Cycles for Rankin-Selberg $L$-functions, I: automorphic periods

Atsushi Ichino; Kartik Prasanna

Cycles for Rankin-Selberg $L$-functions, I: automorphic periods

Number Theory 2026-08-01 v1

Authors: Atsushi Ichino , Kartik Prasanna

Abstract

In this paper, we establish an explicit formula for automorphic periods which will be used in a sequel to study special values of $p$ -adic Rankin-Selberg $L$ -functions. Our motivation is to extend the Bertolini-Darmon-Prasanna formula to the case where the archimedean local sign is opposite to that in the original setting. To this end, we prove a formula for the $\mathrm{GU}(1,1)$ -period of a theta lift from $\mathrm{GSO}(2)$ to $\mathrm{GSp}_4$ , which is adapted to $p$ -adic interpolation. We also introduce a $p$ -depletion Hecke operator for this theta lift, which will play a crucial role in relating the automorphic period to the image of the $p$ -adic Abel-Jacobi map.

Keywords

automorphic periods Rankin-Selberg L-functions theta lift p-adic interpolation Hecke operator Abel-Jacobi map

Cite

@article{arxiv.2608.00807,
  title  = {Cycles for Rankin-Selberg $L$-functions, I: automorphic periods},
  author = {Atsushi Ichino and Kartik Prasanna},
  journal= {arXiv preprint arXiv:2608.00807},
  year   = {2026}
}