Automorphic-twisted summation formulae for pairs of quadratic spaces

TL;DR

This paper establishes a new summation formula for pairs of quadratic spaces aligned with conjectures by prominent mathematicians, providing explicit local factor expressions in unramified cases.

Contribution

It introduces an automorphic-twisted summation formula for quadratic spaces and derives explicit unramified local factors, advancing the understanding of these conjectures.

Findings

Proved a summation formula for pairs of quadratic spaces.

Derived explicit expressions for local factors in unramified cases.

Confirmed conjectural frameworks by Braverman-Kazhdan, Lafforgue, Ngô, and Sakellaridis.

Abstract

We prove a summation formula for pairs of quadratic spaces following the conjectures of Braverman-Kazhdan, Lafforgue, Ng\^{o} and Sakellaridis. We also give an expression of the local factors where all the data are unramified.