tailieunhanh - Đề tài " Automorphic distributions, Lfunctions, and Voronoi summation for GL(3) "

Introduction In 1903 Voronoi [42] postulated the existence of explicit formulas for sums of the form () n≥1 an f (n) , for any “arithmetically interesting” sequence of coefficients (an )n≥1 and every f in a large class of test functions, including characteristic functions of bounded intervals. He actually established such a formula when an = d(n) is the number of positive divisors of n [43]. He also asserted a formula for () an = #{(a, b) ∈ Z2 | Q(a, b) = n} , where Q denotes a positive definite integral quadratic form [44]; . | Annals of Mathematics Automorphic distributions L-functions and Voronoi summation for GL 3 By Stephen D. Miller and Wilfried Schmidn Annals of Mathematics 164 2006 423 488 Automorphic distributions L-functions and Voronoi summation for GL 3 By Stephen D. Miller and Wilfried Schmid 1. Introduction In 1903 Voronoi 42 postulated the existence of explicit formulas for sums of the form an f n n 1 for any arithmetically interesting sequence of coefficients an n 1 and every f in a large class of test functions including characteristic functions of bounded intervals. He actually established such a formula when an d n is the number of positive divisors of n 43 . He also asserted a formula for an a b E Z2 I Q a b n where Q denotes a positive definite integral quadratic form 44 Sierpinski 40 and Hardy 16 later proved the formula rigorously. As Voronoi pointed out this formula implies the bound I a b E Z2 I a2 b2 x nx I O x3 3 for the error term in Gauss classical circle problem improving greatly on Gauss own bound O x1 2 . Though Voronoi originally deduced his formulas from Poisson summation in R2 applied to appropriately chosen test functions one nowadays views his formulas as identities involving the Fourier coefficients of modular forms on GL 2 . modular forms on the complex upper half plane. A discussion of the Voronoi summation formula and its history can be found in our expository paper 28 . The main result of this paper is a generalization of the Voronoi summation formula to GL 3 Z -automorphic representations of GL 3 R . Our technique is quite general we plan to extend the formula to the case of GL n Q GL n A in the future. The arguments make heavy use of representation theory. To illustrate the main idea we begin by deriving the well-known generalization Supported by NSF grant DMS-0122799 and an NSF post-doctoral fellowship. Supported in part by NSF grant DMS-0070714. 424 STEPHEN D. MILLER AND WILFRIED SCHMID of the Voronoi summation formula to .