Introduction 1 2. Special values of the adjoint L-function 4 2.1. Rankin-Selberg integrals for GL n × GL n 4 2.2. An integral representation of L(1, Ad 0 , π) 5 2.3. Ramified calculations 7 3. Whittaker models and automorphic cohomology 8 3.1. The basic set-up to study the cohomology of arithmetic groups 8 3.2. Betti-Whittaker periods 9 3.3. Cohomological pairing and the main theorem on adjoint L-values 11 3.4. On the criticality of the adjoint L-function at s = 1 14 4. Discriminant calculations and cohomological congruences 15 4.1. Some linear algebra 15 4.2. Discriminants and congruence modules 16 4.3. The main theorem on congruences 19 5. The non-vanishing property 20 References 25

No abstract

Abstract. We consider a Hida family of nearly ordinary cusp forms on a quaternion algebra defined over a totally real number field. The aim of this work is to construct a cohomology class with coefficients in a p-adic sheaf over an Iwasawa algebra that can be specialized to cohomology classes attached to classical cusp forms in the given Hida family. Our result extends the work of Greenberg and Stevens on modular symbols attached to ordinary Hida families when the ground field is the field of rational numbers.

We investigate the pair correlation statistics for sequences arising from Hecke eigenvalues with respect to spaces of primitive modular cusp forms. We derive the average pair correlation function of Hecke angles lying in small subintervals of [0, 1]. The averaging is done over non-CM newforms of weight k with respect to Γ0(N ). We also derive similar statistics for Hilbert modular forms and modular forms on hyperbolic 3-spaces.

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.