On the Random Wave Conjecture for Dihedral Maaß Forms

Abstract

We prove two results on arithmetic quantum chaos for dihedral Maaß forms, both of which are manifestations of Berry's random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level 1 forms, these results were previously known for Eisenstein series and conditionally on the generalised Lindelöf hypothesis for Hecke-Maaß eigenforms. A key aspect of the proofs is bounds for certain mixed moments of L-functions that imply hybrid subconvexity.

On the regularized $L^4$-norm for Eisenstein series in the level aspect, Part II

“…The left hand side can be regarded as the fourth moment of E under proper rescaling; the two cases on the right hand side correspond to two models of the Gaussian Moments Conjecture. When T = 0 and χ is quadratic, there exists a complex scalar ǫ, such that ǫE is real-valued, which is similar to classical Eisenstein series [7,8] and dihedral Maaß forms [10] in the t-aspect. So, we expect their moments to behave like a real random wave in the N -aspect.…”

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

“…Suppose also that where and denote . Then is the holomorphic newform 13 of weight , level , and character given by the sum over ideals as We can write down a similar formula when is real; see Appendix A.1 of [HK20]. In this case the hypothesis implies that is a weight 0 Maass form.…”

“…This was generalized in the representation-theoretic language by Shalika and Tanaka [ST69]; see also [HK91, § 13] for a more modern treatment. For an explicit formula for under certain assumptions, see also page 61 of [IK04] for the holomorphic case and Appendix A.1 of [HK20] for the Maass case. The automorphic representation corresponding to is precisely the global automorphic induction of from to .…”

“…We can write down a similar formula when K is real; see Appendix A.1 of [HK20]. In this case the hypothesis χ| A × = 1 implies that θ χ is a weight 0 Maass form.…”

Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications

“…There exist two versions of spectral reciprocities: GL 4 × GL 2 GL 4 × GL 2 reciprocity and GL 2 × GL 2 GL 3 × GL 2 reciprocity. The former one involves work of Andersen-Kıral [1], Blomer-Li-Miller [12], Blomer-Khan [10,11], Humphries-Khan [32], Kuznetsov [45], Nunes [57], and Zacharias [69]. The latter involves work of Blomer et al [9], Nelson [56], Petrow [60], Petrow-Young [61,62], Wu [66], and Young [67].…”