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An Introduction to Tate's Thesis

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An Introduction to Tate's Thesis An introduction to Tate's Thesis James-Michael Leahy Master of Arts Department of Mathematics and Statistics McGill University Montreal, Quebec August, 2010 A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Master of Arts Copyright ⃝c James-Michael Leahy, 2010 ACKNOWLEDGEMENTS First and foremost, I would like to thank my supervisor, Professor Henri Darmon, for his encouragement and support throughout the process of writing my thesis. Andrew Fiori also should be thanked for being a excellent listener and peer educator. I am grateful for the grammatical help and moral support I received from Brynn Smith-Raska. Lastly, I would like to thank Professor Paul Koosis for his meticulous grading of my graduate complex analysis assignments, which certainly improved my abilities in mathematical analysis. iii ABSTRACT In the early 20th century, Erich Hecke attempted to find a further generalization of the Dirichlet L-series and the Dedekind zeta function. In 1920, he [14] introduced the notion of a Gr¨ossencharakter, an ideal class character of a number field, and established the analytic continuation and functional equation of its associated L-series, the Hecke L-series. In 1950, John Tate [27], following the suggestion of his advisor, Emil Artin, recast Hecke's work. Tate provided a more elegant proof of the functional equation of the Hecke L-series by using Fourier analysis on the adeles and employing a reformulation of the Gr¨ossencharakter in terms of a character on the ideles. Tate's work now is generally understood as the GL(1) case of automorphic forms [2]. The thesis provides a thorough analysis of the approach taken by Tate in his own thesis. Background information is furnished by theory concerning topological groups, Pontryagin duality, the restricted-direct topology, and the adeles and ideles. iv ABREG´ E´ Au d´ebutdu 20`emesi`ecle,Erich Hecke a essay´ede trouver une nouvelle g´en´eralisation de la s´erieL de Dirichlet et de la fonction z^etade Dedekind. En 1920, il [14] a introduit la notion de Gr¨ossencharakter, un caract`ere des classes d'id´eauxd'un corps de nombres, et ´etabli le prolongement analytique et l'´equationfonctionnelle de sa s´erie L associ´ee,la s´erie L de Hecke. En 1950, John Tate [27], suivant la suggestion de son directeur de th`ese,Emil Artin, a remani´ele travail de Hecke. Tate a fourni une preuve plus ´el´egante de l'´equation fonctionnelle de la s´erieL de Hecke en employant l'analyse de Fourier sur les ad`eleset en utilisant une reformulation du Gr¨ossencharakter en termes de caract`eresur les id`eles.Le travail de Tate est maintenant g´en´eralement compris comme le cas de GL(1) de la th´eoriedes formes automorphes [2]. Cette th`esedonne un apercu de l'approche adopt´eepar Tate dans sa propre th`ese.Ceci inclut une introduction `ala th´eoriedes groupes topologiques, la dualit´ede Pontryagin, la topologie des produits restreints, et les ad`eleset les id`eles. v TABLE OF CONTENTS ACKNOWLEDGEMENTS . iii ABSTRACT . iv ABREG´ E............................................´ v Introduction . 1 1 Topological Groups, the Haar Measure, and Pontryagin Duality . 14 1.1 Topological Groups and Fields . 14 1.1.1 Definitions and Examples . 14 1.1.2 Some Theory of Topological Groups . 20 1.1.3 Locally Compact Groups and Fields . 25 1.1.4 P-adic Numbers and Topology . 28 1.1.5 Profinite Topology . 33 1.2 Haar Measure . 37 1.3 Pontryagin Duality and the Fourier Inversion Theorem . 45 2 Global and Local Fields . 59 2.1 Global Fields . 59 2.2 Local Fields . 67 3 Restricted Direct Topology and the Adeles and Ideles . 74 3.1 Restricted Direct Topology . 75 3.1.1 Restricted Direct Quasi-Characters and Dual Group . 78 3.1.2 Restricted Direct Integration and Self-Dual Measure . 82 3.2 Adeles and Ideles . 89 4 Tate's Thesis . 105 4.1 Local Quasi-Characters and their Associated Local L-factors . 105 4.2 Local Additive Characters and the Self-Duality of Local Fields . 110 4.3 The Multiplicative Haar Measure . 118 4.4 Local Schwartz-Bruhat Functions . 120 4.5 The Meromorphic Continuation and Functional Equation of the Local Zeta Function . 125 4.6 Local Epsilon Factor and Root Number . 143 4.7 Adelic Schwartz-Bruhat Functions and the Riemann-Roch Theorem . 149 4.8 Idele-Class Characters . 159 vi 4.9 The Meromorphic Continuation and Functional Equation of the Global Zeta Function . 161 4.10 Hecke L-Functions . 169 1 4.11 The Volume of CK and the Regulator . 179 Conclusion . 186 vii Introduction Most students of mathematics have some familiarity with the Riemann zeta function (s), which is defined by the absolutely convergent series X 1 (s) = ns n=1 for complex numbers s such that <(s) > 1. In letting s = 1, the series that results, (1), is the Harmonic series, which diverges. Despite being named the Riemann zeta function, Leonhard Euler was the first to study the function (s) for s 2 R. He established the Euler product expansion, Y 1 (s) = ; 1 − p−s p which is valid in the domain <s > 1. In the third century B.C.E., Euclid understood that there are infinitely many primes. Since the Riemann zeta function is divergent at s = 1, then by the Euler product expansion we obtain Y 1 > 1; 1 − p−1 p which provides an alternate proof of the infinitude of primes. By analyzing the Laurent expansion of cot z, it can be determined that 1 X 1 2 = : n2 6 n=1 The Euler product expansion then yields Y 1 6 1 − = = 0:607927102 ⋅ ⋅ ⋅ : p2 2 p In a non-rigorous fashion, the probability that a natural number, chosen at random, is divisible by a fixed prime p is 1=p. More rigorously, one should choose random natural number from the first N numbers and then let N approach infinity. The probability that two natural numbers, chosen at random, are divisible by p is 1=p2. Then 1 − (1=p2) represents 1 the probability that two natural numbers chosen at random are either both relatively prime to p or, at most, one is not relatively prime to p. For distinct primes p and q, the probability that a natural number chosen at random is divisible by both p and q is 1=pq. Hence, the divisibility events are independent and the product Y 1 1 − p2 p represents the probability that two natural numbers, chosen at random, are relatively prime. It can be determined, by applying the Euler product of the Riemann zeta function, that this 6 probability is 2 . In 1859, Georg Friedrich Bernhard Riemann [25] showed that the function (s) can be analytically continued to the complex plane to a function that is holomorphic for s 6= 1. The residue at the simple pole s = 1 is 1. He also proved the functional equation (s) = (1 − s); (1) −s=2 s where (s) = Γ( 2 )(s) and where Z 1 Γ(s) = e−tts−1dt: 0 It can be shown that Γ(s) is absolutely convergent for <(s) > 1 and can be meromorphically continued to the whole s plane with simple poles at negative integers. Further, Γ(s) 6= 0 for −(1−s)=2 1−s s > 0. Dividing both sides of equation 1 by Γ( 2 ), we obtain (1 − s) = (2)−s2 cos( s)Γ(s)(s): 2 This functional equation shows that in the domain <s < 0, the function (s) has simple zeroes at −2; −4; −6; ⋅ ⋅ ⋅ . In addition to proving the functional equation, Riemann showed that (s) has infinitely many zeros on the critical strip 0 ≤ <s ≤ 1 and hypothesized that all these zeroes lie on the line <s = 1=2. This is known today as the Riemann hypothesis, which has not yet been proved. It is also interesting to note that when evaluating the Riemann zeta 2 function at positive even integer points and at negative integer points, one obtains (2)2n (2n) = (−1)n−1 B ; 2(2n)! 2n and B (−n) = − n+1 ; n + 1 respectively, where Bm for m 2 Z are the Bernoulli numbers. The prime number function (x) is defined for x 2 R to be the number of primes p such that p ≤ x. Riemann provided an explicit formula for (x) that depended on the location of the zeros of (s). See [15], Chapter 27, for the formula. Riemann also proved the prime number theorem, which states that x log x lim = 1: x!1 (x) This can be proved by analyzing the Riemann zeta function and two other functions: X log p X (s) = and (x) = log p: ps p p≤x −s=2 s As a consequence of Tate's thesis, we now are able to realize that the factor of Γ( 2 ), appearing in the functional equation of (s), \originates from" the infinite prime of Q. Let Z −x2 s dx Z1(s) = e jxj : R−{0g jxj Making the change of variable u = x2, we obtain Z Z 1 Z 1 −x2 s dx −x2 s−1 −s=2 −u s=2−1 −s=2 Z1(s) = e jxj = 2 e x dx = e u du = Γ(s=2): jxj 0 0 R−{0g By the definition of (s) and the Euler product of (s), we obtain s Y 1 (s) = −s=2Γ( )(s) = Z (s) : 2 1 1 − p−s p 3 P1 ns Now, let Zp(s) = n=0 1=p . The function Zp(s) is a geometric series and converges to 1=(1 − p−s).
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