Number Theory

The Fourier-Analytic Proof of Quadratic Reciprocity by Michael C. Berg

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By Michael C. Berg

A distinct synthesis of the 3 current Fourier-analytic remedies of quadratic reciprocity.The relative quadratic case used to be first settled via Hecke in 1923, then recast by means of Weil in 1964 into the language of unitary workforce representations. The analytic evidence of the final n-th order case continues to be an open challenge this present day, going again to the tip of Hecke's recognized treatise of 1923. The Fourier-Analytic facts of Quadratic Reciprocity offers quantity theorists drawn to analytic equipment utilized to reciprocity legislation with a special chance to discover the works of Hecke, Weil, and Kubota.This paintings brings jointly for the 1st time in one quantity the 3 present formulations of the Fourier-analytic evidence of quadratic reciprocity. It exhibits how Weil's groundbreaking representation-theoretic remedy is actually similar to Hecke's classical technique, then is going a step additional, providing Kubota's algebraic reformulation of the Hecke-Weil evidence. large commutative diagrams for evaluating the Weil and Kubota architectures also are featured.The writer sincerely demonstrates the price of the analytic process, incorporating probably the most robust instruments of contemporary quantity thought, together with ad?les, metaplectric teams, and representations. eventually, he issues out that the severe universal issue one of the 3 proofs is Poisson summation, whose generalization might eventually give you the answer for Hecke's open challenge.

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Beweis: Im Falle von Z l¨ asst sich eine Bijektion f : N → Z wie folgt angeben. Es sei f (2n) = n und f (2n+1) = −n. Im Falle von Q ist es etwas trickreicher, eine Bijektion g : N → Q anzugeben, welches wir wie folgt leisten. 11), d. h. p0 = 2, p1 = 3, p2 = 5, ur m ∈ N, Pm die Menge aller n ∈ N, n ≥ 2, p4 = 7, . . Sei, f¨ so dass pm die kleinste Primzahl ist, die n teilt. (Z. B. ist ur m ∈ N, m ≥ 1, Tm die P1 = {3, 9, 15, . ) Weiter sei f¨ Menge aller n ∈ N, n ≥ 1, so dass 1 der gr¨ oßte gemeinsame Teiler von m und n ist.

Sei ε > 0. Sei n0 so, dass f¨ Dann gilt f¨ ur alle n ≥ n0 : 2−x2n ≤ yn2 −x2n = (yn −xn )(yn + xn ) ≤ (yn − xn ) · 2yn < 14 · ε · 4 = ε, und damit x2n ≥ 2 − ε. ) Also ur alle ε > 0, woraus x2 ≥ 2 folgt. V¨ollig gilt x2 ≥ 2 − ε f¨ analog zeigt man x2 ≤ 2. Also gilt x2 = 2. 2! 3 besagt, dass Q nicht vollst¨ andig ist: es gibt in Q verlaufende Cauchy-Folgen, die nicht in Q konvergieren. Wir werden nun Q vervollst¨ andigen“. ” ¨ Wir definieren dazu zun¨ achst eine Aquivalenzrelation auf der Menge der (rationalen) Cauchy-Folgen.

2 Die Theorie der nat¨ urlichen Zahlen 29 riablen zweiter Stufe. 3 bei der Axiomatisierung von R wieder begegnen, wo man auch geneigt ist, Variablen erster Stufe (n¨amlich Variablen f¨ ur reelle Zahlen) und Variablen zweiter Stufe (n¨ amlich Variablen f¨ ur Mengen reeller Zahlen) einzuf¨ uhren. Der Preis, der f¨ ur die Einf¨ uhrung von Variablen zweiter Stufe zu zahlen ist, ist allerdings, dass dann Mengenexistenzaxiome erforderlich sind, die nicht ben¨otigt werden, wenn in der Sprache nur Variablen erster Stufe (n¨amlich Variablen f¨ ur Elemente des Bereichs, u ¨ber den man gerade spricht, nicht f¨ ur Mengen derartiger Elemente) verwendet werden.

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