## Mass equidistribution on the torus in the depth aspect.(English)Zbl 1476.58032

It is known that the quantum unique ergodicity (QUE) propery in the arithmetic setting is a special case of the conjecture by Z. Rudnick and P. Sarnak [Commun. Math. Phys. 161, No. 1, 195–213 (1994; Zbl 0836.58043)] regarding the asymptotic behavior of the mass measure associated to a normalized holomorphic modular form or Maass form.
The aim of the present paper is to study the equidistribution of the restriction of the mass of automorphic newforms to a nonsplit torus in the depth aspect. To this direction the author manages to prove a better result than the already known results on the similar problem in the eigenvalue aspect by using a relatively elementary method which makes use of the known effective QUE result in the depth aspect.

### MSC:

 58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity 37A30 Ergodic theorems, spectral theory, Markov operators 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 22E50 Representations of Lie and linear algebraic groups over local fields 37A46 Relations between ergodic theory and harmonic analysis

Zbl 0836.58043
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