato conjecture for the motive attached to . All proofs are based on some relative-trace formulas in p-adic coefficients.

Paper Structure

This paper contains 244 sections, 97 theorems, 569 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $p$-adic $L$-function
  3. Rationality of $\mathscr{L}\xspace$
  4. $p$-adic interpolation
  5. On the $p$-adic Beilinson--Bloch--Kato conjectures for Rankin--Selberg motives
  6. The $p$-adic arithmetic Gan--Gross--Prasad conjecture
  7. Incoherent unitary groups and their representations
  8. Arithmetic diagonal cycles
  9. Gan--Gross--Prasad cycles
  10. The $p$-adic arithmetic Gan--Gross--Prasad conjecture
  11. Dualities
  12. Invariant functionals
  13. $p$-adic heights and main result
  14. $p$-adic relative trace formulas and the proofs
  15. Rationality
  16. ...and 229 more sections

Key Result

Theorem 1

Let $\Pi$ be a trivial-weight hermitian cuspidal automorphic representation of $\mathrm{G}'(\mathbf{A})$ defined over a characteristic-zero field $L$. Then there is an element such that for all $\chi\in Y_{L}(\mathbf{C})$ with underlying embedding $\iota\colon L\hookrightarrow \mathbf{C}$.

Figures (1)

  • Figure 1: Wassily Kandinsky, Diagonal, 1923.

Theorems & Definitions (246)

  • Theorem 1
  • Remark 1.1.1
  • Theorem 2
  • Remark 1.1.2
  • Remark 1.1.3
  • Remark 1.1.4
  • Remark 1.1.5
  • Theorem 3
  • Remark 1.2.1
  • Remark 1.2.2
  • ...and 236 more