A p-adic arithmetic inner product formula

Daniel Disegni, Yifeng Liu

Research output: Contribution to journalArticlepeer-review


Fix a prime number p and let E/F be a CM extension of number fields in which p splits relatively. Let π be an automorphic representation of a quasi-split unitary group of even rank with respect to E/F such that π is ordinary above p with respect to the Siegel parabolic subgroup. We construct the cyclotomic p-adic L-function of π, and a certain generating series of Selmer classes of special cycles on Shimura varieties. We show, under some conditions, that if the vanishing order of the p-adic L-function is 1, then our generating series is modular and yields explicit nonzero classes (called Selmer theta lifts) in the Selmer group of the Galois representation of E associated with π; in particular, the rank of this Selmer group is at least 1. In fact, we prove a precise formula relating the p-adic heights of Selmer theta lifts to the derivative of the p-adic L-function. In parallel to Perrin-Riou’s p-adic analogue of the Gross–Zagier formula, our formula is the p-adic analogue of the arithmetic inner product formula recently established by Chao Li and the second author.

Original languageAmerican English
Pages (from-to)219-371
Number of pages153
JournalInventiones Mathematicae
Issue number1
StatePublished - 1 Apr 2024


  • 11G18
  • 11G40
  • 11G50
  • 11R34

All Science Journal Classification (ASJC) codes

  • General Mathematics


Dive into the research topics of 'A p-adic arithmetic inner product formula'. Together they form a unique fingerprint.

Cite this