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ISSN 1088-6850(e) ISSN 0002-9947(p)

Density results for automorphic forms on Hilbert modular groups II

Author(s): Roelof W. Bruggeman; Roberto J. Miatello
Journal: Trans. Amer. Math. Soc. 362 (2010), 3841-3881.
MSC (2000): Primary 11F30, 11F41, 11F72, 22E30
Posted: 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 , with a discrete subgroup of Hecke type $\Gamma_0(I)$ for a non-zero ideal in the ring of integers of . 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 . The main tool in the derivation is a sum formula of Kuznetsov type (Sum formula for SL over a totally real number field, Theorem 2.1).

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