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On the Drinfeld discriminant function. (English) Zbl 0930.11031

Let \(\mathbb{F}_q\) be a field with \(q\) elements and \(K=\mathbb{F}_q(T)\). Denote by \(\infty\) the place \(1/T\) of \(K\) and by \(K_\infty=\mathbb{F}_q((1/T))\) the completion of \(K\) at \(\infty\). Let \(\Delta\) be the Drinfeld discriminant function. It is a modular form of weight \(q^2-1\) and type 0, defined on the Drinfeld upper half plane \(\Omega=\mathbb{P}^1_C-\mathbb{P}^1_C(K_\infty)\), where \(C\) is the completion of an algebraic closure of \(K_\infty\), and with values in \(C\). The main results of this paper give the behavior of the functions \(|\Delta|\), \(r(\Delta)\), \(|g|\), \(r(g)\dots(g\) is a modular form, which with \(\Delta\) generates the algebra of modular forms of type 0: D. Goss). Here, \(|\cdot|\) is the absolute value of \(K_\infty\) and the function \(r\) can be viewed as a logarithmic derivative (it has been first introduced by M. van der Put [Ann. Fac. Sci. Toulouse (6), Math. 1, 399-438 (1992; Zbl 0789.14020)]).
Applications of the method consist in determining “roots” of \(\Delta/\Delta_n\) \((\Delta_n(z)=\Delta(nz)\), \(n\in\mathbb{F}_q[T]-\{0\})\) in the function field of the modular curve \(X_0(n)\), for \(n\) arbitrary (the case \(n\) prime has been studied earlier by the author [Compos. Math. 57, 219-236 (1986; Zbl 0599.14032)]. Finally, connections with the cuspidal divisor of \(X_0(n)\) are illustrated by examples.

MSC:

11F52 Modular forms associated to Drinfel’d modules
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11G18 Arithmetic aspects of modular and Shimura varieties
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