On the transcendence degree of the differential field generated by Siegel modular forms

Details On the transcendence degree of the differential field generated by Siegel modular forms On the transcendence degree of the differential field generated by Siegel modular forms

2000-06-23

arxiv.org/abs/math/0006176v1

J. Reine Angew. Math. 554 (January 2003), 47--68

Mathematics

Number Theory

21 pages, AmSTeX, uses picture.sty for 1 LaTeX picture; submitted for publication

Scientific paper

10.1515/crll.2003.008

It is a classical fact that the elliptic modular functions satisfies an algebraic differential equation of order 3, and none of lower order. We show how this generalizes to Siegel modular functions of arbitrary degree. The key idea is that the partial differential equations they satisfy are governed by Gauss--Manin connections, whose monodromy groups are well-known. Modular theta functions provide a concrete interpretation of our result, and we study their differential properties in detail in the case of degree 2.

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