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Elliptic Units in Ray Class Fields of Real Quadratic Number Fields

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Elliptic Units in Ray Class Fields of Real Quadratic Number Fields Elliptic units in ray class fields of real quadratic number fields Hugo Chapdelaine McGill University The Faculty of Science Department of Mathematics and Statistics Research in partial fulfilment of the requirements of the degree of Doctor of philosophy April, 2007 Montr´eal, Qu´ebec, Canada Copyright c Hugo Chapdelaine, 2007 Abstract Let K be a real quadratic number field. Let p be a prime which is inert in K. We denote the completion of K at the place p by Kp. Let f > 1 be a positive integer coprime to p. In this thesis we give a p-adic construction of special elements 1 1 u(r, τ) K× for special pairs (r, τ) (Z/fZ)× where = P (C ) P (Q ) ∈ p ∈ × Hp Hp p \ p is the so called p-adic upper half plane. These pairs (r, τ) can be thought of as an analogue of classical Heegner points on modular curves. The special elements u(r, τ) are conjectured to be global p-units in the narrow ray class field of K of conductor f. The construction of these elements that we propose is a generalization of a previous construction obtained in [DD06]. The method consists in doing p-adic integration of certain Z-valued measures on X =(Z Z ) (pZ pZ ). The construction of those p × p \ p × p measures relies on the existence of a family of Eisenstein series (twisted by additive characters) of varying weight. Their moments are used to define those measures. We also construct p-adic zeta functions for which we prove an analogue of the so called Kronecker’s limit formula. More precisely we relate the first derivative at s =0ofa certain p-adic zeta function with log N u(r, τ). Finally we also provide some − p Kp/Qp evidence both theoretical and numerical for the algebraicity of u(r, τ). Namely we relate a certain norm of our p-adic invariant with Gauss sums of the cyclotomic field Q(ζf ,ζp). The norm here is taken via a conjectural Shimura reciprocity law. We also have included some numerical examples at the end of section 18. 1 R´esum´e Soit K un corps de nombre quadratique r´eel. Soit p un nombre premier inerte dans K. Nous noterons par Kp la compl´etion de K en p. Soit f > 1 un entier positif copremier `a p. Dans cette th`ese nous donnons une construction p-adique de certains ´el´ements 1 1 u(r, τ) K× pour certaines paires (r, τ) (Z/fZ)× o`u = P (C ) P (Q ). ∈ p ∈ × Hp Hp p \ p Ces paires (r, τ) sont en quelque sorte des analogues des points de Heegner classiques sur les courbes modulaires. Nous avons conjectur´eque les ´el´ements u(r, τ) sont des p- unit´es dans le corps de classe de K au sens restreint de conducteur f. La construction de ces ´el´ements que nous proposons est une g´en´eralisation d’une construction obtenue dans [DD06]. La m´ethode consiste essentiellement a faire de l’integration p-adique de certaines mesures sur X =(Z Z ) (pZ pZ ) `avaleurs dans Z. La construction de p × p \ p × p ces mesures repose essentiellement sur l’existence d’une famille de s´eries d’Eisenstein (tordues par des caract`eres additifs) avec le poids k 2 qui varie. Les moments de ces ≥ s´eries d’Eisenstien sont utilis´es pour d´efinir ces mesures. Nous construisons aussi une fonction zeta p-adique pour laquelle nous prouvons un analogue de la formule limite de Kronecker. Plus pr´ecis´ement, nous relions la premi`ere d´eriv´ee en s = 0 d’une certaine fonction zeta p-adique avec log N u(r, τ). Finalement nous donnons − p Kp/Qp une bonne raison th´eorique de croire en l’alg´ebricit´ede u(r, τ). A` savoir, nous relions une certaine norme de notre invariant p-adique avec des sommes de Gauss contenues dans le corps cyclotomique Q(ζf ,ζp). La norme ici est d´efinie `al’aide d’une loi de r´eciprocit´ede Shimura conjecturale. Nous avons aussi inclu quelques r´esultats num´eriques `ala fin de la section 18. 2 Aknowledgements First of all, I would like to thank my supervisor, Prof. Henri Darmon, for all his devotion and enthusiasm regarding the field of mathematics. As a mathematician, he was a great source of inspiration and at the same time a great director. Second of all, I would like thank my former professor, Claude Levesque, who gave me my first opportunity to start a career in mathematics. Without all his good advices my mathematical journey would probably be very different. I would also like to thank all the administration staff of the Mathematics and Statistics department of university McGill. I would like to give a special thanks to Carmen for all her patience and all her energy to make sure that I would meet all the deadlines and the bureaucratic requirements. I would also like to thank all the students that I met in the last four years. They were a great source of entertainment and more than anything else they taught me thousands of new things about life. Among all of those people I have a few special thanks to give. I would like thank Mak Trifkovich who taught me hundred of new things on a wide range of topics: English pronunciation and grammar, political and economical right wing ideology, history of mankind and mathematics to name a few. I would also like to thank Matthew Greenberg with whom I shared many interests: watching hockey, beer drinking, black sense of humour and of course mathematics. I would also like to thank my two office mates in the last two years, Andrew Archibald and Gabriel Chenevert, who had to put up with me. I had great and interesting discussions with both of them. Finally and not the least, I want to thank my family, Pierre, Maryse and Gene- vi`eve, who were always very supportive and encouraging with me. I know that all of them are happy for me since I was lucky enough to find a passion in life: the passion of doing mathematics. 3 Contents 1 The Z-rank of strong p-units 13 1 2 Distributions on P (Qp) and holomorphic functions on the upper half plane 15 3 A review of the classical setting 24 4 Modular units and Eisenstein series 30 4.1 TheSiegelfunction ............................ 30 4.2 Modular units associated to a good divisor . ... 31 4.3 Thedualmodularunit .......................... 33 12 ∆(τ) 4.4 From g k (Nτ) to ....................... 34 ( N ,0) ∆(Nτ) 4.5 The p-stabilizationofmodularunits. 35 4.6 The involution ιN on X1(N)(C)..................... 38 4.7 Bernoulli polynomials and Eisenstein series . ...... 39 4.8 Hecke operators on modular forms twisted by an additive character . 42 4.9 The q-expansion of Ek,p(r, τ)....................... 45 4.10 Relation between Eisenstein series and modular units . ........ 46 5 The Z-valued measures µ c c and the invariant u(δ , τ) 49 r{ 1 → 2} r 1 5.1 Z-valued measures on P (Qp) ...................... 49 5.2 Periods of modular units and Dedekind sums . .. 52 5.3 Themodularsymbolsareodd . 57 4 5.4 From to O(N , f)........................... 58 H Hp 0 5.5 Construction of the Kp×-points u(δr, τ) ................. 63 6 The measures µ c c 70 { 1 → 2} 6.1 From P1(Q ) to (Q Q ) (0, 0) ................... 70 pe p × p \{ } 6.2 Splittingofthe2-cocycle . 74 7 Archimedean zeta functions attached to totally real number fields 77 7.1 Zeta functions twisted by additive characters . ...... 77 7.2 Gauss sums for Hecke characters and Dirichlet characters....... 80 7.3 Relation between Ψ( a , χ ,s) and L(χ, s)................ 80 df ∞ 1 a 7.4 Partial zeta functions ζ(a− , f,w,s) as the dual of Ψ( df ,w,s)..... 83 8 Archimedean zeta functions attached to real quadratic number fields 87 8.1 Involution on X (fN )(C)revisited ................ 87 ∗fN0 1 0 8.2 Zeta functions Ψ and Ψ∗ ......................... 89 8.3 Proof of the functional equation of Ψ for K quadraticreal . 94 9 Relation between special values of zeta functions and Eisenstein se- ries 100 9.1 Archimedean zeta function associated to a class in O(N , f)/Γ ............................... 100 Hp 0 0 9.2 Special valuese of ζ(δr, (A, τ), 1 k) as integrals of Eisenstein series of − even weight F2k .............................. 103 9.3 Some explicit formulas for ζ∗(δr, (A, τ), 0)................ 108 5 9.4 Relation between ζ(δ, (1, τ),s) and ζ∗(δ, (1, τ),s)............ 108 10 P-adic zeta functions and p-adic Kronecker limit formula 110 11 Dedekind sums and periods of Eisenstein series 111 11.1Dedekindsums .............................. 111 11.2Atechnicallemma ............................ 113 11.3 Moments of Eisenstein series . 117 11.4 Moments of µ c c ......................... 125 r{ 1 → 2} e 12 Proof of theorem 6.1 126 12.1 Measures on Z Z ........................... 126 p × p 12.2 A partial modular symbol of measures on Z Z ........... 132 p × p 12.3 From Z Z to X ............................ 135 p × p 12.4 The measure µ c c is Γ invariant ................ 139 r{ 1 → 2} 0 e e 13 The measure µ c c is Z-valued 141 r{ 1 → 2} e 14 Explicit formulas of µ c c on balls of X 146 r{ 1 → 2} e 15 Stability property of µ c c on balls of decreasing radius marked j{ 1 → 2} at an integral center 149 e 16 Explicit formulas of ζ(δ, (A, τ), 1 k) in terms of the measures µ c − r{ 1 → c2 153 } e 17 Some evidence for the algebraicity of the u(r, τ) invariant 158 6 17.1 The reciprocity map of Class field theory applied to K(f ) ....
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