DOCSLIB.ORG

Microfilmed 1994 Information to Users

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Microfilmed 1994 Information to Users U-M-I MICROFILMED 1994 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely, event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the bade of the book. Photographs induded in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 04209G0 On the M2-part” of the Birch and Swlnnerton-Dyer Conjecture for elliptic curves with complex multiplication Gonzalez-Aviles, Cristian D., Ph.D. The Ohio State University, 1004 UMI 300 N. Zeeb Rd. Ann Aibor, MI 48106 O n t h e “2 - p a r t ” o f t h e B ir c h a n d S w i n n e r t o n -D y e r c o n j e c t u r e f o r ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Cristian D. Gonzalez-Aviles, * * * * * The Ohio State University 1994 Dissertation Committee: Approved by Professor Karl Rubin Professor Robert Gold Adviser Professor Warren Sinnott Department of Mathematics A cknowledgements I thank my advisor Prof. Karl Rubin for his guidance and help; 1 have been very fortunate to be able to share in his wisdom and superb talent. My thanks go to Prof. Warren Sinnott as well, who helped me overcome many of the technical difficulties involved in this work. I also benefited from conversations with Professor Robert Gold and Mr. Cristian Popcscu. I thank them heartily for their help. Eslc trabajo csta dcdicado a mis queridos padres , quicncs me han apoyado sicmpre. V it a June 25, 1961 ...............Born in Arica, Chile. 1985 ...............................B. Sc., Univcrsidad dc Chile, Santiago, Chile. 1986 ...............................M. Sc., Univcrsidad dc Chile, Santiago, Chile. Thesis: “Class numbers of quadratic function fields and continued fractions” (J. Number Theory 40, no. 1, January 1992, pp. 38*59). Thesis advisor: Prof. Ricardo Bacza. 1987-1994 .................... Graduate Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio. F i e l d s o f S t u d y Major Field: Mathematics T a b l e o f C o n t e n t s ACKNOWLEDGEMENTS ........................................................................................ ii VITA ............................................................................................................................... iii CHAPTER PAGE I Introduction ............................................................................................................ I II The primes of bad reduction ............................................................................... 5 2.1 Preliminaries for Chapter II ...................................................................... 5 2.2 Sclmcr groups ............................................................................................... 8 2.3 Sclmer groups, continued ........................................................................... 11 III The “main conjecture” of Iwasawa t h e o r y ..................................................... 18 3.1 Preliminaries for Chapter III .................................................................... 18 3.2 Elliptic units ................................................................... 21 3.3 Euler systems of elliptic units ................................................................... 26 3.4 A logarithm map ......................................................................................... 29 3.5 Statement of the main conjecture ............................................................ 33 3.6 Results from Iwasawa theory. ................................................................. 35 3.7 Iwasawa modules of global and elliptic units ......................................... 39 3.8 A divisibility relation .................................................................................. 44 3.9 The Iwasawa invariants of X * ................................................................. 54 3.10 The Iwasawa invariants of the p>adic L-function. Proof of the main conjecture ...................................................................................................... 60 iv IV The Birch and Swinnerton-Dyer conjecture ..................................................... 67 4.1 The conjecture over A' ................................................................................... 67 4.2 The conjecture over Q ..................................... 71 BIBLIOGRAPHY ........................................................................................................ 79 v C H A PT E R I Introduction Let E be an elliptic curve defined over K — Q (\/—7) (one of the 9 imaginary quadratic fields of class number 1), with complex multiplication by the ring of integers Ok of K and with minimal period lattice generated by SI € Cx. Suppose that L (E fK , 1) ^ 0. Then E(K) is finite (Coates and Wiles [8]) and the Talc-Shafarcvich group ]M{EjK} is finite (Rubin [22]). Further, for each prime q of J(t let Cq = [E(I<q) : £b(/fq)], where Eo(Kq) C E(K q) is the subgroup of points with non-singular reduction modulo q. In this dissertation we prove the following Theorem A. If E has good reduction at 2 then L(E/I<, 1) = m i. (# £ (/0 )-2 m• ( E / K ) • n^q. (1.1) q In other words, the full Birch and Swinnerton-Dyer conjecture is true for E. Remarks, (a) Rubin [23] has shown that (1.1) is true up to multiplication by an element of K divisible only by primes dividing #0% (for all 9 imaginary quadratic fields K of class number 1). Thus when 0j[* = { ± 1} (as is the case for K — Q (\/—7)), the verification of (1.1) reduces, in a sense to be made precise shortly, to “checking 2-components”. The case K — Q (\/—7) treated in this paper is, for reasons that are explained below, the simplest of all. 2 (b) With more work it should be possible to remove the restriction on E in The­ orem A. We hope to be able to resolve this issue in the near future. (c) Other authors have addressed the problem studied in this dissertation. Sec [1] and [11]. The effectiveness of their methods is limited to those cases where the elliptic curve in question has few primes of bad reduction (at most 2). We hasten to remark, however, that the first of .the references given addresses this problem for other choices of K as well. This author is currently working on these remaining cases. In this thesis wc also prove the following result. For any non-zero integer d let Ed denote the elliptic curve y2 = x3 -f-21dx3 + 112d3x, which has complex multiplication by Ok = Z[(l + V=7)/2). Theorem B. Suppose d — l(m od4) and L(Ed/Q t 1) / 0. Then the full Birch and Swinnerton-Dyer conjecture is true for Ed/Q. The following is a brief outline of the paper. Fix, once and for all, an embedding of K = Q(\/-7) in C. Assume that E has good reduction at 2 and L (E /K t 1) ^ 0, and let B denote the set of primes of K where E has a bad reduction. We will sec below (Proposition 2.1.7) that cq = 4 for each q € B. Further, let 4> denote the Heckc character of K attached to E and 2/(tf>,s) the corresponding Hecke //-function. Then the //-function of E over K factors as L(E/K , s) = L{4>y s)L{$t s) (1.2) and L(4i, l)/Q € J<> so L{E/K, l)/fm = Nk/q(Z,(& l)/«) € Q+. Thus formula (1.1) is true up to multiplication by a power of 2 (cf. Remark (a) above), and therefore Theorem A is equivalent to the statement that L (E /K }l)/SlO, and 46*(^£?(/C)3»)*'a* 3 # m 2°° differ by a 2-adic unit. Here b = #B. Since 2 splits in K , say 2 = pp with p ^ p, one can show, using the factorization (1,2) and the fact that = 4 (Corollary 2.1.2), that the result sought will follow from l(0,l)/ft - 2t-2.#UIp» (1.3) where ~ means “equal up to a unit of /fp”. To establish (1.3) we first show (in §2.3) that #Hom(-Voo, Ep«>)** = 2h~2 • #£(/fp)p- • #UIP« (1.4) where A',*, is the Galois group of the maximal abelian 2-extension of = K(Ep«>) which is unramificd away from the prime above p, and *£ = Gal(KcafE). The essential ingredient in the derivation of this result is Theorem 2.2.6 below, which gives the index of the p°°-Sclmcr group of E in the larger group of 1-cohomology classes which arc locally trivial outside of B U {p}. The proof of Theorem 2.2.6 depends crucially on work of Bashmakov [2]. The rest of the argument leading to (1.4) is likely to seem familiar to those readers acquainted with Coates' paper [6] (but also a little more complicated, because here we deal with the troublesome prime 2). Now Rubin ([23], §11) has shown how to relate the integer on the left-hand side of formula (1.4) to ordp(Xf($, l)/fl) when p^f 2, as an application of Theorem 4.1 of [23], the “main conjecture" of Iwasawa theory for K . In Chapter III we prove a “main conjecture" for the extension KoojK (Theorem 3.5.1 below) which is well-suited for similar applications. The fact that 2 splits in I( means that we are in the setting of a one-variable main conjecture, which is the reason why the case K = Q (\/—7) is the 4 simplest of ail.
Load more

Recommended publications
  • Heegner Points, Stark-Heegner Points, and Values of L-Series
    Heegner points, Stark-Heegner points, and values of L-series Henri Darmon∗ Abstract. Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian exten- sions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank 1, arising from the work of Gross-Zagier ≤ and Kolyvagin. In [Da2], it is suggested that Heegner points admit a host of conjectural generalisations, referred to as Stark-Heegner points because they occupy relative to their classical counterparts a position somewhat analogous to Stark units relative to elliptic or circular units. A better understanding of Stark-Heegner points would lead to progress on two related arithmetic questions: the explicit construction of global points on elliptic curves (a key issue arising in the Birch and Swinnerton-Dyer conjecture) and the analytic construction of class fields sought for in Kronecker's Jugendtraum and Hilbert's twelfth problem. The goal of this article is to survey Heegner points, Stark-Heegner points, their arithmetic applications and their relations (both proved, and conjectured) with special values of L-series attached to modular forms. Mathematics Subject Classification (2000). Primary 11G05; Secondary 11G15. Keywords. Elliptic curves, modular forms, L-series, Heegner points, Stark-Heegner points. 1. Introduction Elliptic curves are distinguished among projective algebraic curves by the fact that they alone are endowed with the structure of a (commutative) algebraic group.
    [Show full text]
  • Arithmetic Theory of Elliptic ~Urves
    J. Coates R. Greenberg K. A. Ribet K. Rubin Arithmetic Theory of Elliptic ~urves- Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, July 12-19, 1997 Editor: C. Viola Fonduiione C.I.M.E. Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Springer Tokyo Authors John H. Coates Ralph Greenberg Department of Pure Mathematics Department of Mathematics Preface and Mathematical Statistics University of Washington University of Cambridge Seattle, WA 98195, USA 16 Mill Lane Cambridge CB2 1 SB, UK The C.I.M.E. Session "Arithmetic Theory of Elliptic Curves" was held at Kenneth A. Ribet Karl Rubin Cetraro (Cosenza, Italy) from July 12 to July 19, 1997. Department of Mathematics Department of Mathematics University of California Stanford University The arithmetic of elliptic curves is a rapidly developing branch of Berkeley CA 94720, USA Stanford CA 94305, USA mathematics, at the boundary of number theory, algebra, arithmetic alge- braic geometry and complex analysis. ~fterthe pioneering research in this Editor field in the early twentieth century, mainly due to H. Poincar6 and B. Levi, Carlo Viola the origin of the modern arithmetic theory of elliptic curves goes back to Dipartimento di Matematica L. J. Mordell's theorem (1922) stating that the group of rational points on Universiti di Pisa an elliptic curve is finitely generated. Many authors obtained in more re- Via Buonarroti 2 56127 Pisa, Italy cent years crucial results on the arithmetic of elliptic curves, with important connections to the theories of modular forms and L-functions.
    [Show full text]
  • P-ADIC L-FUNCTIONS and EULER SYSTEMS: a TALE in TWO TRILOGIES
    p-ADIC L-FUNCTIONS AND EULER SYSTEMS: A TALE IN TWO TRILOGIES MASSIMO BERTOLINI, FRANCESC CASTELLA, HENRI DARMON SAMIT DASGUPTA, KARTIK PRASANNA, VICTOR ROTGER Abstract. This article surveys six different special value formulae for p-adic L-functions, stressing their common features and their eventual arithmetic applications via Kolyvagin's theory of \Euler systems", in the spirit of Coates{Wiles and Kato{Perrin-Riou. Contents Introduction 1 1. Classical examples 3 1.1. Circular units 3 1.2. Elliptic units 8 1.3. Heegner points 15 2. Euler systems of Garrett-Rankin-Selberg type 19 2.1. Beilinson-Kato elements 20 2.2. Beilinson-Flach elements 25 2.3. Gross-Kudla-Schoen cycles 29 Conclusion 35 References 38 Introduction This article surveys six different special value formulae for p-adic L-functions, stressing their common features and their eventual arithmetic applications via Kolyvagin's theory of \Euler systems", in the spirit of Coates{Wiles and Kato{Perrin-Riou. The most classical instances are: (1) Leopoldt's formula for the value at s = 1 of the Kubota-Leopoldt p-adic L-function in terms of p-adic logarithms of circular units; (2) Katz's p-adic Kronecker limit formula for values of the two variable p-adic L- function of a quadratic imaginary field at finite order characters in terms of p-adic logarithms of associated elliptic units. They are reviewed in Sections 1.1 and 1.2 respectively. Section 1.3 describes the more recent formula of [BDP] and explains why it is a direct generalisation of the formulae of Leopoldt and Katz in the setting where special units are replaced by Heegner points.
    [Show full text]
  • GROSS-STARK UNITS and P-ADIC ITERATED INTEGRALS ATTACHED to MODULAR FORMS of WEIGHT ONE
    GROSS-STARK UNITS AND p-ADIC ITERATED INTEGRALS ATTACHED TO MODULAR FORMS OF WEIGHT ONE HENRI DARMON, ALAN LAUDER AND VICTOR ROTGER To Glenn Stevens on his 60th birthday Abstract. This article can be read as a companion and sequel to [DLR], which proposes a conjectural expression for the so-called p-adic iterated integrals attached to a triple (f; g; h) of classical eigenforms of weights (2; 1; 1). When f is a cusp form, this expression involves the p-adic logarithms of so-called Stark points: distinguished points on the modular abelian variety attached to f, defined over the number field cut out by the Artin representations attached to g and h. The goal of this paper is to formulate an analogous conjecture when f is a weight two Eisenstein series rather than a cusp form. The resulting formula involves the p-adic logarithms of units and p-units in suitable number fields, and can be seen as a new variant of Gross's p-adic analogue of Stark's conjecture on Artin L-series at s = 0. Contents Introduction 1 1. The conjecture 2 2. Group-theoretic preliminaries 6 3. Circular units and Eisenstein series of weight one 10 4. Elliptic units and theta series of imaginary quadratic fields 16 5. Gross-Stark units and theta series of real quadratic fields 20 6. Exotic forms 23 References 26 Introduction This article can (and probably should) be read as a companion and sequel to [DLR], which describes an arithmetic formula for the so-called p-adic iterated integrals attached to a triple (f; g; h) of classical modular eigenforms of weights (2; 1; 1).
    [Show full text]
  • Heegner Points, Stark–Heegner Points, and Values of L-Series
    Heegner points, Stark–Heegner points, and values of L-series Henri Darmon∗ Abstract. Elliptic curves over Q are equipped with a systematic collection of Heegner points aris- ing from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1, arising from the work of Gross–Zagier and Kolyvagin. In [Da2], it is suggested that Heegner points admit a host of conjectural generalisations, referred to as Stark–Heegner points because they occupy relative to their classical counterparts a position somewhat analo- gous to Stark units relative to elliptic or circular units. A better understanding of Stark–Heegner points would lead to progress on two related arithmetic questions: the explicit construction of global points on elliptic curves (a key issue arising in the Birch and Swinnerton-Dyer conjecture) and the analytic construction of class fields sought for in Kronecker’s Jugendtraum and Hilbert’s twelfth problem. The goal of this article is to survey Heegner points, Stark–Heegner points, their arithmetic applications and their relations (both proved, and conjectured) with special values of L-series attached to modular forms. Mathematics Subject Classification (2000). Primary 11G05; Secondary 11G15. Keywords. Elliptic curves, modular forms, L-series, Heegner points, Stark–Heegner points. 1. Introduction Elliptic curves are distinguished among projective algebraic curves by the fact that they alone are endowed with the structure of a (commutative) algebraic group.
    [Show full text]
  • STARK POINTS and HIDA-RANKIN P-ADIC L-FUNCTION 3
    STARK POINTS AND HIDA-RANKIN p-ADIC L-FUNCTION DANIELE CASAZZA AND VICTOR ROTGER Abstract. This article is devoted to the elliptic Stark conjecture formulated in [DLR], which proposes a formula for the transcendental part of a p-adic avatar of the leading term at s = 1 of the Hasse-Weil-Artin L-series L(E,̺1 ⊗̺2,s) of an elliptic curve E/Q twisted by the tensor product ̺1 ⊗ ̺2 of two odd 2-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a 2 × 2 p-adic regulator involving the p-adic formal group logarithm of suitable Stark points on E. This conjecture was proved in [DLR] in the setting where ̺1 and ̺2 are induced from characters of the same imaginary quadratic field K. In this note we prove a refinement of this result, that was discovered experimentally in [DLR, Remark 3.4] in a few examples. Namely, we are able to determine the algebraic constant up to which the main theorem of [DLR] holds in a particular setting where the Hida- Rankin p-adic L-function associated to a pair of Hida families can be exploited to provide an alternative proof of the same result. This constant encodes local and global invariants of both E and K. Contents 1. Introduction 1 2. Hecke characters, theta series and Katz’s p-adic L-function 6 3. Classical and p-adic Rankin L-fuctions 8 4. Proof of the main theorem 13 References 16 1. Introduction Let E/Q be an elliptic curve of conductor NE and let f S2(NE) denote the eigenform associated to it by modularity.
    [Show full text]
  • Alan Lauder and Victor Rotger
    STARK POINTS AND p-ADIC ITERATED INTEGRALS ATTACHED TO MODULAR FORMS OF WEIGHT ONE HENRI DARMON, ALAN LAUDER AND VICTOR ROTGER To our families Abstract. Let E be an elliptic curve over Q and let %[ and %] be odd two-dimensional Artin representations for which %[ ⊗ %] is self-dual. The progress on modularity achieved in the last decades ensures the existence of normalised eigenforms f, g and h of respective weights 2, 1 and 1, giving rise to E, %[, and %] via the constructions of Eichler-Shimura and Deligne-Serre. When the Hasse-Weil-Artin L-series L(E; %[ ⊗ %]; s) has a double zero at the center, this article examines certain p-adic iterated integrals attached to the triple (f; g; h), and proposes a formula for their value involving the formal group logarithms of global points on E|referred to as Stark points|which are defined over the number field cut out by %[ ⊗ %]. This formula can be viewed as an elliptic curve analogue of Stark's conjecture on units attached to weight one forms. It is proved when g and h are binary theta series attached to a common imaginary quadratic field in which p splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing p-adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell-Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark-Heegner points attached to Shintani-type cycles on Hp ×H), and extensions of Q with Galois group a central extension of the dihedral group D2n or of one of the exceptional subgroups A4, S4, and A5 of PGL2(C).
    [Show full text]
  • Arithmetic Applications of the Euler Systems of Beilinson–Flach Elements and Diagonal Cycles
    Universitat Politecnica` de Catalunya Programa de doctorat de matematica` aplicada Facultat de Matematiques` i Estad´ıstica Arithmetic applications of the Euler systems of Beilinson{Flach elements and diagonal cycles Author: Advisor: Oscar´ RIVERO SALGADO Victor ROTGER CERDA` PhD dissertation submitted for the degree of Doctor in Mathematics Barcelona, February 2021 2 Aos meus pais, Alberto e Josefa, por quererme sempre de xeito incondicional. \Tra bufalo e locomotiva la differenza salta agli occhi: la locomotiva ha la strada segnata, il bufalo pu`oscartare di lato e cadere." Francesco De Gregori. Contents Abstract 5 Resum 7 Acknowledgements and funding 9 Introduction 11 The Birch and Swinnerton-Dyer conjecture and beyond . 11 A taste of the main results: a p-adic Gross{Stark formula . 15 Exceptional zeros of Euler systems . 18 Artin formalism and Euler systems . 19 Organization of the thesis . 20 1 Background material I 21 1.1 BSD-type conjectures . 21 1.2 Euler systems . 26 1.3 p-adic L-functions . 33 2 Background material II 43 2.1 Gross{Stark units . 43 2.2 The exceptional zero phenomenon . 53 2.3 Congruences between modular forms . 58 3 Beilinson{Flach elements and exceptional zeros 65 3.1 Introduction . 65 3.2 Preliminary concepts . 72 3.3 Derived Beilinson{Flach elements . 76 3.4 Derivatives of Fourier coefficients via Galois deformations . 83 3.5 Proof of main results . 88 3.6 Darmon{Dasgupta units and p-adic L-functions . 91 4 Beilinson{Flach elements and Stark units 95 4.1 Introduction . 95 4.2 The main conjecture of Darmon, Lauder and Rotger .
    [Show full text]
  • The Coates-Wiles Theorem
    University of Cambridge Department of Pure Mathematics and Mathematical Statistics The Coates-Wiles Theorem Alexandre Daoud Homerton College Submitted in part fulfilment of the requirements for the degree of Master in Advanced Studies in Mathematics of the University of Cambridge, May 2017 Abstract Let E be an elliptic curve with complex multiplication by an order in an imaginary quadratic field K of class number 1 and E its associated Hecke character. The Coates-Wiles Theorem states that if the Hecke L-function L( E; s) is non-vanishing at s = 1 then E(K) is finite. This theorem fits into the larger framework of the Birch and Swinnerton-Dyer conjecture which remains open to this day. In this essay we will explore Rubin's proof of the Coates-Wiles Theorem via the machinery of abstract Euler systems. In particular, we will construct the Euler system of elliptic units by using torsion points of E to generate global units in abelian extensions of K. We will then use a modified Selmer group to reduce the problem down to studying a particular ideal class group and the group of global units of K. Using the Euler system, we will be able to annihilate this class group whence the Coates-Wiles Theorem will follow via an application of the well-known Chebotarev Density Theorem and Mordell-Weil Theorem. i Contents 1 Introduction 1 2 Calculation of the Selmer Group 3 2.1 Galois Cohomology of Torsion Points . .3 2.2 The Relaxed Selmer Group . .5 2.3 The Kummer Pairing . .7 2.4 Establishing p-adic control .
    [Show full text]
  • Arxiv:Math/0510310V5 [Math.NT] 9 Jun 2007
    Euler complexes and geometry of modular varieties A.B. Goncharov To Iosif Bernstein for his 60th birthday Contents 1 Introduction 2 1.1 Summary ......................................... 2 1.2 The double logarithm at roots of unity and modular curves ([G2]) ........... 3 1.3 Mixed Tate motives and the Bloch group . ......... 4 1.4 Relation with the motivic fundamental group of G µ ................ 5 m − N 1.5 Eulercomplexesonmodularcurves. .......... 6 1.6 Euler complexes, CM points, and modular hyperbolic 3-folds .............. 7 2 Euler complexes on modular curves 10 2.1 The weight two motivic complex and the dilogarithm . .............. 10 M 2.2 A reciprocity law for K3 of the function field of an elliptic curve ([G1]) . 12 2.3 The modular and elliptic units revisited . ............. 14 2.4 The elements θE(a1, a2, a3) ................................. 15 2.5 Eulercomplexesonmodularcurves. .......... 18 3 Specialization of Euler complexes at a cusp 19 3.1 Some identities in the Bloch group . .......... 19 3.2 Thespecializationmap. ........ 20 3.3 Specialization of the Euler complex at a cusp on Y1(N) ................. 21 4 Imaginary quadratic fields and tessellations of the hyperbolic space 22 arXiv:math/0510310v5 [math.NT] 9 Jun 2007 4.1 Bianchi tessellations of the hyperbolic space . ................ 22 4.2 Bianchi tessellations for Euclidean fields . ............... 23 4.3 Describing the modular complexes for d = 1, 3....................... 25 5 Euler complexes, CM elliptic curves, and modular hyperbolic 3-folds 26 5.1 Euler complexes for CM elliptic curves and geometry of modular orbifolds . 26 5.2 The rational cohomology of Γ ( ) ............................. 30 1 P 5.3 Applications.................................... ...... 31 5.4 Relations between the elements θE(a1, a2, a3) ......................
    [Show full text]
  • Arxiv:1910.13310V3 [Math.NT] 1 Nov 2020 O.Ct N Scoe Nsii Otesuyo .G Cs and [Cas] G
    THE EXCEPTIONAL ZERO PHENOMENON FOR ELLIPTIC UNITS OSCAR´ RIVERO Abstract. The exceptional zero phenomenon has been widely studied in the realm of p-adic L-functions, where the starting point lies in the foundational works of Ferrero–Greenberg, Katz, Gross, or Mazur–Tate–Teitelbaum, among others. This phenomenon also appears in the study of Euler systems, and in this case one is led to study higher order derivatives of cohomology classes in order to extract the arithmetic information which is usually encoded in the explicit reciprocity laws which make the connection with p-adic L-funtions. In this work, we focus on the elliptic units of an imaginary quadratic field and study this exceptional zero phenomenon, proving an explicit formula relating the logarithm of a derived elliptic unit either to special values of Katz’s two variable p-adic L-function or to its derivatives. Further, we interpret this fact in terms of an L-invariant and relate this result to other approaches to the exceptional zero phenomenon, most notably to the work of Bley, and we also compare this setting with other scenarios concerning Heegner points and Beilinson–Flach elements. Contents 1. Introduction 1 2. Circular units 5 2.1. Leopoldt’s formula 5 2.2. Exceptional zeros and circular units 8 3. Elliptic units 10 3.1. Elliptic units 10 3.2. Katz’s two-variable p-adic L-function of an imaginary quadratic field 11 3.3. A reciprocity law for elliptic units 12 4. Exceptional zeros and elliptic units 14 4.1. Specialization at the character ψ 14 4.2.
    [Show full text]
  • CM Elliptic Curves and the Coates–Wiles Theorem
    Mathematics and Engineering Physics CM elliptic curves and the Coates{Wiles Theorem Advisor: Francesc Fit´e Author: Mart´ıRoset Advisor (UPC): V´ıctorRotger February 8, 2019 Abstract We present one of the only known cases of the Birch and Swinnerton-Dyer Conjecture, the so called Coates{Wiles Theorem. Let K be an imaginary quadratic field with ring of integers O and let E be an elliptic curve defined over Q with complex multiplication by O. The Coates{Wiles Theorem states that if the L-series attached to E=Q does not vanish at 1 then the set of rational points of E is finite. We explain a proof given by Karl Rubin, which uses the theory of Euler systems. We begin the exposition reviewing the basic theory of elliptic curves over local fields and elliptic curves with complex multiplication. After that, we study the Euler system of elliptic units and its connection with the L-series attached to E. Keywords BSD Conjecture, Coates{Wiles Theorem, L-series, elliptic curves with CM, Euler systems, elliptic units. 1 Contents Acknowledgments . .4 Introduction . .5 1 Elliptic curves over local fields and the formal group 8 1.1 Formal groups . .8 1.2 The formal group associated to an elliptic curve . 15 1.3 Reduction modulo π ........................... 18 1.4 Applications to the study E1(K) and the torsion subgroup of E=K . 20 2 Elliptic curves with complex multiplication 24 2.1 Review of elliptic curves over C ..................... 24 2.2 Elliptic curves with complex multiplication over C ........... 27 2.3 Torsion subgroups .
    [Show full text]