On derivatives of Siegel-Eisenstein series over global function fields

TL;DR

This paper investigates the derivatives of incoherent Siegel-Eisenstein series over function fields, relating their Fourier coefficients to quadratic form arithmetic and special cycles on moduli schemes of Drinfeld modules.

Contribution

It explicitly computes Fourier coefficients of the derivatives for dimension 2 quadratic spaces and links them to special cycles on moduli schemes, extending the understanding of Eisenstein series in function fields.

Findings

Explicit formulas for Fourier coefficients in dimension 2 case

Connection between derivatives of Eisenstein series and special cycles

Relation to arithmetic of quadratic forms over function fields

Abstract

The aim of this article is to study the derivative of "incoherent" Siegel-Eisenstein series on symplectic groups over function fields. By the Siegel-Weil formula for "coherent" Siegel-Eisenstein series, we can relate the non-singular Fourier coefficients of the derivative in question to the arithmetic of quadratic forms. Restricting to the special case when the incoherent quadratic space has dimension 2, we explicitly compute all the Fourier coefficients, and connect the derivative with the special cycles on the coarse moduli schemes of rank 2 Drinfeld modules with "complex multiplication."