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Equidistribution in Shrinking Sets and L^4-Norm Bounds for Automorphic Forms. (arXiv:1705.05488v1 [math.NT])

来源于:arXiv
We study two closely related problems stemming from the random wave conjecture for Maa\ss{} forms. The first problem is bounding the $L^4$-norm of a Maa\ss{} 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 $\Gamma \backslash \mathbb{H}$, quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the Planck scale. This result is conditional on the generalised Lindel\"{o}f hypothesis for Maa\ss{} eigenforms but is unconditional for Eisenstein series. We also show that equidistribution for Maa\ss{} 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 clas 查看全文>>