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Linear independence of Poincaré series of exponential type via non-analytic methods


Authors: Siegfried Böcherer and Soumya Das
Journal: Trans. Amer. Math. Soc. 367 (2015), 1329-1345
MSC (2010): Primary 11F30; Secondary 11F46
Published electronically: July 25, 2014
MathSciNet review: 3280046
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Abstract: Given a finite set $ \{T\}$ of symmetric, positive definite, half-integral $ n$ by $ n$ matrices over $ \mathbf {Z}$ which are inequivalent under the action of $ \textup {GL}(n, \mathbf {Z})$, we show that the corresponding set of Poincaré series $ \{ P_k^n(T) \}$ attached to them are linearly independent for weights $ k$ in infinitely many arithmetic progressions. We also give a quite explicit description of those arithmetic progressions for all even degrees, when the matrices $ T$ have no improper automorphisms and their level is an odd prime. Our main tools are theta series with simple harmonic polynomials as coefficients and techniques familiar from the theory of modular forms mod $ p$.


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Additional Information

Siegfried Böcherer
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan
Email: boecherer@t-online.de

Soumya Das
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India
Email: somu@math.tifr.res.in, soumya.u2k@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2014-06204-1
Keywords: Congruences, Siegel modular forms, Fourier coefficients, Poincar\'e series
Received by editor(s): September 30, 2012
Received by editor(s) in revised form: May 25, 2013
Published electronically: July 25, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.