The Ramanujan Journal

, Volume 27, Issue 2, pp 235–284

Mass equidistribution of Hilbert modular eigenforms



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.


Modular forms Hilbert modular forms Number theory Quantum chaos Quantum unique ergodicity 

Mathematics Subject Classification (2000)

58J51 11M36 


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.PasadenaUSA

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