Non-reductive cycles and twisted arithmetic transfers for Shimura curves

TL;DR

This paper discusses recent advances in the study of special cycles on Shimura varieties, the twisted arithmetic fundamental lemma, and their connections to the arithmetic Langlands program, providing new formulations and proofs.

Contribution

It introduces new formulations of arithmetic twisted Gan-Gross-Prasad conjecture and proves the twisted arithmetic fundamental lemma and related arithmetic transfers.

Findings

Proof of twisted AFL established

New formulations of arithmetic twisted Gan-Gross-Prasad conjecture

Progress on arithmetic analogs of the relative Langlands program

Abstract

In this largely expository note, we explain some recent progress on new cycles on Shimura varieties and Rapoport-Zink spaces, (twisted) arithmetic fundamental lemma, and arithmetic analogs of relative Langlands program. We explain related formulations of arithmetic twisted Gan-Gross-Prasad conjecture, the proof of twisted AFL and certain arithmetic transfers.