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Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula

Authors: Marco Matone and Roberto Volpato
Journal: Proc. Amer. Math. Soc. 141 (2013), 2575-2587
MSC (2010): Primary 14H42; Secondary 14H40, 14H55
Published electronically: November 14, 2012
MathSciNet review: 3056547
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Abstract: Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for $ g\geq 4$, a new class of vector-valued modular forms, defined on the Teichmüller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus $ 4$ is proportional to the square root of the products of Thetanullwerte $ \chi _{68}$, which is a proof of the recently rediscovered Klein ``amazing formula''. Furthermore, it turns out that the coefficients of such a quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian locus, implying new theta relations involving the latter, $ \chi _{68}$ and the theta series corresponding to the even unimodular lattices $ E_8\oplus E_8$ and $ D_{16}^+$. We also find, for $ g=4$, a functional relation between the singular component of the theta divisor and the Riemann period matrix.

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Additional Information

Marco Matone
Affiliation: Dipartimento di Fisica “G. Galilei” and Istituto Nazionale di Fisica Nucleare, Università di Padova, Via Marzolo, 8, 35131 Padova, Italy

Roberto Volpato
Affiliation: Institut für Theoretische Physik, ETH Zurich, CH-8093 Zürich, Switzerland
Address at time of publication: Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institute, 14476 Potsdam, Germany

Received by editor(s): June 26, 2011
Received by editor(s) in revised form: October 27, 2011
Published electronically: November 14, 2012
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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