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**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.

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.

Reviewer: David Goss (Columbus/Ohio)

### 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 |

### Keywords:

Drinfeld discriminant; cyclotomic polynomials; Drinfeld module; Carlitz module; elliptic modular forms; rigid analytic space; product expansion; division values
Full Text:
DOI

### References:

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