Petersson’s trace formula and the Hecke eigenvalues of Hilbert modular forms
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AbstractUsing an explicit relative trace formula, we obtain a Petersson trace formula for holomorphic Hilbert modular forms. Our main result expresses a sum (over a Hecke eigenbasis) of products of Fourier coefficients and Hecke eigenvalues in terms of generalized Kloosterman sums and Bessel functions. As an application we show that the normalized Hecke eigenvalues for a fixed prime p have an asymptotic weighted equidistribution relative to a polynomial times the Sato-Tate measure, as the norm of the level goes to ∞. Mathematics Subject Classification: 11F70, 11F72, 11F30, 11L05 Introduction Let h ∈ Sk(Γ0(N)) be a Hecke eigenform (k even), and for a prime p ∤ N define the normalized Hecke eigenvalue by The Ramanujan-Petersson conjecture asserts that. This is a theorem of Deligne. Because we have assumed trivial central character, the operator Tp is self-adjoint, so its eigenvalues are real numbers, and thus For a fixed non-CM newform h, the Sato-Tate conjecture predicts that the set is equidistributed in [-2, 2] relative to the Sato-Tate measure Taylor has recently proven this in many cases when k = 2 [Ta]. When the prime p is fixed, the normalized eigenvalues of Tp on the space Sk(Γ0(N)) are asymptotically equidistributed relative to the measure as k + N → ∞.
All Author(s) ListKnightly A., Li C.
All Editor(s) Listed. by Bas Edixhoven, Gerard van der Geer and Ben Moonen .
Detailed descriptioned. by Bas Edixhoven, Gerard van der Geer and Ben Moonen .
Pages145 - 188
LanguagesEnglish-United Kingdom

Last updated on 2020-30-05 at 00:54