Zeroes of Eisenstein series for principal congruence subgroups over rational function fields
A classical result of \textit{F. K. C. Rankin} and \textit{H. P. F. Swinnerton-Dyer} [Bull. Lond. Math. Soc. 2, 169--170 (1970; Zbl 0203.35504)] says that Eisenstein series for the full modular group have all their zeros in the standard fundamental domain on the unit circle. An analogue in rational function fields was proven by the reviewer [C. R. Acad. Sci., Paris, Sér. I 321, No. 7, 817--820 (1995; Zbl 0847.11033)]. In this paper, Gekeler studies what happens for Eisenstein series for principal congruence subgroups over a rational function field. He locates the zeros inside certain very specific rational subdomains of a standard fundamental domain inside the Drinfeld upper half space (not necessarily the strict function field analogue of the unit circle). The proofs are based on studying the Newton polygons of some polynomials introduced by \textit{D. Goss} [Compos. Math. 41, 3--38 (1980; Zbl 0422.10020)].
scientific article; zbMATH DE number 5966269
scientific article