Growth order and congruences of coefficients of the Drinfeld discriminant function
The analogue of the discriminant of an elliptic curve in the arithmetic of function fields is given by the so-called Drinfeld discriminant function \(\Delta\), which is the leading coefficient of a rank two Drinfeld module over \(A={\mathbb F}_q[T]\), which can be seen as a modular form on \(GL(2,A)\) depending on a rank-two lattice in an algebraically closed complete field containing the completion of the quotient field of \(A\) w.r.t.\ the valuation \(-\deg\). As such, it admits a series expansion in the parameter \(t\) at the cusp, and this paper contains some properties of the (largely mysterious) expansion coefficients of \(\Delta\). It turns out that \(\Delta\) admits a \((q-1)\)-th root \(h\); and \(h/t\) admits a \((q^2-1)\)-th root \(U\). Let \([i]:=T^{q^i}-T\). The main results of this paper are as follows: (a) it is shown that the coefficients \(c_k\) in the \(t\)-expansion of \(U\) have degree \(\leq k\) in \(T\), and the exact coefficient of \(c_k\) in \(T^k\) is determined (in particular, \(\deg(c_k)=k\) occurs infinitely often); (b) \(U\) is ``balanced'', i.e., each \(c_k\) is a polynomial in \([1]\) and the coefficient of \(c_k\) in \([1]^j\) is only non-zero if \(j=k\) mod \(q-1\); (c) The coefficients \(b_k\) of \(-h/t\) in \(T\) satisfy \(b_{k q^d} = b_k\) mod \([d]\). The starting point for the proofs is an expansion of \(\Delta\) as an infinite product previously obtained by the author [J. Number Theory 21, 135-140 (1985; Zbl 0572.10021)]. For (c), Hecke operators are used. The paper finishes with some further conjectures about the coefficients.
scientific article; zbMATH DE number 1375112