Let A=F q [T ] be the polynomial ring over the finite field F q of q elements. D. Goss remarks in [13, (2.1)] that the algebra of (Drinfeld ) modular forms for GL(2, A) is the free ring generated by the two Eisenstein series of weights q&1 and q 2 &1. For a more general congruence subgroup, an abstract presentation of the corresponding ring of modular forms is not known, and is marked an open question by E.-U. Gekeler VIII.3.1]). The aim of this paper is to develop tools from algebraic geometry to study the case of a principal congruence subgroup 1(N) of level N in GL(2, A).
The paper is organized as follows: Eisenstein series of weight one are defined, and it is shown that they generated the vector space of modular forms of weight one (which is in sharp contrast with the classical situation, where e.g. the square of Dedekind's eta-function ' 2 is a cusp form of weight one for 1(12)). Using Castelnuovo Mumford regularity, we show that the ring of modular forms is generated by these Eisenstein series and the cusp forms of weight two, thus improving slightly a bound obtained by D. Goss. We then study embeddings of Drinfeld modular curves via Eisenstein series, and the normality of rings of modular forms. The next paragraph interprets Eisenstein series as torsion points of a generic Drinfeld module, to obtain a reduced set of equations for the image of the embedded modular curve under the group of permutations of the variables. This result will allow the use of computational commutative algebra for solving the original problem. In the final paragraph, we calculate explicitly the degrees of various embeddings of Drinfeld modular curves, and give applications to automorphisms of modular curves and the original problem Article No. NT972185