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Transactions of the American Mathematical Society

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A note on automorphic forms of weight one and weight three


Author: Peter F. Stiller
Journal: Trans. Amer. Math. Soc. 291 (1985), 503-518
MSC: Primary 11F12; Secondary 14D05
DOI: https://doi.org/10.1090/S0002-9947-1985-0800250-2
MathSciNet review: 800250
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Abstract: In this paper the author develops an interesting relationship between classical automorphic forms of weights one and three, and the solutions of certain second order differential equations related to elliptic (modular) surfaces. In particular for a cusp form of weight three, it is shown that the special values of the associated Dirichlet series can be determined from the periods of an inhomogeneous differential equation, or what is the same thing, the monodromy of an associated third order differential equation. Explicit examples are provided for principal congruence subgroups $\Gamma (N)$ with $N \equiv 0 \operatorname {mod} 4$.


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Additional Information

Keywords: Elliptic surface, automorphic form, <IMG WIDTH="15" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$q$">-expansion, <IMG WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$K$">-equations, Gauss-Manin connection, monodromy
Article copyright: © Copyright 1985 American Mathematical Society