Estimates of automorphic cusp forms over quaternion algebras

Abstract

In this article, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [3] and [9] to higher dimensions.The Bergman kernel, which is the reproducing kernel for L 2 -holomorphic functions defined on a domain in C n has been extensively studied in complex analysis. The generalization of the Bergman kernel to complex manifolds as the reproducing kernel for the space of global holomorphic sections of a vector bundle carries the information on the algebraic and geometric structures of the underlying manifolds.Automorphic forms defined over quaternion algebras are global sections of a holomorphic vector bundle. In this article, we derive asymptotic estimates of the Bergman kernel associated to cuspidal holomorphic vector-valued modular forms of large weight defined over quaternion algebras. We prove an average version of the holomorphic QUE conjecture, when the associated complex analytic space is compact. We also derive quantitative estimates of the Bergman kernel associated to classical Hilbert modular cusp forms.In the first half of this article, utilizing certain results from geometric analysis, we derive asymptotic estimates of the Bergman kernel associated to automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture.Jorgenson and Kramer have derived optimal estimates of heat kernels defined over hyperbolic Riemann surfaces of finite volume, and used the estimates to derive optimal estimates of the Bergman kernel associated to cusp forms. In the second half of the article, we extend the heat kernel estimates from [9] to the setting of Hilbert modular varieties, to derive quantitative estimates of the Bergman kernel associated to classical Hilbert modular cusp forms. Our estimates for the Bergman kernel are optimal, and complement the asymptotic estimates derived in the first half of the article.Some of the results of this article are known to experts, and the others are very much expected. So this article serves also as a comprehensive collection of estimates for the Bergman kernel associated to automorphic forms defined over quaternion algebras.

Estimates of Picard modular cusp forms

“…Let Γ ⊂ SU (n, 1), C be a discrete, cofinite subgroup. Following results from [FJK16], [AB18] and [Ma22], the following estimate is expected, which has now assumed the status of a folk-lore conjecture…”

“…In this article, we extend methods from [AM18], [AM20], [Ar16], and [AB18], to derive estimates of Picard modular cusp forms. For n ≥ 2, we prove the asymptotic, qualitative, and quantitative estimates of the Bergman kernel associated to Picard modular cusp forms defined with respect to cocompact subgroups of SU (n, 1), C .…”

Bounds for the Bergman kernel and the sup-norm of holomorphic Siegel cusp forms

“…From (1.5), it is clear that B k (Z, Z) is invariant under Sp(n, Z), is bounded on H n , and that this bound should depend only on n and k. Moreover, there are hints towards a conjecture (see [AB18,JK11,Kra21], even though not mentioned explicitly anywhere) that B k satisfies the following bound.…”

“…Remarks and discussions about proofs. When n = 1, Conjecture 1.2 is known from the works [FJK19,JK04] by the heat-kernel method (also see [AB18] for the case of Hilbert modular forms), but it appears with O ǫ (k 3/2−ǫ ) as the lower bound. It also follows from (1.2) by summing over a basis, with k ±ǫ defects in the upper and lower bound respectively.…”

Global Automorphic Sobolev Theory and The Automorphic Heat Kernel

“…Heat kernels arise in many contexts in mathematics and physics, so many that Jorgenson and Lang have called the heat kernel "ubiquitous" [23]. Automorphic heat kernels are important in physics and number theory: applications include multiloop amplitudes of certain bosonic strings [8], asymptotic formulas for spectra of arithmetic quotients [7,10,14], a relationship between eta invariants of certain manifolds and closed geodesics using the Selberg trace formula [30], systematic construction of zeta-type functions via heat Eisenstein series [23][24][25][26], sup-norm bounds for automorphic forms [2,3,13,21,22], limit formulas for period integrals and a Weyl-type asymptotic law for a counting function for period integrals, via identities that can be considered as a special cases of Jacquet's relative trace formula [36,37], and an average version of the holomorphic QUE conjecture for automorphic cusp forms associated to quaternion algebras [3]. Heat asymptotics on spaces of automorphic forms are of continuing interest; see [32] and its references.…”