A product expansion for the discriminant function of Drinfeld modules of rank two. (English) Zbl 0572.10021

One of the most beautiful aspects of the classical theory of elliptic modular forms is the product formula for the form \(\Delta\), \(\Delta =(2\pi i)^{12} q \prod (1-q^ n)^{24}\). Let \(A={\mathbb F}_ q(T)\). Associated to \(\text{GL}(2,A)\) one has a theory of modular forms on the rigid analytic space \(\Omega^ 2\) complete with expansions at cusps, etc. On \(\Omega^ 2\) it was known that there was a cusp form of weight \((q^ 2- 1)\), which was never zero on \(\Omega^ 2\) and which was the unique normalized cusp form of that weight; this form was also denoted “\(\Delta\) ”.
In this paper the author establishes the wonderful fact that this \(\Delta\) also has a product expansion associated to it. Moreover, the terms of this expansion are related to division values (and a “Fermat equation”) in a manner exactly similar to the above classical formula.


11F52 Modular forms associated to Drinfel’d modules
11G09 Drinfel’d modules; higher-dimensional motives, etc.
14G15 Finite ground fields in algebraic geometry
14M99 Special varieties
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