On derivatives of Siegel–Eisenstein series over global function fields

Abstract

The aim of this article is to study the derivative of ‘incoherent’ Siegel–Eisenstein series on symplectic groups over function fields. Comparing the Fourier coefficients of incoherent Siegel–Eisenstein series with ‘coherent’ ones, the Siegel–Weil formula enables us to understand the non‐singular Fourier coefficients of the derivative in question by theta series (together with a local quantity coming from the corresponding Whittaker functions). Restricting to the special case when the incoherent quadratic space has dimension 2, we explicitly compute all the Fourier coefficients, and express the derivative in terms of the degree of the special cycles on the coarse moduli schemes of rank 2 Drinfeld modules with ‘complex multiplication’.