Finite modular forms
The author develops a theory of modular forms for the fractional linear action of \(\Gamma:= \text{GL}(2,K)\) on the ``upper half plane'' \(\Omega:={\mathbf P}^1_K - {\mathbf P}^1(K)\), where \(K\) is a finite field. The theory looks like a shadow of the theory of classical or Drinfeld modular forms and, indeed, occurs naturally as the reduction of the latter kind of objects at the infinite prime. A modular form of weight \(k\) and type \(m\) is a rational function in a uniformizer \(z\) of \({\mathbf P}^1\) without poles in \(\Omega\) and at \(\infty\), which has the usual transformation behaviour under the action of \(\Gamma\) (with character \(\det^{-m}\)). One considers \(t:=(z-z^q)^{-1}\) as a uniformizer at infinity, and a modular form \(f\) has a series development in \(t\), which turns out to be a polynomial. Now the usual theory can be developed: Eisenstein series with their explicit series expansions (whence results on the reduction of their zeros), \(j\)-invariant, dimension formulae for vector spaces of modular forms, construction of a basis in terms of two special forms, linear relations between forms of different weight (which can exist!), Serre-derivatives, Eichler-Shimura map. Finally, the dimension of the space of cusp forms of weight \(k+2\) and type \(m+1\) turns out to equal the multiplicity of \(\text{St} \otimes \det^{-m}\) in \(\text{Sym}^k \;V\) , where \(\text{St}\) is the Steinberg representation of \(\Gamma\) and \(V\) is a two-dimensional vector space over an algebraic closure of \(K\) with obvious linear \(\Gamma\)-action. The proof is essentially a ``finite'' version of Teitelbaum's construction of a ``residue map''.
scientific article; zbMATH DE number 1763657