An Arithmetic Transfer Identity

TL;DR

This paper proves a specific case of the Arithmetic Fundamental Lemma conjecture for n=2, linking deformation lengths of quasi-homomorphisms to orbital integrals of a test function on a symmetric space.

Contribution

It establishes a new case of the Arithmetic Fundamental Lemma by connecting deformation lengths with orbital integrals for GL_2.

Findings

Proves the conjecture for n=2.

Constructs a test function with matching orbital integrals.

Links deformation lengths to orbital integrals in a new setting.

Abstract

We prove a variant of the Arithmetic Fundamental Lemma conjecture of Wei Zhang for n=2. More precisely, we consider the deformation lengths of certain quasi-homomorphisms of quasi-canonical lifts in the sense of Gross. We prove the existence of a test function on a symmetric space related to GL_2 whose orbital integrals over GL_1 equal the deformation lengths in question.