The FourierAnalytic Proof of Quadratic ReciprocityISBN: 9780471358305
118 pages
March 2000

A unique synthesis of the three existing Fourieranalytic
treatments 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 group representations. The analytic proof of the general nth order case is still an open problem today, going back to the end of Hecke's famous treatise of 1923. The FourierAnalytic Proof of Quadratic Reciprocity provides number theorists interested in analytic methods applied to reciprocity laws with a unique opportunity to explore the works of Hecke, Weil, and Kubota.
This work brings together for the first time in a single volume the three existing formulations of the Fourieranalytic proof of quadratic reciprocity. It shows how Weil's groundbreaking representationtheoretic treatment is in fact equivalent to Hecke's classical approach, then goes a step further, presenting Kubota's algebraic reformulation of the HeckeWeil proof. Extensive commutative diagrams for comparing the Weil and Kubota architectures are also featured.
The author clearly demonstrates the value of the analytic approach, incorporating some of the most powerful tools of modern number theory, including adèles, metaplectric groups, and representations. Finally, he points out that the critical common factor among the three proofs is Poisson summation, whose generalization may ultimately provide the resolution for Hecke's open problem.
The relative quadratic case was first settled by Hecke in 1923, then recast by Weil in 1964 into the language of unitary group representations. The analytic proof of the general nth order case is still an open problem today, going back to the end of Hecke's famous treatise of 1923. The FourierAnalytic Proof of Quadratic Reciprocity provides number theorists interested in analytic methods applied to reciprocity laws with a unique opportunity to explore the works of Hecke, Weil, and Kubota.
This work brings together for the first time in a single volume the three existing formulations of the Fourieranalytic proof of quadratic reciprocity. It shows how Weil's groundbreaking representationtheoretic treatment is in fact equivalent to Hecke's classical approach, then goes a step further, presenting Kubota's algebraic reformulation of the HeckeWeil proof. Extensive commutative diagrams for comparing the Weil and Kubota architectures are also featured.
The author clearly demonstrates the value of the analytic approach, incorporating some of the most powerful tools of modern number theory, including adèles, metaplectric groups, and representations. Finally, he points out that the critical common factor among the three proofs is Poisson summation, whose generalization may ultimately provide the resolution for Hecke's open problem.
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Hecke's Proof of Quadratic Reciprocity.
Two Equivalent Forms of Quadratic Reciprocity.
The StoneVon Neumann Theorem.
Weil's "Acta" Paper.
Kubota and Cohomology.
The Algebraic Agreement Between the Formalisms of Weil and Kubota.
Hecke's Challenge: General Reciprocity and Fourier Analysis on the March.
Bibliography.
Index.
Two Equivalent Forms of Quadratic Reciprocity.
The StoneVon Neumann Theorem.
Weil's "Acta" Paper.
Kubota and Cohomology.
The Algebraic Agreement Between the Formalisms of Weil and Kubota.
Hecke's Challenge: General Reciprocity and Fourier Analysis on the March.
Bibliography.
Index.
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MICHAEL C. BERG, PhD, is Professor of Mathematics at Loyola Marymount University, Los Angeles, California.
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"Provides number theorists interested in analytic methods applied
to reciprocity laws with an opportunity to explore the work of
Hecke, Weil, and Kubota and their Fourieranalytic treatments..."
(SciTech Book News, Vol. 24, No. 4, December 2000)
"The content of the book is very important to number theory and is wellprepared...this book will be found to be very interesting and useful by number theorists in various areas." (Mathematical Reviews, 2002a)
"The content of the book is very important to number theory and is wellprepared...this book will be found to be very interesting and useful by number theorists in various areas." (Mathematical Reviews, 2002a)
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