The Euler System of Generalized Heegner Cycles
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Manjul Bhargava
The Work of Manjul Bhargava Manjul Bhargava's work in number theory has had a profound influence on the field. A mathematician of extraordinary creativity, he has a taste for simple problems of timeless beauty, which he has solved by developing elegant and powerful new methods that offer deep insights. When he was a graduate student, Bhargava read the monumental Disqui- sitiones Arithmeticae, a book about number theory by Carl Friedrich Gauss (1777-1855). All mathematicians know of the Disquisitiones, but few have actually read it, as its notation and computational nature make it difficult for modern readers to follow. Bhargava nevertheless found the book to be a wellspring of inspiration. Gauss was interested in binary quadratic forms, which are polynomials ax2 +bxy +cy2, where a, b, and c are integers. In the Disquisitiones, Gauss developed his ingenious composition law, which gives a method for composing two binary quadratic forms to obtain a third one. This law became, and remains, a central tool in algebraic number theory. After wading through the 20 pages of Gauss's calculations culminating in the composition law, Bhargava knew there had to be a better way. Then one day, while playing with a Rubik's cube, he found it. Bhargava thought about labeling each corner of a cube with a number and then slic- ing the cube to obtain 2 sets of 4 numbers. Each 4-number set naturally forms a matrix. A simple calculation with these matrices resulted in a bi- nary quadratic form. From the three ways of slicing the cube, three binary quadratic forms emerged. -
Sir Andrew J. Wiles
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society March 2017 Volume 64, Number 3 Women's History Month Ad Honorem Sir Andrew J. Wiles page 197 2018 Leroy P. Steele Prize: Call for Nominations page 195 Interview with New AMS President Kenneth A. Ribet page 229 New York Meeting page 291 Sir Andrew J. Wiles, 2016 Abel Laureate. “The definition of a good mathematical problem is the mathematics it generates rather Notices than the problem itself.” of the American Mathematical Society March 2017 FEATURES 197 239229 26239 Ad Honorem Sir Andrew J. Interview with New The Graduate Student Wiles AMS President Kenneth Section Interview with Abel Laureate Sir A. Ribet Interview with Ryan Haskett Andrew J. Wiles by Martin Raussen and by Alexander Diaz-Lopez Allyn Jackson Christian Skau WHAT IS...an Elliptic Curve? Andrew Wiles's Marvelous Proof by by Harris B. Daniels and Álvaro Henri Darmon Lozano-Robledo The Mathematical Works of Andrew Wiles by Christopher Skinner In this issue we honor Sir Andrew J. Wiles, prover of Fermat's Last Theorem, recipient of the 2016 Abel Prize, and star of the NOVA video The Proof. We've got the official interview, reprinted from the newsletter of our friends in the European Mathematical Society; "Andrew Wiles's Marvelous Proof" by Henri Darmon; and a collection of articles on "The Mathematical Works of Andrew Wiles" assembled by guest editor Christopher Skinner. We welcome the new AMS president, Ken Ribet (another star of The Proof). Marcelo Viana, Director of IMPA in Rio, describes "Math in Brazil" on the eve of the upcoming IMO and ICM. -
Arxiv:1407.1093V1 [Math.NT]
MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER Abstract. We show that the cyclotomic Iwasawa–Greenberg Main Conjecture holds for a large class of modular forms with multiplicative reduction at p, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument. 1. Introduction The cyclotomic Iwasawa–Greenberg Main Conjecture was established in [18], in com- bination with work of Kato [13], for a large class of newforms f ∈ Sk(Γ0(N)) that are ordinary at an odd prime p ∤ N, subject to k ≡ 2 (mod p − 1) and certain conditions on the mod p Galois representation associated with f. The purpose of this note is to extend this result to the case where p | N (in which case k is necessarily equal to 2). ∞ n Recall that the coefficients an of the q-expansion f = n=1 anq of f at the cusp at infinity (equivalently, the Hecke eigenvalues of f) are algebraic integers that generate a finite extension Q(f) ⊂ C of Q. Let p be an odd primeP and let L be a finite extension of the completion of Q(f) at a chosen prime above p (equivalently, let L be a finite extension of Qp in a fixed algebraic closure Qp of Qp that contains the image of a chosen embedding Q(f) ֒→ Qp). Suppose that f is ordinary at p with respect to L in the sense that ap is a unit in the ring of integers O of L. -
An Introduction to the Birch and Swinnerton-Dyer Conjecture
Rose-Hulman Undergraduate Mathematics Journal Volume 16 Issue 1 Article 15 An Introduction to the Birch and Swinnerton-Dyer Conjecture Brent Johnson Villanova University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Johnson, Brent (2015) "An Introduction to the Birch and Swinnerton-Dyer Conjecture," Rose-Hulman Undergraduate Mathematics Journal: Vol. 16 : Iss. 1 , Article 15. Available at: https://scholar.rose-hulman.edu/rhumj/vol16/iss1/15 Rose- Hulman Undergraduate Mathematics Journal An Introduction to the Birch and Swinnerton-Dyer Conjecture Brent A. Johnson a Volume 16, No. 1, Spring 2015 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal Villanova University Rose-Hulman Undergraduate Mathematics Journal Volume 16, No. 1, Spring 2015 An Introduction to the Birch and Swinnerton-Dyer Conjecture Brent A. Johnson Abstract. This article explores the Birch and Swinnerton-Dyer Conjecture, one of the famous Millennium Prize Problems. In addition to providing the basic theoretic understanding necessary to understand the simplest form of the conjecture, some of the original numerical evidence used to formulate the conjecture is recreated. Recent results and current problems related to the conjecture are given at the end. Acknowledgements: I would like to thank Professor Robert Styer and Professor Alice Deanin for their incredible mentorship, patience, and friendship. RHIT Undergrad. Math. J., Vol. 16, No. 1 Page 271 1 Introduction An elliptic curve is a projective, nonsingular curve given by the general Weierstrass equation 2 3 2 E : y + a1xy + a3y = x + a2x + a4x + a6: There is no doubt that elliptic curves are amongst the most closely and widely studied objects in mathematics today. -
On Functional Equations of Euler Systems
ON FUNCTIONAL EQUATIONS OF EULER SYSTEMS DAVID BURNS AND TAKAMICHI SANO Abstract. We establish precise relations between Euler systems that are respectively associated to a p-adic representation T and to its Kummer dual T ∗(1). Upon appropriate specialization of this general result, we are able to deduce the existence of an Euler system of rank [K : Q] over a totally real field K that both interpolates the values of the Dedekind zeta function of K at all positive even integers and also determines all higher Fitting ideals of the Selmer groups of Gm over abelian extensions of K. This construction in turn motivates the formulation of a precise conjectural generalization of the Coleman-Ihara formula and we provide supporting evidence for this conjecture. Contents 1. Introduction 1 1.1. Background and results 1 1.2. General notation and convention 4 2. Coleman maps and local Tamagawa numbers 6 2.1. The local Tamagawa number conjecture 6 2.2. Coleman maps 8 2.3. The proof of Theorem 2.3 10 3. Generalized Stark elements and Tamagawa numbers 13 3.1. Period-regulator isomorphisms 13 3.2. Generalized Stark elements 16 3.3. Deligne-Ribet p-adic L-functions 17 3.4. Evidence for the Tamagawa number conjecture 18 arXiv:2003.02153v1 [math.NT] 4 Mar 2020 4. The functional equation of vertical determinantal systems 18 4.1. Local vertical determinantal sytems 19 4.2. The functional equation 20 5. Construction of a higher rank Euler system 24 5.1. The Euler system 25 5.2. The proof of Theorem 5.2 26 6. -
Multiplicative Reduction and the Cyclotomic Main Conjecture for Gl2
Pacific Journal of Mathematics MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER Volume 283 No. 1 July 2016 PACIFIC JOURNAL OF MATHEMATICS Vol. 283, No. 1, 2016 dx.doi.org/10.2140/pjm.2016.283.171 MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER We show that the cyclotomic Iwasawa–Greenberg main conjecture holds for a large class of modular forms with multiplicative reduction at p, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument. 1. Introduction The cyclotomic Iwasawa–Greenberg main conjecture was established in[Skinner and Urban 2014], in combination with work of Kato[2004], for a large class of newforms f 2 Sk.00.N// that are ordinary at an odd prime p - N, subject to k ≡ 2 .mod p − 1/ and certain conditions on the mod p Galois representation associated with f . The purpose of this note is to extend this result to the case where p j N (in which case k is necessarily equal to 2). P1 n Recall that the coefficients an of the q-expansion f D nD1 anq of f at the cusp at infinity (equivalently, the Hecke eigenvalues of f ) are algebraic integers that generate a finite extension Q. f / ⊂ C of Q. Let p be an odd prime and let L be a finite extension of the completion of Q. f / at a chosen prime above p (equivalently, let L be a finite extension of Qp in a fixed algebraic closure Qp of Qp that contains the image of a chosen embedding Q. -
Euler Systems and Refined Conjectures of Birch Swinnerton
Contemporary Mathematics Volume 00, 0000 Euler Systems and Refined Conjectures of Birch Swinnerton-Dyer Type HENRI DARMON Abstract. The relationship between arithmetic objects (such as global fields, or varieties over global fields) and the analytic properties of their L- functions poses many deep and difficult questions. The theme of this paper is the Birch and Swinnerton Dyer conjecture, and certain refinements that were proposed by Mazur and Tate. We will formulate analogues of these conjectures over imaginary quadratic fields involving Heegner points, and explain how the fundamental work of V.A. Kolyvagin sheds light on these new conjectures. 1 Preliminaries. x The relationship between arithmetic objects (such as global fields, or varieties over global fields) and the analytic properties of their L-functions poses many deep and subtle questions. The theme of this paper is the Birch Swinnerton-Dyer conjecture, which concerns the case where the arithmetic object in question is an elliptic curve defined over a global field. Let E be an elliptic curve defined over the rational numbers. The conjecture of Shimura-Taniyama-Weil asserts that E is modular, i.e., is equipped with a rational map ' : X (N) E; 0 −! where X0(N) is the modular curve of level N, defined over Q, which parameter- izes elliptic curves with a distinguished cyclic N-isogeny. We assume that E has this property. (For a specific E this can be checked by a finite computation.) 1991 Mathematics Subject Classification. Primary 11G40; Secondary 11G05. The ideas for this paper are part of the author's Harvard PhD thesis; he gratefully ac- knowledges the support of Harvard University, and in particular of his advisor, B.H. -
Anticyclotomic Euler Systems for Unitary Groups
ANTICYCLOTOMIC EULER SYSTEMS FOR UNITARY GROUPS ANDREW GRAHAM AND SYED WAQAR ALI SHAH Abstract. Let n ≥ 1 be an odd integer. We construct an anticyclotomic Euler system for certain cuspidal automorphic representations of unitary groups with signature (1; 2n − 1). Contents 1. Introduction 1 2. Preliminaries 7 3. Cohomology of unitary Shimura varieties 13 4. Horizontal norm relations 15 5. Interlude: cohomology functors 25 6. Vertical norm relations at split primes 31 7. The anticyclotomic Euler system 41 Appendix A. Continuous ´etalecohomology 49 Appendix B. Shimura{Deligne varieties 55 Appendix C. Ancona's construction for Shimura{Deligne varieties 60 References 64 1. Introduction In [Kol89], Kolyvagin constructs an anticyclotomic Euler system for (modular) elliptic curves over Q which satisfy the so-called \Heegner hypothesis". One can view the classes in this construction as images under the modular parameterisation of certain divisors on modular curves arising from the embedding of reductive groups (1.1) ResE=Q Gm ,! GL2 where E is an imaginary quadratic number field. These divisors (Heegner points) are defined over ring class fields of E and satisfy trace compatibility relations as one varies the field of definition. In particular, Kolyvagin shows that if the bottom Heegner point is non-torsion, then the group of E-rational points has rank equal to one. Combining this so-called \Kolyvagin step" with the Gross{Zagier formula that relates the height of this point to the derivative of the L-function at the central value, one obtains instances of the Birch{ Swinnerton-Dyer conjecture in the analytic rank one case. The above construction has been generalised to higher weight modular forms and to situations where a more general hypothesis is placed on the modular form (see [Sch86], [Nek92], [BDP13] and [Zha01]). -
CV of Xin Wan
CV of Xin Wan September 19, 2016 1 Personal • Born: November 6, 1985, China. • Address: 560 Riverside Dr. APT 15E, New York, NY, 10027. • Phone: 609-955-2892. • Email: [email protected]. • Homepage: http://www.math.columbia.edu/~xw2295. 2 Education • B.S. Peking University, China, 2003-2007. • Ph.D Princeton University, 2007-2012. Thesis Advisor: Christopher Skinner. • Research Interest: Number Theory. More precisely: automorphic forms, Galois representations, Iwasawa theory, Birch and Swinnerton- Dyer conjecture and Bloch-Kato conjecture. 3 Work Experience • September 2012-June 2013, Member, Institute for Advance Study, Princeton. • September 2013-June 2016, Ritt Assistant Professor, Columbia University, New York. • July 2016-Now, Associate Professor, Academy of Mathematics and Systems Science, Chinese Academy of Science. 4 Publication All papers are available at http://www.mcm.ac.cn/faculty/wx. 1 • p-adic Eisenstein Series and L-Functions of certain cusp forms on denite unitary groups (joint with E.Eischen), Journal of the Institute of Mathematics of Jussieu, volume 15, issue 03, pp. 471-510, July 2016. • Families of Nearly Ordinary Eisenstein Series on Unitary Groups (with an Appendix by Kai- Wen Lan), Algebra and Number Theory, 2015, 9-9, pp 1955-2054. • Iwasawa Main Conjecture for Hilbert Modular Forms, Forum of Mathematics, Sigma, Volumn 3 (2015), e18, 95 Pages. • Introduction to Skinner-Urban's Work on Iwasawa Main Conjecture, in Iwasawa Theory 2012, State of the Art and Recent Advances, Contributions in Mathematical and Computational Sciences, Vol. 7, Springer, 2014. • Iwasawa Main Conjecture for Non-Ordinary Modular Forms, arXiv:1607.07729, 2016. • Λ-adic Gross-Zagier formula for elliptic curves at supersingular primes, (joint with F.Castella), arXiv:1607.02019, 2016. -
On the $ P $-Part of the Birch-Swinnerton-Dyer Formula For
ON THE p-PART OF THE BIRCH–SWINNERTON-DYER FORMULA FOR MULTIPLICATIVE PRIMES FRANCESC CASTELLA Abstract. Let E/Q be a semistable elliptic curve of analytic rank one, and let p> 3 be a prime for which E[p] is irreducible. In this note, following a slight modification of the methods of [JSW15], we use Iwasawa theory to establish the p-part of the Birch and Swinnerton-Dyer formula for E. In particular, we extend the main result of loc.cit. to primes of multiplicative reduction. Contents 1. Introduction 1 2. Selmer groups 3 3. A p-adic Waldspurger formula 8 4. Main Conjectures 10 5. Proof of Theorem A 12 References 14 1. Introduction Let E/Q be a semistable elliptic curve of conductor N, and let L(E,s) be the Hasse–Weil L-function of E. By the celebrated work of Wiles [Wil95] and Taylor–Wiles [TW95], L(E,s) is known to admit analytic continuation to the entire complex plane, and to satisfy a functional equation relating its values at s and 2 s. The purpose of this note is to prove the following result towards the Birch and Swinnerton-Dyer− formula for E. Theorem A. Let E/Q be a semistable elliptic curve of conductor N with ords=1L(E,s) = 1, and let p> 3 be a prime such that the mod p Galois representation ρ¯ : Gal(Q/Q) AutF (E[p]) E,p −→ p is irreducible. If p N, assume in addition that E[p] is ramified at some prime q = p. Then | 6 arXiv:1704.06608v2 [math.NT] 14 Jun 2017 L′(E, 1) ordp = ordp #Ш(E/Q) cℓ(E/Q) , Reg(E/Q) ΩE · Yℓ|N where Reg(E/Q) is the discriminant of the N´eron–Tate height pairing on E(Q) R; • ⊗ ΩE is the N´eron period of E; • Ш(E/Q) is the Tate–Shafarevich group of E; and • c (E/Q) is the Tamagawa number of E at the prime ℓ. -
AN INTRODUCTION to P-ADIC L-FUNCTIONS
AN INTRODUCTION TO p-ADIC L-FUNCTIONS by Joaquín Rodrigues Jacinto & Chris Williams Contents General overview......................................................................... 2 1. Introduction.............................................................................. 2 1.1. Motivation. 2 1.2. The Riemann zeta function. 5 1.3. p-adic L-functions. 6 1.4. Structure of the course.............................................................. 11 Part I: The Kubota–Leopoldt p-adic L-function.................................... 12 2. Measures and Iwasawa algebras.......................................................... 12 2.1. The Iwasawa algebra................................................................ 12 2.2. p-adic analysis and Mahler transforms............................................. 14 2.3. An example: Dirac measures....................................................... 15 2.4. A measure-theoretic toolbox........................................................ 15 3. The Kubota-Leopoldt p-adic L-function. 17 3.1. The measure µa ..................................................................... 17 3.2. Restriction to Zp⇥ ................................................................... 18 3.3. Pseudo-measures.................................................................... 19 4. Interpolation at Dirichlet characters..................................................... 21 4.1. Characters of p-power conductor. 21 4.2. Non-trivial tame conductors........................................................ 23 4.3. Analytic -
L-FUNCTIONS and CYCLOTOMIC UNITS 1. Introduction
L-FUNCTIONS AND CYCLOTOMIC UNITS TOM WESTON, UNIVERSITY OF MICHIGAN 1. Introduction Let K be a number field with r1 real embeddings and r2 pairs of complex con- jugate embeddings. If ζK (s) is the Dedekind zeta function of K, then ζK (s) has a zero of order r1 + r2 1 at s = 0, and the value of the first non-zero derivative is given by the Dirichlet− class number formula: (r1+r2 1) hK RK ζK − (0) = : − wK Here RK is the regulator of K, wK is the number of roots of unity in K and hK is the class number of K. This formula is a striking connection between arithmetic and analysis, and there have been many attempts to generalize it to other L-functions: one expects that the value of a \motivic" L-function at an integer point should involve a transcendental factor, a boring rational factor, and an interesting rational factor. In the case of the Dedekind zeta function, these roles are played by RK , wK , and hK , respectively. In the general case one expects that the interesting rational factor is the order of a certain Selmer group. 2. Selmer groups and Kolyvagin systems Let us recall the definition of the Selmer group of a p-adic Galois representation. Let T be a free Zp-module with a continuous action of the absolute Galois group ¯ GK = Gal(K=K). We define the Cartier dual of T by T ∗ = HomZp (T; µp1 ) with the adjoint Galois action. Let be a Selmer structure on T . Recall that consists of choices of local Selmer structuresF F 1 1 H (Kv;T ) H (Kv;T ) F ⊆ for each place v of K; these are assumed to coincide with the unramified choice at almost all places v.