The analytic theory of vectorial Drinfeld modular forms

TL;DR

This paper extends the theory of Drinfeld modular forms to a vector-valued setting, introducing new structures like a 'field of uniformisers' and analyzing their properties, with applications to arithmetical problems.

Contribution

It generalizes Drinfeld modular forms to vector-valued forms with new algebraic structures and provides detailed analysis and conjectures for special cases.

Findings

Spaces of vector-valued modular forms are finite-dimensional.

Introduction of a 'field of uniformisers' for expansion analysis.

A harmonic product formula leading to conjectural formulas for Eisenstein series.

Abstract

In this paper we generalise the notion of Drinfeld modular form for the group $\Gamma$ := GL2(Fq[$\theta$]) to a vector-valued setting, where the target spaces are certain modules over positive characteristic Banach algebras over which are defined what we call the 'representations of the first kind'. Under quite reasonable restrictions, we show that the spaces of such modular forms are finite-dimensional, are endowed with certain generalisations of Hecke operators, with differential operators{\`a} la Serre etc. The crucial point of this work is the introduction of a 'field of uniformisers', a valued field in which we can study the expansions at the cusp infinity of our modular forms and which is wildly ramified. Examples of such modular forms are given through the construction of Poincar{\'e} and Eisenstein series. After this the paper continues with a more detailed analysis of the special case of modular forms associated to a restricted class of representations $\rho$ * $\Sigma$ of $\Gamma$ which has more importance in arithmetical applications. More structure results are given in this case, and a harmonic product formula is obtained which allows, with the help of three conjectures on the structure of an Fp-algebra of A-periodic multiple sums, to produce conjectural formulas for Eisenstein series. Some of these formulas can be proved.