Berg The Fourier-Analytic Proof of Quadratic Reciprocity

1. Auflage 2011
ISBN: 978-1-118-03119-3
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 118 Seiten, E-Book

Reihe: Wiley Series in Pure and Applied Mathematics

A unique synthesis of the three existing Fourier-analytictreatments of quadratic reciprocity.
The relative quadratic case was first settled by Hecke in 1923,then recast by Weil in 1964 into the language of unitary grouprepresentations. The analytic proof of the general n-th order caseis still an open problem today, going back to the end of Hecke'sfamous treatise of 1923. The Fourier-Analytic Proof of QuadraticReciprocity provides number theorists interested in analyticmethods applied to reciprocity laws with a unique opportunity toexplore the works of Hecke, Weil, and Kubota.
This work brings together for the first time in a single volume thethree existing formulations of the Fourier-analytic proof ofquadratic reciprocity. It shows how Weil's groundbreakingrepresentation-theoretic treatment is in fact equivalent to Hecke'sclassical approach, then goes a step further, presenting Kubota'salgebraic reformulation of the Hecke-Weil proof. Extensivecommutative diagrams for comparing the Weil and Kubotaarchitectures are also featured.
The author clearly demonstrates the value of the analytic approach,incorporating some of the most powerful tools of modern numbertheory, including adèles, metaplectric groups, andrepresentations. Finally, he points out that the critical commonfactor among the three proofs is Poisson summation, whosegeneralization may ultimately provide the resolution for Hecke'sopen problem.

Autoren/Hrsg.

Berg, Michael C.

Inhaltsverzeichnis

Hecke's Proof of Quadratic Reciprocity.
Two Equivalent Forms of Quadratic Reciprocity.
The Stone-Von Neumann Theorem.
Weil's "Acta" Paper.
Kubota and Cohomology.
The Algebraic Agreement Between the Formalisms of Weil andKubota.
Hecke's Challenge: General Reciprocity and Fourier Analysis on theMarch.
Bibliography.
Index.

Über Autor(innen)

MICHAEL C. BERG, PhD, is Professor of Mathematics at Loyola Marymount University, Los Angeles, California.