We prove integrality of the ratio 〈f, f〉/〈g, g〉 (outside an explicit finite set of primes), where g is an arithmetically normalized holomorphic newform on a Shimura curve, f is a normalized Hecke eigenform on GL(2) with the same Hecke eigenvalues as g and 〈, 〉 denotes the Petersson inner product. The primes dividing this ratio are shown to be closely related to certain level-lowering congruence...

In modern number theory there are famous theorems on the modularity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke’s converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat’s Last Theorem to name a few. In this article in the spirit of the Langlands philosophy we raise the question on the modularity of the GL2-twisted spinor L-function ZG⊗h(...

In this paper we will prove bounds for the fourth power moment in the t aspect over a short interval of automorphic L-functions L(s, g) for GL(2) on the central critical line Re s = 1/2. Here g is a fixed holomorphic or Maass Hecke eigenform for the modular group SL2(Z), or in certain cases, for the Hecke congruence subgroup Γ0(N ) with N > 1. The short interval is from a large K to K + K103/13...

We prove explicit rationality-results for Asai- $L$-functions, $L^S(s,\Pi',{\rm As}^\pm)$, and Rankin-Selberg $L^S(s,\Pi\times\Pi')$, over arbitrary CM-fields $F$, relating critical values to powers of $(2\pi i)$. Besides determining the contribution archimedean zeta-integrals our formulas as concrete i)$, it is one advantages approach, that applies very general non-cuspidal isobaric automorphi...

Abstract In this paper, we prove the Gan-Gross-Prasad conjecture and Ichino-Ikeda for unitary groups $U_{n}\times U_{n+1}$ U n × + 1 in all endoscopic cases. Our main technical innovation is compu...

Introduction 2 1. Preliminaries 6 1.1. Algebraic modular forms 6 1.2. Modular forms over C 9 1.3. p-adic modular forms 11 1.4. Elliptic curves with complex multiplication 12 1.5. Values of modular forms at CM points 14 2. Generalised Heegner cycles 15 2.1. Kuga-Sato varieties 15 2.2. The variety Xr and its cohomology 18 2.3. Definition of the cycles 19 2.4. Relation with Heegner cycles and L-se...

Let F be a number field and K a quadratic algebra over F , i.e., either F × F or a quadratic field extension of F . Denote by G the F -group defined by GL(2)/K. Then, given any cuspidal automorphic representation π of G(AF ), one has (cf. [8], [9]) a transfer to an isobaric automorphic representation Π of GL4(AF ) corresponding to the L-homomorphism LG → LGL(4). Usually, Π is called the Rankin-...

The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, has given explicit formulas for central critical values of a large class Rankin-Selberg tensor products. Although the conjecture is not proved in full generality, there been considerable progress, especially $L$-values form $L(1/2,BC(\pi) \times BC(\pi'))$, where $\pi$ $\pi'$ are cohomological automorphic repres...