×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Drinfeld, V.G, Elliptic modules, Math sb., Math. USSR-sb., 23, No. 4, 594-627, (1974), [Russian]; Engl. transl.
[2] Gekeler, E, Drinfeld-moduln und modulare formen über rationalen funktionenkörpern, Bonner math. schriften, (1980) · Zbl 0446.14018
[3] Gekeler, E, Zur arithmetik von Drinfeld-moduln, Math. ann., 262, 167-182, (1983) · Zbl 0536.14028
[4] Goss, D, Von staudt for \(F\)_{q}[T], Duke math. J., 45, 885-910, (1978) · Zbl 0404.12013
[5] Goss, D, Π-adic Eisenstein series for function fields, Comp. math., 41, 3-38, (1980) · Zbl 0422.10020
[6] Goss, D, Modular forms for \(F\)_{r}[T], Crelle’s J., 31, 16-39, (1980) · Zbl 0422.10021
[7] Goss, D, The Algebraist’s upper half-plane, Bull. amer. math. soc., 233, 391-415, (1980) · Zbl 0433.14017
[8] Hayes, D, Explicit class field theory for rational function fields, Trans. amer. math. soc., 189, 77-91, (1974) · Zbl 0292.12018
[9] Lang, S, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.