The subject of this thesis is the theory of nonholomorphic modular forms of non-integral weight, and its applications to arithmetical functions involving Dedekind sums and Kloosterman sums. As was discovered by Andre Weil, automorphic forms of non-integral weight correspond to invariant funtions on Metaplectic groups. We thus give an explicit description of Meptaplectic groups corresponding to ...

We characterize the space of the so-called planar mixed automorphic forms of type (ν, μ)with respect to an equivariant pair (ρ, τ) as the image, by an appropriate transform, of the usual (Landau) automorphic forms involving special multiplier.

In this paper, we obtain the existence of almost automorphic solutions to some classes of nonautonomous higher order abstract differential equations with Stepanov almost automorphic forcing terms. A few illustrative examples are discussed at the very end of the paper.

We consider the higher integral moments for automorphic L-functions in short intervals and give a proof for the conjecture of Conrey et al. under Generalized Riemann Hypothesis for automorphic L-function.

PREFACE This article follows the format of five lectures that we gave on automorphic Lfunctions. The lectures were intended to be a brief introduction for number theorists to some of the main ideas in the subject. Three of the lectures concerned the general properties of automorphic L-functions, with particular reference to questions of spectral decomposition. We have grouped these together as ...

We introduce the notion of automorphic symbol generalizing the classical modular symbol and use it to attach very general p-adic L-functions to nearly ordinary Hilbert automorphic forms. Then we establish an exact control theorem for the p-adically completed cohomology of a Hilbert modular variety localized at a suitable nearly ordinary maximal ideal of the Hecke algebra. We also show its freen...

We compute the arithmetic L-invariants (of Greenberg–Benois) of twists of symmetric powers of p-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on p-adic families of automorphic forms over symplectic and unitary groups. Combining thes...

In this paper, we propose a conjectural identity between the Fourier–Jacobi periods on symplectic groups and the central value of certain Rankin–Selberg L-functions. This identity can be viewed as a refinement to the global Gan–Gross–Prasad conjecture for Sp(2n)×Mp(2m). To support this conjectural identity, we show that when n = m and n = m±1, it can be deduced from the Ichino–Ikeda’s conjectur...

The goal of this paper is to provide a complete and refined study the standard L-functions $$L(\pi ,{\text {Std}},s)$$ for certain non-generic cuspidal automorphic representations $$\pi $$ $$G_2({\mathbb {A}})$$ . For representation that corresponds modular form $$\varphi level one even weight on $$G_2$$ , we explicitly define completed L-function, $$\Lambda (\pi Assuming Fourier coefficient no...

For s ∈ P(Q) \ {0, 1}, we study the compatible system of Galois representations (ρ l (3) : Gal(Q̄/Q) → G2(Ql))l introduced in [4], where G2(Ql) ≤ GL7(Ql) is the simple group of type G2. We prove that, under some mild condition on s, the image of the Tate twisted Galois representation ρ l (3) coincides with G2(Zl) for all l up to a set of primes having density zero, and we show that ρ l is potent...