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Abstract. A basic idea of Dirichlet is to study a collection of interesting quantities {an}n≥1 by means of its Dirichlet series in a complex variable w: n≥1 ann−w. In this paper we examine this construction when the quantities an are themselves infinite series in a second complex variable s, arising from number theory or representation theory. We survey a body of recent work on such series and present a new conjecture concerning them. The object of this paper is to give a survey of a body of recent work applying methods from automorphic forms to problems in number theory. Generalizing this work, we shall also formulate a new conjecture concerning Langlands L-functions, which implies such results as the Lindelöf Hypothesis in twisted aspect. This

2. A review of classical modular forms 3. A review of Maass forms 4. Hecke L-functions

Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, χ a primitive character of conductor q, and s a point on the critical line ℜs = 1. It is proved that 2 L(g ⊗ χ, s) ≪ε,g,s q 1 2 − 1 8 (1−2θ)+ε, where ε> 0 is arbitrary and θ = 7 is the current known approximation towards the Ramanujan– 64 Petersson conjecture (which would allow θ = 0); moreover, the dependence on s and all the parameters of g is polynomial. This result is an analog of Burgess ’ classical subconvex bound for Dirichlet L-functions. In Appendix 2 the above result is combined with a theorem of Waldspurger and the adelic calculations of Baruch–Mao to yield an improved uniform upper bound for the Fourier coefficients of holomorphic half-integral weight cusp forms.

The best bound known for character sums was given independently by G. Pólya and I.M. Vinogradov in 1918 (see [4], p.135-137). For any non-principal Dirichlet character χ (mod q) we let M(χ): = max

This article gives an introduction to the multiple Dirichlet series arising from sums of twisted automorphic L-functions. We begin by explaining how such series arise from Rankin-Selberg constructions. Then more recent work, using Hartogs ’ continuation principle as extended by Bochner in place of such constructions, is described. Applications to the nonvanishing of L-functions and to other problems are also discussed, and a multiple Dirichlet series over a function field is computed in detail.

Let M be a complete Riemannian manifold with finite volume which we initially assume to be compact. Then since M is compact, L 2 (M) is spanned by the eigenfunctions of the Laplacian ∆ on M. Many interesting questions can be asked about these eigenfunctions

It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D ? 3705, there is always at least one prime p ! p D=2 such that the Kronecker symbol (D=p) = \Gamma1. 1991 Mathematics Subject Classification 11R11, 11Y40 The first author is a Presidential Faculty Fellow. His research is partiallly supported by the NSF. The research of the second two authors is partially supported by NSERC of Canada 1 1 Introduction Let D be the fundamental discriminant of a real quadratic field and let S = f5; 8; 12; 13; 17; 24; 28; 33; 40; 57; 60; 73; 76; 88; 97; 105; 124; 129; 136; 145; 156; 184; 204; 249; 280; 316; 345; 364; 385; 424; 456; 520; 609; 616; 924; 940; 984; 1065; 1596; 2044; 2244; 3705g: At the end of Chapter 6 of [5], the second author made the following conjecture. Conjecture. The values of D for which the least prime p such that the Kronecker symbol (D=p) = \Gamma1 satisfies p ? p D=2 are precisely those in S. He also veri...

Let E be an elliptic curve defined over the rationals. For any prime p of good reduction, let Ep be the elliptic curve over Fp obtained by reducing E mod p. Let ap(E) be the trace of the Frobenius morphism of Ep. Then, Hasse proved that #E(Fp) = p + 1 − ap(E) with |ap(E) | ≤ 2 √ p. The case ap(E) = 0 corresponds to supersingular reduction mod p.

In this paper we report some progress towards a conjecture of Rudnick and Sarnak regarding eigenfunctions of the Laplacian ∆ on a compact manifold M for certain special arithmetic surfaces M of constant curvature (see below for definitions):