Arithmetic volume of Shtukas and Langlands duality

TL;DR

This paper generalizes the relationship between the arithmetic volume of Shtukas and derivatives of zeta functions by incorporating arbitrary coweights and reveals a structural role for the Langlands dual group in these formulas.

Contribution

It introduces uniform formulas for eigenweights in terms of the Langlands dual group, extending previous work to more general settings.

Findings

Derived formulas involving eigenweights and the dual group.

Established a structural role for the Langlands dual group.

Extended the relation to arbitrary coweights for split semisimple groups.

Abstract

We extend the work of Feng--Yun--Zhang relating the arithmetic volume of Shtukas with derivatives of zeta functions by allowing arbitrary coweights for split semisimple algebraic groups. As in their original work, the formula involves some numbers called eigenweights. We obtain uniform formulas for the eigenweights in terms of the Langlands dual group, marking the first structural role for the dual group in such formulas governing derivatives of L-functions.