Mass equidistribution of Hilbert modular eigenforms. (English) Zbl 1282.11046
Summary: Let \(\mathbb{F}\) be a totally real number field, and let \(f\) traverse a sequence of non-dihedral holomorphic eigen cusp forms on \(\mathrm{GL}_{2}/\mathbb{F}\) of weight \(\left(k_{1},\dots,k_{[\mathbb{F}:\mathbb{Q}]}\right)\), trivial central character and full level. We show that the mass of \(f\) equidistributes on the Hilbert modular variety as \(\max\left(k_{1},\dots,k_{[\mathbb{F}:\mathbb{Q}]}\right) \rightarrow \infty\).
Our result answers affirmatively a natural analog of a conjecture of Z. Rudnick and P. Sarnak [Commun. Math. Phys. 161, No. 1, 195–213 (1994; Zbl 0836.58043)]. Our proof generalizes the argument of R. Holowinsky and K. Soundararajan [Ann. Math. (2) 172, No. 2, 1517–1528 (2010; Zbl 1211.11050)] 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.
Our result answers affirmatively a natural analog of a conjecture of Z. Rudnick and P. Sarnak [Commun. Math. Phys. 161, No. 1, 195–213 (1994; Zbl 0836.58043)]. Our proof generalizes the argument of R. Holowinsky and K. Soundararajan [Ann. Math. (2) 172, No. 2, 1517–1528 (2010; Zbl 1211.11050)] 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.
MSC:
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11M36 | Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) |
| 58J51 | Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity |
Keywords:
modular forms; Hilbert modular forms; number theory; quantum chaos; quantum unique ergodicityReferences:
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