# Bounds on Maass spectra from holomorphic forms

I will discuss new constraints on the spectra of Maass forms on compact hyperbolic 2-orbifolds. The constraints arise from integrals of products of four functions in discrete series representations realized in L2(Γ∖G), where Γ is a cocompact lattice in G=PSL2(ℝ). Such integrals can be expanded using the spectral data of either holomorphic forms, or of Maass forms. The resulting spectral identities implicitly determine the Maass spectrum in terms of the spectrum and triple integrals of holomorphic forms. I will use these identities to prove upper bounds on the first positive Maass eigenvalue λ1 for all hyperbolic 2-orbifolds of a fixed genus. The bounds are often nearly saturated by known surfaces. For example, the bound for genus two is λ1 <3.838898, while the Bolza surface has λ1≈3.838887. I will also discuss the image of X↦λ1(X) when X ranges over all hyperbolic 2-orbifolds. The method has extensions to non-compact surfaces, and to higher-dimensional hyperbolic manifolds. The work was inspired by recent developments in the conformal bootstrap, which I will sketch. It was pointed out to me that closely related methods have been used to prove bounds on L-functions.

The talk will be based on https://arxiv.org/pdf/2111.12716.pdf which is joint work with P. Kravchuk and S. Pal.

### Affiliation

Member, School of Natural Sciences, IAS