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Automorphic distributions, \(L\)-functions, and Voronoi summation for \(\text{GL}(3)\). (English) Zbl 1162.11341

From the text: “In 1903 Voronoi postulated the existence of explicit formulas for sums of the form \(\sum_{n\geq 1}a_nf(n)\) for any ‘arithmetically interesting’ sequence of coefficients \((a_n)_{n\geq 1}\) and every \(f\) in a large class of test functions ... . He actually established ... . He also asserted ... .” “The main result of this paper is a generalization of the Voronoi summation formula to \(\text{GL}(3,\mathbb Z)\)-automorphic representations of \(\text{GL}(3,\mathbb R)\). Our technique is quite general ... . The arguments make heavy use of representation theory. To illustrate the main idea, we begin by deriving the well-known generalization of the Voronoi summation formula to the coefficients of modular forms on \(\text{GL}(2)\) ...” – this occupies the next four pages of the introduction, and the next page hosts the statement of the main result, “an analogue of the \(\text{GL}(2)\) Voronoi summation formula for cusp forms on \(\text{GL}(3)\)”. This summation formula involves “Fourier coefficients of a cuspidal \(\text{GL}(3,\mathbb Z)\)-automorphic representations of \(\text{GL}(3,\mathbb R)\) as in (5.9)”, and twisted values of “a Schwartz function \(f\) which vanishes to infinite order at the origin, or more generally ...”, on one side, and Kloosterman sums and certain integrals in \(f\) on the other. Anyway, “Only very special types of cusp forms on \(\text{GL}(3,\mathbb Z)\backslash \text{GL}(3,\mathbb R)\) have been constructed explicitly; these all come from the” “symmetric square functorial lift of cusp forms on \(\text{SL}(2,\mathbb Z)\backslash H\)”. A discussion of \(L\)-functions follows, as well as statements: “In the past, the problem of converting multiplicative to additive information was the main obstacle to proving a Voronoi summation formula for \(\text{GL}(3)\). Our methods bypass this difficulty entirely by dealing with the automorphic representation directly, without any input from the Hecke action” and “Section 7 concludes with a proof of the \(\text{GL}(3)\) converse theorem of J. A. Shalika [Ann. Math. (2) 100, 171–193 (1974; Zbl 0316.12010)]. Though this theorem has been long known, of course, our arguments provide the first proof for \(\text{GL}(3)\) that can be couched in classical language, i.e., without adèles.”

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11M41 Other Dirichlet series and zeta functions

Citations:

Zbl 0316.12010
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