On Central Critical Values of the Degree Four $L$-Functions for GSp(4): The Fundamental Lemma. III
On Central Critical Values of the Degree Four $L$-Functions for GSp(4): The Fundamental Lemma. III

Author: Masaaki Furusawa

Publisher: American Mathematical Soc.

Published: 2013-08-23

Total Pages: 150

ISBN-13: 0821887424

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Some time ago, the first and third authors proposed two relative trace formulas to prove generalizations of Böcherer's conjecture on the central critical values of the degree four -functions for , and proved the relevant fundamental lemmas. Recently, the first and second authors proposed an alternative third relative trace formula to approach the same problem and proved the relevant fundamental lemma. In this paper the authors extend the latter fundamental lemma and the first of the former fundamental lemmas to the full Hecke algebra. The fundamental lemma is an equality of two local relative orbital integrals. In order to show that they are equal, the authors compute them explicitly for certain bases of the Hecke algebra and deduce the matching.

On Central Critical Values of the Degree Four $L$-functions for $\mathrm {GSp}(4)$: The Fundamental Lemma
On Central Critical Values of the Degree Four $L$-functions for $\mathrm {GSp}(4)$: The Fundamental Lemma

Author: Masaaki Furusawa

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 158

ISBN-13: 0821833286

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Proves two equalities of local Kloosterman integrals on $\mathrm{GSp}\left(4\right)$, the group of $4$ by $4$ symplectic similitude matrices. This book conjectures that both of Jacquet's relative trace formulas for the central critical values of the $L$-functions for $\mathrm{g1}\left(2\right)$ in [{J1}] and [{J2}].

Transfer of Siegel Cusp Forms of Degree 2
Transfer of Siegel Cusp Forms of Degree 2

Author: Ameya Pitale

Publisher: American Mathematical Soc.

Published: 2014-09-29

Total Pages: 120

ISBN-13: 0821898566

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Let be the automorphic representation of generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and be an arbitrary cuspidal, automorphic representation of . Using Furusawa's integral representation for combined with a pullback formula involving the unitary group , the authors prove that the -functions are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations have a functorial lifting to a cuspidal representation of . Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of to a cuspidal representation of . As an application, the authors obtain analytic properties of various -functions related to full level Siegel cusp forms. They also obtain special value results for and

Recent Advances in Hodge Theory
Recent Advances in Hodge Theory

Author: Matt Kerr

Publisher: Cambridge University Press

Published: 2016-02-04

Total Pages: 533

ISBN-13: 1316531392

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In its simplest form, Hodge theory is the study of periods – integrals of algebraic differential forms which arise in the study of complex geometry and moduli, number theory and physics. Organized around the basic concepts of variations of Hodge structure and period maps, this volume draws together new developments in deformation theory, mirror symmetry, Galois representations, iterated integrals, algebraic cycles and the Hodge conjecture. Its mixture of high-quality expository and research articles make it a useful resource for graduate students and seasoned researchers alike.

A Generating Function Approach to the Enumeration of Matrices in Classical Groups over Finite Fields
A Generating Function Approach to the Enumeration of Matrices in Classical Groups over Finite Fields

Author: Jason Fulman

Publisher: American Mathematical Soc.

Published: 2005

Total Pages: 104

ISBN-13: 0821837060

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Generating function techniques are used to study the probability that an element of a classical group defined over a finite field is separable, cyclic, semisimple or regular. The limits of these probabilities as the dimension tends to infinity are calculated in all cases, and exponential convergence to the limit is proved. These results complement and extend earlier results of the authors, G. E. Wall, and Guralnick & Lubeck.

Positive Definite Functions on Infinite-Dimensional Convex Cones
Positive Definite Functions on Infinite-Dimensional Convex Cones

Author: Helge Glöckner

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 150

ISBN-13: 0821832565

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A memoir that studies positive definite functions on convex subsets of finite- or infinite-dimensional vector spaces. It studies representations of convex cones by positive operators on Hilbert spaces. It also studies the interplay between positive definite functions and representations of convex cones.