On the transcendence degree of the differential field generated by Siegel modular forms
Number Theory
2009-02-24 v1 Algebraic Geometry
Abstract
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.
Keywords
Cite
@article{arxiv.math/0006176,
title = {On the transcendence degree of the differential field generated by Siegel modular forms},
author = {Daniel Bertrand and Wadim Zudilin},
journal= {arXiv preprint arXiv:math/0006176},
year = {2009}
}
Comments
21 pages, AmSTeX, uses picture.sty for 1 LaTeX picture; submitted for publication