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This book introduces the method of automorphic descent, providing an explicit inverse map to the (weak) Langlands functorial lift from generic, cuspidal representations on classical groups to general linear groups. The essence of this method is the study of certain Fourier coefficients of Gelfand–Graev type, or of Fourier–Jacobi type when applied to certain residual Eisenstein series. This book contains a complete account of this automorphic descent, with complete, detailed proofs. The book will be of interest to graduate students and mathematicians, who specialize in automorphic forms and in representation theory of reductive groups over local fields. Relatively self-contained, the content of some of the chapters can serve as topics for graduate students seminars.

In this paper we characterize irreducible generic representations of$\\text{SO}_{2n+1}(k)$(where k is a p-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of$\\text{SO}_{2n+1}({\\Bbb A})$(where${\\Bbb A}$is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem); and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of$\\text{SO}_{2n+1}(k)$.

This volume contains the proceedings of the workshop on ``Advances in the Theory of Automorphic Forms and Their $L$-functions'' held in honor of James Cogdell's 60th birthday, held from October 16-25, 2013, at the Erwin Schrodinger Institute (ESI) at the University of Vienna.The workshop and the papers contributed to this volume circle around such topics as the theory of automorphic forms and their $L$-functions, geometry and number theory, covering some of the recent approaches and advances to these subjects. Specifically, the papers cover aspects of representation theory of $p$-adic groups, classification of automorphic representations through their Fourier coefficients and their liftings, $L$-functions for classical groups, special values of $L$-functions, Howe duality, subconvexity for $L$-functions, Kloosterman integrals, arithmetic geometry and cohomology of arithmetic groups, and other important problems on $L$-functions, nodal sets and geometry.

This work studies the local theory for certain Rankin-Selberg convolutions for the standard $L$-function of degree $21n$ of generic representations of $\\textnormal{SO}_{2\\ell +1}(F)\\times \\textnormal{GL}_n(F)$ over a local field $F$. The local integrals converge in a half-plane and continue meromorphically to the whole plane. One main result is the existence of local gamma and $L$-factors. The gamma factor is obtained as a proportionality factor of a functional equation satisfied by the local integrals. In addition, Soudry establishes the multiplicativity of the gamma factor ($1

This volume contains the proceedings of the conference Automorphic Forms and Related Geometry: Assessing the Legacy of I.I. Piatetski-Shapiro, held from April 23-27, 2012, at Yale University, New Haven, CT. Ilya I. Piatetski-Shapiro, who passed away on 21 February 2009, was a leading figure in the theory of automorphic forms. The conference attempted both to summarize and consolidate the progress that was made during Piatetski-Shapiro's lifetime by him and a substantial group of his co-workers, and to promote future work by identifying fruitful directions of further investigation. It was organized around several themes that reflected Piatetski-Shapiro's main foci of work and that have promise for future development: functoriality and converse theorems; local and global L -functions and their periods; p -adic L -functions and arithmetic geometry; complex geometry; and analytic number theory. In each area, there were talks to review the current state of affairs with special attention to Piatetski-Shapiro's contributions, and other talks to report on current work and to outline promising avenues for continued progress. The contents of this volume reflect most of the talks that were presented at the conference as well as a few additional contributions. They all represent various aspects of the legacy of Piatetski-Shapiro.

This paper is a continuation of our previous work (D. Jiang and D. Soudry, On the genericity of cuspidal automorphic forms on${\\rm SO}_{2n+1}$, J. reine angew. Math., to appear). We extend Moeglin's results (C. Moeglin, J. Lie Theory 7 (1997), 201–229, 231–238) from the even orthogonal groups to old orthogonal groups and complete our proof of the CAP conjecture for irreducible cuspidal automorphic representations of $\\mathrm{SO}_{2n+1}(\\mathbb{A})$ with special Bessel models. We also give a characterization of the vanishing of the central value of the standard $L$-function of $\\mathrm{SO}_{2n+1}(\\mathbb{A})$ in terms of theta correspondence. As a result, we obtain the weak Langlands functorial transfer from $\\mathrm{SO}_{2n+1}(\\mathbb{A})$ to $\\mathrm{GL}_{2n}(\\mathbb{A})$ for irreducible cuspidal automorphic representations of $\\mathrm{SO}_{2n+1}(\\mathbb{A})$ with special Bessel models.

This book is the first of two volumes, which represent leading themes of current research in automorphic forms and representation theory of reductive groups over local fields. Articles in this volume mainly represent global aspects of automorphic forms. Among the topics are the trace formula; functoriality; representations of reductive groups over local fields; the relative trace formula and periods of automorphic forms; Rankin - Selberg convolutions and L-functions; and, p-adic L-functions. The articles are written by leading researchers in the field, and bring the reader, advanced graduate students and researchers alike, to the frontline of the vigorous research in these deep, vital topics. The companion volume (\"\"Contemporary Mathematics, Volume 489\"\") is devoted to local aspects of automorphic forms.