Abstract Let F be a totally real field, and let $E/F$ CM quadratic extension. We construct p -adic L -function attached to Hida families for the group $\mathrm {GL}_{2/F}\times \mathrm {Res}_{E/F}\mathrm {GL}_{1}$ . It is characterized by an exact interpolation property critical Rankin–Selberg -values, at classical points corresponding representations $\pi \boxtimes \chi $ with weights of $\chi...

Let q be a prime and K = Q( √ −D) be an imaginary quadratic field such that q is inert in K. If q is a prime above q in the Hilbert class field of K, there is a reduction map rq : E``(OK) −→ E``(Fq2) from the set of elliptic curves over Q with complex multiplication by the ring of integers OK to the set of supersingular elliptic curves over Fq2 . We prove a uniform asymptotic formula for the nu...

This is an announcement of results on rank-one Eisenstein cohomology of GLN , with N ≥ 3 an odd integer, and algebraicity theorems for ratios of successive critical values of certain Rankin–Selberg L-functions for GLn ×GLn′ when n is even and n′ is odd. Résumé: Cette note est une annonce de résultats sur la cohomologie d’Eisenstein de rang un de GLN , avec N ≥ 3 un entier impair, et de théorème...

a r t i c l e i n f o a b s t r a c t Let f and g be Maass cusp forms for SL m (Z) and SL m (Z), respectively, with 2 ≤ m ≤ m. Let λ f ×g (n) be the normalized coefficients of L(s, f × g), the Rankin–Selberg L-function for f and g. In this paper the asymptotics of a Voronoi-type summation formula for λ f ×g (n) are derived. As an application estimates are obtained for the smoothly weighted aver...

One way of characterizing Atle Selberg’s mathematical genius is that he had a “golden touch”. In those domains he thought about in depth, he saw further than generations before him, repeatedly uncovering truths lying below the surface. His breakthroughs on long-standing problems were based on imaginative and novel ideas which, once digested, were appreciated as simple and decisive. The impact o...

This paper develops the study by analytic methods of generalized principal series Maass forms on GL(3). These occur as an infinite sequence one-parameter families in two-parameter spectrum GL(3) forms, analogous to relationship between holomorphic modular and spherical cusp GL(2). We develop a Kuznetsov trace formula attached these at each weight use it prove arithmetically-weighted Weyl law, d...

A theorem of W. Kohnen states that the generalized Riemann hypothesis (GRH) holds on an average for holomorphic cusp forms on the upper half plane for the full modular group SL2(Z). In this article we prove a couple of generalizations of this theorem of Kohnen that the GRH holds on an average for holomorphic cusp forms on the upper half plane for arbitrary level, weight and primitive nebentypus...

We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d, 2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Maass forms, and it involves the S...