Atkin-Lehner theory for Drinfeld modular forms and applications

TL;DR

This paper develops Atkin-Lehner theory for Drinfeld modular forms, providing new definitions, commutativity results, and applications including decomposition criteria and construction of p-adic forms.

Contribution

It introduces an equivalent definition of p-newforms for Drinfeld modular forms and establishes their properties and applications.

Findings

New definition of p-newforms simplifies computations

Proves commutativity between Hecke operators and Atkin-Lehner involutions

Provides criteria for decomposing cusp forms and constructing p-adic forms

Abstract

The present paper deals with Atkin-Lehner theory for Drinfeld modular forms. We provide an equivalent definition of $\mathfrak{p}$-newforms (which makes computations easier) and commutativity results between Hecke operators and Atkin-Lehner involutions. As applications we show a criterion for a direct sum decomposition of cusp forms, we exibit $\mathfrak{p}$-newforms arising from lower levels and we provide $\mathfrak{p}$-adic Drinfeld modular forms of level greater than 1.