Drinfeld modular forms of weight one
Let \(A=\mathbb{F}_q[T]\) be the polynomial ring over the finite field \(\mathbb{F}_q\), \(K= \mathbb{F}_q(T)\) its quotient field, \(K_\infty= \mathbb{F}_q((T^{-1}))\) the \(\infty\)-adic completion, \(C\) the completed algebraic closure of \(K_\infty\), and \(\Omega= C-K_\infty\) the Drinfeld upper half-plane. The modular group \(\Gamma(1)= GL(2,A)\) acts on \(\Omega\), and the associated \(C\)-algebras \(M(\Gamma(1))\) of Drinfeld modular forms is a polynomial ring \(C[g,\Delta]\) in two modular forms \(g\) and \(\Delta\) of respective weights \(q-1\) and \(q^2-1\). It is a basic albeit unsolved problem to determine the structure of the algebra of modular forms for convergence subgroups \(\Gamma\) of \(\Gamma(1)\), especially for full congruence subgroups \(\Gamma(N)= \{\gamma\in\Gamma\mid \gamma\equiv 1\pmod N\}\) of \(\Gamma(1)\). Fix such a \(\Gamma=\Gamma(N)\) with \(N\in A-\mathbb{F}_q\), and let \(M(\Gamma)=\oplus M_k(\Gamma)\) be its algebra of modular forms (\(k\)= weight). \(M_k\) certainly contains the Eisenstein series \(E_u^{(k)}\) \((u\in(N^{-1}/ A)^2)\), where \[ E_u^{(k)}(z)= \mathop{{\sum}'}_{\substack{ (a,b)\in A^2\\ (a,b)\equiv u\pmod A}} \frac{1} {(az+b)^k}. \tag \(*\) \] Using an argument of Hecke, the author first shows that the \(E_u^{(k)}\) generate a vector space complement in \(M_k\) of the cusp forms of weight \(k\) (Prop. 1.12). Since there are no cusp forms of weight one, the result of Mumford may be applied to conclude that \(M(\Gamma)\) is generated by the \(E_u^{(1)}\) and the cusp forms of weight two (Prop. 1.15). Two natural questions arise: (1) Do we already have \[ \begin{aligned} M(\Gamma) & = C\left[E_u^{(1)}\mid u \text{ as in (*)}\right]\\ & =: E(\Gamma)?\end{aligned} \] (2) What are the relations between the different \(E_u^{(1)}\)? At least, \(M(\Gamma)\) is the integral closure of \(E(\Gamma)\) in their common field of fractions (Prop. 1.17). Now the main result of the paper is Theorem 2.2, too complicated to be stated here in detail, which relates \(E(\Gamma)\) with some ring \(R/J\) that admits a simple and explicit description through Drinfeld modules. Finally, some examples are given \((\Gamma= \Gamma(N)\) with \(N\in A\) linear, or \((q,\text{deg }N)= (2,2))\), where the above questions can be answered.
modular forms for full convergence subgroups
scientific article; zbMATH DE number 1109648
scientific article