Mass equidistribution of Hilbert modular eigenforms
Identifieur interne : 000286 ( Main/Exploration ); précédent : 000285; suivant : 000287Mass equidistribution of Hilbert modular eigenforms
Auteurs : Paul D. Nelson [États-Unis]Source :
- The Ramanujan Journal [ 1382-4090 ] ; 2012-02-01.
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Abstract
Abstract: Let $\mathbb{F}$ be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on $\operatorname{GL}_{2}/\mathbb{F}$ of weight $(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]})$ , trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as $\max(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}) \rightarrow \infty$ . Our result answers affirmatively a natural analog of a conjecture of Rudnick and Sarnak (Commun. Math. Phys. 161(1), 195–213, 1994). Our proof generalizes the argument of Holowinsky–Soundararajan (Ann. Math. 172(2), 1517–1528, 2010) who established the case $\mathbb{F} = \mathbb{Q}$ . 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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DOI: 10.1007/s11139-011-9319-9
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