Mass equidistribution of Hilbert modular eigenforms

Details Mass equidistribution of Hilbert modular eigenforms Mass equidistribution of Hilbert modular eigenforms

2010-11-04

arxiv.org/abs/1011.1291v2

Ramanujan Journal, volume 27, issue 2, year 2012, pages 235-284

Mathematics

Number Theory

40 pages; typos corrected, nearly accepted form

Scientific paper

10.1007/s11139-011-9319-9

Let F be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on GL(2)/F of weight (k_1,...,k_n), trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as max(k_1,...,k_n) tends to infinity. Our result answers affirmatively a natural analogue of a conjecture of Rudnick and Sarnak (1994). Our proof generalizes the argument of Holowinsky-Soundararajan (2008) who established the case F = Q. The essential difficulty in doing so is to adapt Holowinsky's bounds for the Weyl periods of the equidistribution problem in terms of manageable shifted convolution sums of Fourier coefficients to the case of a number field with nontrivial unit group.

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