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The cubic moment of central values of automorphic Lfunctions
 Ann. of Math
, 2000
"... 2. A review of classical modular forms 3. A review of Maass forms 4. Hecke Lfunctions ..."
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Cited by 22 (1 self)
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2. A review of classical modular forms 3. A review of Maass forms 4. Hecke Lfunctions
On some applications of automorphic forms to number theory
 Bull. Amer. Math. Soc. (N.S
, 1996
"... 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 ..."
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Cited by 14 (8 self)
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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 Lfunctions, which implies such results as the Lindelöf Hypothesis in twisted aspect. This
A BURGESSLIKE SUBCONVEX BOUND FOR TWISTED LFUNCTIONS
, 2005
"... 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 ..."
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Cited by 13 (6 self)
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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 Lfunctions. 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 halfintegral weight cusp forms.
Large character sums: Pretentious characters and the PolyaVinogradov theorem
 J. Amer. Math. Soc
"... The best bound known for character sums was given independently by G. Pólya and I.M. Vinogradov in 1918 (see [4], p.135137). For any nonprincipal Dirichlet character χ (mod q) we let M(χ): = max ..."
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Cited by 11 (4 self)
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The best bound known for character sums was given independently by G. Pólya and I.M. Vinogradov in 1918 (see [4], p.135137). For any nonprincipal Dirichlet character χ (mod q) we let M(χ): = max
Multiple Dirichlet Series and Automorphic Forms
 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS
"... This article gives an introduction to the multiple Dirichlet series arising from sums of twisted automorphic Lfunctions. We begin by explaining how such series arise from RankinSelberg constructions. Then more recent work, using Hartogs ’ continuation principle as extended by Bochner in place of s ..."
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Cited by 8 (6 self)
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This article gives an introduction to the multiple Dirichlet series arising from sums of twisted automorphic Lfunctions. We begin by explaining how such series arise from RankinSelberg 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 Lfunctions and to other problems are also discussed, and a multiple Dirichlet series over a function field is computed in detail.
Entropy of quantum limits
 Comm. Math. Phys
"... 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): ..."
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Cited by 8 (3 self)
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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):
An Upper Bound on the Least Inert Prime in a Real Quadratic Field
"... 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 Pre ..."
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Cited by 7 (2 self)
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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...
DISTRIBUTION OF PERIODIC TORUS ORBITS AND DUKE’S THEOREM FOR CUBIC FIELDS
"... We study periodic torus orbits on space of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits; for rank 3 lattices, we show that the equivalence classes become uniformly distributed. This is a cubic analogue of D ..."
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Cited by 3 (3 self)
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We study periodic torus orbits on space of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits; for rank 3 lattices, we show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke’s theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular 5fold SL3(Z)\SL3(R)/SO3. In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL3(Z)\SL3(R)/SO3 of volume ≤ V becomes equidistributed as V → ∞. The proof combines subconvexity estimates, measure classification, and local harmonic analysis.