On the characteristic $p$ valued measure associated to Drinfeld discriminant

TL;DR

This paper explores the structure of tempered distributions on projective spaces over function fields of characteristic p, focusing on measures linked to Drinfeld discriminants and Poincaré series, advancing understanding in characteristic p modular forms.

Contribution

It introduces the structure of tempered distributions in characteristic p and computes associated measures for Drinfeld discriminants and Poincaré series, expanding the theory of modular forms in this setting.

Findings

Calculated characteristic p valued measures for Drinfeld discriminant and Poincaré series.

Provided structural insights into tempered distributions over projective spaces.

Reviewed foundational results on functions over Bruhat-Tits trees and modular forms.

Abstract

In this paper, after reviewing known results on functions over Bruhat-Tits trees and the theory of characteristic $p$ valued modular forms, we present some structure of the tempered distributions on the projective space ${\P1}(\bk_\infty)$ over a complete function field $\bk_\infty$ of characteristic $p$, and calculate the characteristic $p$ valued measure associated to the Drinfeld discriminant and the characteristic $p$ valued measure associated to the Poinca\'e series.