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Transactions of the American Mathematical Society

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Density results for automorphic forms on Hilbert modular groups II

Authors: Roelof W. Bruggeman and Roberto J. Miatello
Journal: Trans. Amer. Math. Soc. 362 (2010), 3841-3881
MSC (2000): Primary 11F30, 11F41, 11F72, 22E30
Published electronically: February 24, 2010
MathSciNet review: 2601612
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Abstract: We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for $ \mathrm{SL}_2$ over a totally real number field $ F$, with a discrete subgroup of Hecke type $ \Gamma_0(I)$ for a non-zero ideal $ I$ in the ring of integers of $ F$. The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many cuspidal automorphic representations with multi-eigenvalues in various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips (see §1.2.4-1.2.13) and products of prescribed small intervals for all but one of the infinite places of $ F$. The main tool in the derivation is a sum formula of Kuznetsov type (Sum formula for SL$ _2$ over a totally real number field, Theorem 2.1).

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Additional Information

Roelof W. Bruggeman
Affiliation: Mathematisch Instituut, Universiteit Utrecht, Postbus 80010, NL-3508 TA Utrecht, Nederland

Roberto J. Miatello
Affiliation: FaMAF-CIEM, Universidad Nacional de Córdoba, Córdoba 5000, Argentina

Received by editor(s): October 17, 2008
Published electronically: February 24, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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