Equidistribution in shrinking sets and \(L^4\)-norm bounds for automorphic forms. (English) Zbl 1448.11084
Summary: We study two closely related problems stemming from the random wave conjecture for Maaß forms. The first problem is bounding the \(L^4\)-norm of a Maaß form in the large eigenvalue limit; we complete the work of Spinu to show that the \(L^4\)-norm of an Eisenstein series \(E(z,1/2+it_g)\) restricted to compact sets is bounded by \(\sqrt{\log t_g}\). The second problem is quantum unique ergodicity in shrinking sets; we show that by averaging over the centre of hyperbolic balls in \(\Gamma \setminus \mathbb {H}\), quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the Planck scale. This result is conditional on the generalised Lindelöf hypothesis for Hecke-Maaß eigenforms but is unconditional for Eisenstein series. We also show that equidistribution for Hecke-Maaß eigenforms need not hold at or below the Planck scale. Finally, we prove similar equidistribution results in shrinking sets for Heegner points and closed geodesics associated to ideal classes of quadratic fields.
MSC:
| 11F12 | Automorphic forms, one variable |
| 58J51 | Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity |
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