1. Introduction 1.1. Background. L-Functions and Modular Forms
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On Modular Forms for the Paramodular Group
On Modular Forms for the Paramodular Group Brooks Roberts and Ralf Schmidt Contents 1 Definitions 3 2 Linear independence at different levels 6 3 The level raising operators 8 4 Oldforms and newforms 13 5 Saito–Kurokawa liftings 16 6 Two theorems 23 Appendix 26 References 28 Introduction Let F be a p–adic field, and let G be the algebraic F –group GSp(4). In our paper [RS] we presented a conjectural theory of local newforms for irreducible, admissible, generic representations of G(F ) with trivial central character. The main feature of this theory is that it considers fixed vectors under the paramodular groups K(pn), a certain family of compact-open subgroups. The group K(p0) is equal to the standard maximal compact subgroup G(o), where o is the ring of integers of F . In fact, K(p0) and K(p1) represent the two conjugacy classes of maximal compact subgroups of G(F ). In general K(pn) can be conjugated into K(p0) if n is even, and into K(p1) if n is odd. Our theory is analogous to Casselman’s well-known theory for representations of GL(2,F ); see [Cas]. The main conjecture made in [RS] states that for each irreducible, admissible, generic representation (π, V ) of PGSp(4,F ) there exists an n such that the space V (n) of K(pn) invariant vectors is non-zero; if n0 is the minimal such n then dimC(V (n0)) = 1; and the Novodvorski zeta integral of a suitably normalized vector in V (n0) computes the L–factor L(s, π) (for this last statement we assume that V is the Whittaker model of π). -
Notices of the American Mathematical Society
• ISSN 0002-9920 March 2003 Volume 50, Number 3 Disks That Are Double Spiral Staircases page 327 The RieITlann Hypothesis page 341 San Francisco Meeting page 423 Primitive curve painting (see page 356) Education is no longer just about classrooms and labs. With the growing diversity and complexity of educational programs, you need a software system that lets you efficiently deliver effective learning tools to literally, the world. Maple® now offers you a choice to address the reality of today's mathematics education. Maple® 8 - the standard Perfect for students in mathematics, sciences, and engineering. Maple® 8 offers all the power, flexibility, and resources your technical students need to manage even the most complex mathematical concepts. MapleNET™ -- online education ,.u A complete standards-based solution for authoring, nv3a~ _r.~ .::..,-;.-:.- delivering, and managing interactive learning modules \~.:...br *'r¥'''' S\l!t"AaITI(!\pU;; ,"", <If through browsers. Derived from the legendary Maple® .Att~~ .. <:t~~::,/, engine, MapleNefM is the only comprehensive solution "f'I!hlislJer~l!'Ct"\ :5 -~~~~~:--r---, for distance education in mathematics. Give your institution and your students cornpetitive edge. For a FREE 3D-day Maple® 8 Trial CD for Windows®, or to register for a FREE MapleNefM Online Seminar call 1/800 R67.6583 or e-mail [email protected]. ADVANCING MATHEMATICS WWW.MAPLESOFT.COM I [email protected]\I I WWW.MAPLEAPPS.COM I NORTH AMERICAN SALES 1/800 267. 6583 © 2003 Woter1oo Ma')Ir~ Inc Maple IS (J y<?glsterc() crademork of Woterloo Maple he Mar)leNet so troc1ema'k of Woter1oc' fV'lop'e Inr PII other trcde,nork$ (ye property o~ their respective ('wners Generic Polynomials Constructive Aspects of the Inverse Galois Problem Christian U. -
Congruences Between Modular Forms
CONGRUENCES BETWEEN MODULAR FORMS FRANK CALEGARI Contents 1. Basics 1 1.1. Introduction 1 1.2. What is a modular form? 4 1.3. The q-expansion priniciple 14 1.4. Hecke operators 14 1.5. The Frobenius morphism 18 1.6. The Hasse invariant 18 1.7. The Cartier operator on curves 19 1.8. Lifting the Hasse invariant 20 2. p-adic modular forms 20 2.1. p-adic modular forms: The Serre approach 20 2.2. The ordinary projection 24 2.3. Why p-adic modular forms are not good enough 25 3. The canonical subgroup 26 3.1. Canonical subgroups for general p 28 3.2. The curves Xrig[r] 29 3.3. The reason everything works 31 3.4. Overconvergent p-adic modular forms 33 3.5. Compact operators and spectral expansions 33 3.6. Classical Forms 35 3.7. The characteristic power series 36 3.8. The Spectral conjecture 36 3.9. The invariant pairing 38 3.10. A special case of the spectral conjecture 39 3.11. Some heuristics 40 4. Examples 41 4.1. An example: N = 1 and p = 2; the Watson approach 41 4.2. An example: N = 1 and p = 2; the Coleman approach 42 4.3. An example: the coefficients of c(n) modulo powers of p 43 4.4. An example: convergence slower than O(pn) 44 4.5. Forms of half integral weight 45 4.6. An example: congruences for p(n) modulo powers of p 45 4.7. An example: congruences for the partition function modulo powers of 5 47 4.8. -
Benedict Gross Harvard University, Professor
THE ALBERT LEON WHITEMAN MEMORIAL MATHEMATICS LECTURES February 22 and 23, 2016 Benedict Gross Harvard University, Professor Benedict Gross is the George Vasmer Leverett Professor of Mathematics at Harvard University and the former Dean of Harvard College. He is very well known for his research in number theory. His many honors and awards include his election as a Fellow of the American Academy of Arts and Sciences, Membership of the National Academy of Sciences and a Fellow of the American Mathematical Society. He received a MacArthur Fellowship and the Cole Prize in number theory from the AMS. How large is n! = n(n-1)(n-2)…3.2.1 ? Monday, February 22, 2016 Andrus Gerontology Center Time: 4:00-4:30 pm: Reception in Gerontology Courtyard Time: 4:30 pm: LECTURE - Gerontology: Leonard Davis Auditorium located in 124 Short abstract: The number n! (pronounced "n factorial") occurs in many counting problems. For example, that 52! is the number of ways to shuffle a deck of cards. This number grows very rapidly with n, and mathematicians of the 17th century used the new methods of calculus to estimate it. After reviewing some of this work, I'll discuss Euler's Gamma function, which interpolates the function F(n) = (n-1)! to the real numbers, as well as a more recent analog. The rank of elliptic curves Tuesday, February 23, 2016 3:00-3:30 pm: Reception in Kaprielian Hall 410 3:30-4:30 pm: LECTURE in Kaprielian Hall 414 Abstract: Elliptic curves, which are given by cubic equations in two variables, have been a central object of study in number theory since the time of Fermat. -
Henri Darmon
Henri Darmon Address: Dept of Math, McGill University, Burnside Hall, Montreal, PQ. E-mail: [email protected] Web Page: http://www.math.mcgill.ca/darmon Telephone: Work (514) 398-2263 Home: (514) 481-0174 Born: Oct. 22, 1965, in Paris, France. Citizenship: Canadian, French, and Swiss. Education: 1987. B.Sc. Mathematics and Computer Science, McGill University. 1991. Ph.D. Mathematics, Harvard University. Thesis: Refined class number formulas for derivatives of L-series. University Positions: 1991-1994. Princeton University, Instructor. 1994-1996. Princeton University, Assistant Professor. 1994-1997. McGill University, Assistant Professor. 1997-2000. McGill University, Associate Professor. 2000- . McGill University, Professor. 2005-2019. James McGill Professor, McGill University. Other positions: 1991-1994. Cercheur hors Qu´ebec, CICMA. 1994- . Chercheur Universitaire, CICMA. 1998- . Director, CICMA (Centre Interuniversitaire en Calcul Math´ematique Alg´ebrique). 1999- . Member, CRM (Centre de Recherches Math´ematiques). 2005-2014. External member, European network in Arithmetic Geometry. Visiting Positions: 1991. IHES, Paris. 1995. Universit´a di Pavia. 1996. Visiting member, MSRI, Berkeley. 1996. Visiting professor and guest lecturer, University of Barcelona. 1997. Visiting Professor, Universit´e Paris VI (Jussieu). 1997. Visitor, Institut Henri Poincar´e. 1998. Visiting Professor and NachDiplom lecturer, ETH, Zuric¨ h. 1999. Visiting professor, Universit`a di Pavia. 2001. Visiting professor, Universit`a di Padova. 2001. Korea Institute for Advanced Study. 2002. Visiting professor, RIMS and Saga University (Japan). 1 2003. Visiting Professor, Universit´e Paris VI, Paris. 2003. Visiting professor, Princeton University. 2004. Visiting Professor, Universit´e Paris VI, Paris. 2006. Visiting Professor, CRM, Barcelona, Spain. 2008. Visiting Professor, Universit´e Paris-Sud (Orsay). -
The Rings of Hilbert Modular Forms for Q( 29) and Q( 37)
p p THE RINGS OF HILBERT MODULAR FORMS FOR Q( 29) AND Q( 37) BRANDON WILLIAMS Abstract. We use Borcherds products and their restrictions to Hirzebruch-Zagier curvesp to deter- minep generators and relations for the graded rings of Hilbert modular forms for the fields Q( 29) and Q( 37). These seem to be the first cases where the graded ring can be computed despite obstructions to the existence of Borcherds products with arbitrary divisors. 1. Introduction The problem considered in this note is to determine generators and relations for the graded rings M∗(ΓK ) of Hilbert modular forms over real quadraticp fields K. Therep is some history to this.p The structure has been worked out in thep cases Q( 5) (Gundlach, [12]), Q( 2) (Hammond, [13]), Q( 13) (van der Geer and Zagier, [10]), Q( 17) (Hermann, [14] determinedp the structure of the subring of symmetric forms; the full ring appears in section 6.5 of [18]), and Q( 6) (van derp Geer,p [8]). Sectionp 6.2 of [18] gives a more detailed overview. Simpler derivations for the fields Q( 5); Q( 13); Q( 17) were given by Mayer [18] using a reduction process against Borcherds products with simple zeros on the diagonal, and the work here extends this idea. p We only consider fields K = Q( p) of prime discriminant p ≡ 1 (4) and class number one as this makes a number of aspects of the theory simpler (but there are ultimately workarounds for both of these assumptions). The first few such primes are p = 5; 13; 17; and the corresponding graded rings have the form p M∗(Γ ) = [X2;X5;X6;X15]=(R30); Q( 5) C p M∗(Γ ) = [X2;X3;X4;X5;X6;X8;X9]=(R9;R10;R12;R18); Q( 13) C p (1) (2) (1) (2) (1) (2) M∗(Γ ) = [X2;X3;X ;X ;X ;X ;X ;X ;X7;X8;X9]=I; Q( 17) C 4 4 5 5 6 6 (1) (2) I = (R9;R10;R11;R12 ;R12 ;R13;R14;R18); (i) (i) where Xk;Xk denote modular forms of weight k and where Rk;Rk denote relations that are homo- geneous of weight k, see [18]. -
Paramodular Forms of Degree 2 with Particular Emphasis on Level T = 5
Paramodular Forms of Degree 2 with Particular Emphasis on Level t = 5 Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der Rheinisch-Westfälischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Mathematiker Axel Marschner aus Wertheim/Main B ERICHTER Universitätsprofessor Dr. Aloys Krieg Privat-Dozent Dr. Jürgen Müller Tag der mündlichen Prüfung: 17.12.2004 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar Preface Modular forms are important objects in Number Theory. In a simplified way, a modular form is a function on an open subset of Cn with an invariance property under a sufficiently large discrete group and a nice Fourier expansion. The theory of modular forms was initi- ated in the 19th century and was intimately related to the theory of integrals over algebraic functions (e. g. elliptic integrals). A systematic theory of modular forms with respect to the symplectic group Sp(n,Z) was developed by C. L. Siegel (1939). Modular forms can be used to determine for example numbers of representations of a quadratic form (Siegel’s Hauptsatz) or Abelian extensions of (real or imaginary) number fields. Recently they were part in Wiles’ proof of Fermat’s last Theorem. One possible generalization of symplectic modular forms are paramodular forms. First results about paramodular forms were shown by Christian [Ch1] and Köhler ([Kö1], [Kö2] and [Kö3]). Generators of a graded ring of paramodular forms of degree 2 were first determined by Igusa for Γ1 (the paramodular group of degree 2 and level 1), i. e. the ring of Siegel modular forms of genus 2. -
2008 Conant Prize
2008 Conant Prize The 2008 Levi L. Conant Prize was awarded at the history, known partial results and blind alleys, 114th Annual Meeting of the AMS in San Diego in various threads of numerical and theoretical evi- January 2008. dence, and suggestive connections with disparate The Conant Prize is awarded annually to recog- branches of mathematics and theoretical physics. nize an outstanding expository paper published The mathematical exposition is enhanced by the in either the Notices of the AMS or the Bulletin of judicious use of anecdotes illustrating the human the AMS in the preceding five years. Established drama of the quest for a proof and of figures that in 2001, the prize honors the memory of Levi L. help the reader visualize the zeta function as a Conant (1857–1916), who was a mathematician at function of a complex variable and the key connec- Worcester Polytechnic University. The prize carries tions between the distribution of prime numbers, a cash award of US$1,000. the distribution of the zeros of the Riemann zeta The Conant Prize is awarded by the AMS Coun- function, and conjecturally also the distribution of cil acting on the recommendation of a selection the eigenvalues of random Hermitian operators. committee. For the 2008 prize the members of Conrey remarks on one of those fascinating the selection committee were: Noam D. Elkies, connections (Gauss’s class number problem and Stephen J. Greenfield, and Carl R. Riehm (chair). a “conspiracy of L-functions”) that “we seem to Previous recipients of the Conant Prize are: Carl be players in the middle of a mystery novel.” The Pomerance (2001), Elliott Lieb and Jakob Yngvason same can be said of the status of the Riemann (2002), Nicholas Katz and Peter Sarnak (2003), Hypothesis itself. -
Arxiv:Math/0605346V2 [Math.AG] 21 May 2007 Ru SL(2 Group Fhceoeaos Rmtefuircoefficients
SIEGEL MODULAR FORMS GERARD VAN DER GEER Abstract. These are the lecture notes of the lectures on Siegel modular forms at the Nordfjordeid Summer School on Modular Forms and their Applications. We give a survey of Siegel modular forms and explain the joint work with Carel Faber on vector-valued Siegel modular forms of genus 2 and present evidence for a conjecture of Harder on congruences between Siegel modular forms of genus 1 and 2. 1. Introduction Siegel modular forms generalize the usual modular forms on SL(2, Z) in that the group SL(2, Z) is replaced by the automorphism group Sp(2g, Z) of a unimodular symplectic form on Z2g and the upper half plane is replaced by the Siegel upper half plane g. The integer g 1 is called the degree or genus. Siegel pioneeredH the generalization≥ of the theory of elliptic modular forms to the modular forms in more variables now named after him. He was motivated by his work on the Minkowski-Hasse principle for quadratic forms over the rationals, cf., [95]. He investigated the geometry of the Siegel upper half plane, determined a fundamental domain and its volume and proved a central result equating an Eisenstein series with a weighted sum of theta functions. No doubt, Siegel modular forms are of fundamental importance in number theory and algebraic geometry, but unfortunately, their reputation does not match their importance. And although vector-valued rather than scalar-valued Siegel modular forms are the natural generalization of elliptic modular forms, their reputation amounts to even less. A tradition of ill-chosen notations may have contributed to this, but the lack of attractive examples that can be handled decently seems to be the main responsible. -
January 2006
THE LONDON MATHEMATICAL SOCIETY NEWSLETTER No. 344 January 2006 Forthcoming COUNCIL DIARY Independently of these dis- 18 November 2005 cussions, October Council had Society highlighted the need for possi- Meetings The main business at this, the ble radical revision of the last Council meeting of Frances Society's organisation. To 2006 Kirwan's presidency, was con- progress this, a Council Retreat Friday 10 February sideration of the Framework will be held in January 2006 to London Studies Initiative on the rela- discuss matters such as the G. Segal tionship between the London Society's core values and objec- U. Tillmann Mathematical Society and the tives in the modern world, and (Mary Cartwright Institute for Mathematics and to identify and prioritise the Lecture) its Applications. Council was activities that are central to very grateful for the carefully members’ perception of the 1 Monday 15 May considered views that had been LMS. This will lead on to consid- Leicester received from many members eration of the governance and Midlands Regional during the consultation period. management best suited to Meeting Council authorised further con- delivering these outcomes. M. Bridson sideration of versions of the Reporting from the N. Hitchin H-framework and the inverted November meeting of the H. Kraft Y-framework, as a route to uni- Council for the Mathematical A. Zelevinsky fication of the two societies. Sciences, the President and the A choice between the two alter- Education Secretary expressed Friday 16 June natives will be made in March concern at the momentum London 2006. It was recognised that the building up from the Bologna Yu Manin H-framework was viable only if agreement, intended to unify (Hardy Lecture) it had a life of at most 10 years qualifications across Europe. -
Introduction to the Tate Issue
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 4, October 2017, Pages 541–543 http://dx.doi.org/10.1090/bull/1585 Article electronically published on July 10, 2017 INTRODUCTION TO THE TATE ISSUE SUSAN FRIEDLANDER Last year the AMS published the Collected Works of John Tate1 whose bril- liant mathematics revolutionized areas of number theory, algebraic geometry, and algebra. His scientific accomplishments place him among the most significant math- ematicians of the 20th century. This issue of the Bulletin of the American Mathe- matical Society celebrates John Tate and the publication of his collected works. For over more than six decades, Tate’s work has been recognized by many awards and distinctions including the Abel prize in 2010 “for his vast and lasting impact on the theory of numbers and the wealth of essential mathematical ideas and constructions that he initiated”. The articles in this issue of the Bulletin illustrate some of these ideas and give a historical overview of Tate’s prolific collaboration with Jean Pierre Serre. The influence of Tate’s work was spread not only through his groundbreaking publications but also through his many PhD students, mainly during his professor- ship at Harvard from 1954 to 1990 after which he became the Sid W. Richardson Chair in Mathematics at the University of Texas in Austin. These mathematicians are listed in Table 1. Table 1. List of former students2 Edward Assmus, Jr. Harvard University 1958 Leonard Evens Harvard University 1960 James Cohn Harvard University 1961 Andrew Ogg Harvard University 1961 Stephen Shatz Harvard University 1962 Jonathan Lubin Harvard University 1963 Saul Lubkin Harvard University 1963 Judith Obermayer Harvard University 1963 Stephen Lichtenbaum Harvard University 1964 J. -
Vitae Ken Ono Citizenship
Vitae Ken Ono Citizenship: USA Date of Birth: March 20,1968 Place of Birth: Philadelphia, Pennsylvania Education: • Ph.D., Pure Mathematics, University of California at Los Angeles, March 1993 Thesis Title: Congruences on the Fourier coefficients of modular forms on Γ0(N) with number theoretic applications • M.A., Pure Mathematics, University of California at Los Angeles, March 1990 • B.A., Pure Mathematics, University of Chicago, June 1989 Research Interests: • Automorphic and Modular Forms • Algebraic Number Theory • Theory of Partitions with applications to Representation Theory • Elliptic curves • Combinatorics Publications: 1. Shimura sums related to quadratic imaginary fields Proceedings of the Japan Academy of Sciences, 70 (A), No. 5, 1994, pages 146-151. 2. Congruences on the Fourier coefficeints of modular forms on Γ0(N), Contemporary Mathematics 166, 1994, pages 93-105., The Rademacher Legacy to Mathe- matics. 3. On the positivity of the number of partitions that are t-cores, Acta Arithmetica 66, No. 3, 1994, pages 221-228. 4. Superlacunary cusp forms, (Co-author: Sinai Robins), Proceedings of the American Mathematical Society 123, No. 4, 1995, pages 1021-1029. 5. Parity of the partition function, 1 2 Electronic Research Annoucements of the American Mathematical Society, 1, No. 1, 1995, pages 35-42 6. On the representation of integers as sums of triangular numbers Aequationes Mathematica (Co-authors: Sinai Robins and Patrick Wahl) 50, 1995, pages 73-94. 7. A note on the number of t-core partitions The Rocky Mountain Journal of Mathematics 25, 3, 1995, pages 1165-1169. 8. A note on the Shimura correspondence and the Ramanujan τ(n)-function, Utilitas Mathematica 47, 1995, pages 153-160.