Schedule and Abstracts
|
|
- Wendy Parks
- 5 years ago
- Views:
Transcription
1 Schedule and Abstracts All talks will take place at the Institute of Mathematics. = Expository Monday, August 13 9:00-9:10 Welcome words 9:10-10:40 Gerhard Frey Galois Theory: the Key to Numbers and Cyphers I 11:00-12:45 Joachim von zur Gathen Computational complexity and cryptography 14:30-15:20 Roberto Avanzi Arithmetic of supersingular Koblitz curves in characteristic three 15:45-17:30 Joachim von zur Gathen Computational complexity and cryptography 9:00-10:45 11:00-12:45 14:30-15:30 15:45-16:45 17:00-17:30 Tuesday, August 14 Gerhard Frey Galois Theory: the Key to Numbers and Cyphers II Jim Massey TBA Lenny Taelman Cyclotomic function fields and values of L-functions Douglas Ulmer Arithmetic of Jacobians in Artin-Schreier extensions Marc Palm Which factors can occur in an automorphic representations? 1
2 9:00-10:45 11:00-12:00 12:10-13:00 14:30-15:30 15:45-16:45 17:00-18:00 Wednesday, August 15 Jim Massey TBA Thomas Mittelholzer Coding techniques for data storage systems Jan Tuitman Computing zeta functions with p-adic cohomology Alexei Panchishkin Applications of Drinfeld modules in cryptography Matthew Papanikolas Explicit class field theory for function fields Ralf Butenuth Quaternionic Drinfeld modular forms and harmonic cocycles 9:00-10:00 10:15-11:15 11:30-12:30 14:30-15:30 15:45-16:45 17:00-18:00 Thursday, August 16 Federico Pellarin On the values of Goss zeta function Alina C. Cojocaru Frobenius distributions for generic Drinfeld modules of rank 2 Hans-Georg Rück On uniformizable t-modules, an example Dinesh Thakur Automata: Tools and appplications to function field arithmetic Florian Breuer An invitation to the Andre-Oort Conjecture Douglas Ulmer Birch and Swinnerton-Dyer conjecture 2
3 9:00-10:00 10:15-11:15 11:30-12:30 14:30-15:30 15:45-16:45 Friday, August 17 Yuri Zarhin Brauer-Grothendieck groups and Brauer-Manin sets Matthew Papanikolas Hyperderivative power sums and log-algebraicity identities Michael Spiess p-adic L-functions via group cohomology Cécile Armana Explicit bases of modular symbols over function fields Fu-Tsun Wei On Rankin triple product L-functions and function field analogue of Gross-Kudla formula 9:00-10:00 10:15-11:15 11:30-12:30 Saturday, August 18 Dinesh Thakur Higher multiplicities and arithmetic derivatives Florian Breuer On Drinfeld modular polynomials Gebhard Böckle Independence of l-adic Galois representations over finitely generated fields 3
4 Abstracts Cécile Armana (University of Münster) Explicit bases of modular symbols over function fields. We will discuss Teitelbaum s theory of modular symbols over a rational function field of positive characteristic. They are an essential tool for computations of certain automorphic forms, Drinfeld modular forms and elliptic curves over such a field. As in the classical setting the space of modular symbols is described in a very combinatorial way by a finite presentation. We will explain how explicit bases may be derived for several congruences subgroups (such a statement is not known in the classical case). We will also mention applications. Roberto Avanzi (Qualcomm Research) Arithmetic of supersingular Koblitz curves in characteristic three. In this presentation we show how knowledge from algebraic geometry, algebraic number theory and low brow fiddling with explicit aspects of the arithmetic of curves come together to bring significant practical performance enhancements. The main topic here are digital expansions of scalars for supersingular Koblitz curves in characteristic three curves which can be used in pairing-based cryptography. Since in pairing-based protocols besides pairing computations also scalar multiplications are required, and the performance of the latter is not negligible, improving it is clearly important as well. The digital expansions are used in a kind of double-and-add scalar multiplication algorithm, where doubling is replaced by a the Frobenius endomorphism. The considered expansions of integers are therefore made not to an integral base, but to the algebraic base of τ, where τ is a zero of a polynomial τ 2 ± 3τ + 3. We consider several properties of these expansions and apply them to scalar multiplication on the aforementioned curves. For instance, we construct digit sets for these expansions as products of powers of algebraic integers associated to certain endomorphisms of the curve. The starting point is a structure theorem for unit groups of residue classes of a quadratic order associated to the Frobenius endomorphism. The algebraic integers we choose are associated to generators of these unit groups. There are of course several choices for these generators: we chose generators associated to endomorphisms for which we could find efficient explicit formulae in a suitable coordinate system. This is a joint work with Clemens Heuberger (Klagenfurt, Austria) and Helmut Prodinger (Stellenbosch, South Africa). 4
5 Gebhard Böckle (Heidelberg University) Independence of l-adic Galois representations over finitely generated fields. Let K be a finitely generated field (over its prime field) and let X be a scheme of finite type over K. We consider the family of l-adic cohomology groups V l = H q c (X, Q l ) while l varies over the rational primes. For each l this yields a Galois representation of the absolute Galois group G K of K on the finite dimensional vector spaces V l. Two basic questions are: (1) What is the image of G K in Aut(V l )? (2) Are the images independent? In the simplest case, K a number field and X an elliptic curve, both questions were answered by Serre in In general Question (1) is very difficult, already in the case that X is an abelian variety over K. The talk will focus on Question (2). It was recently answered by Serre if K is a number fields and by Gajda-Petersen if K is of characteristic zero. The talk is on joint work with W. Gajda and S. Petersen, which provides the expected answer for K of positive characteristic. Florian Breuer (Stellenbosch University) On Drinfeld modular polynomials. We construct Drinfeld modular polynomials which relate the isomorphism invariants of rank r Drinfeld F q [T ]-modules which are linked by an isogeny of a given type. We also prove a generalization of the Kronecker congruence relations for these poynomials. Joint work with Hans-Georg Rueck. Ralf Butenuth (Heidelberg University) Quaternionic Drinfeld modular forms and harmonic cocycles. Drinfeld modular forms for congruence subgroups of GL 2 (F q [T ]) were introduced in the 1980 s by Goss as an analogue for function fields of classical modular forms. By a result of Teitelbaum they are related to combinatorial objects, called harmonic cocycles, on a Bruhat- Tits tree. We study Drinfeld modular forms for inner forms on GL 2 related to unit groups in maximal orders of quaternion algebras. We obtain a result analogous to Teitelbaum s and give an explicit description of the space of harmonic cocycles, which is useful for computations. Alina C. Cojocaru (University of Illinois at Chicago) Frobenius distributions for generic Drinfeld modules of rank 2. Let q be an odd prime power, A = F q [T ], and k = F q (T ). Let ψ be a generic Drinfeld A-module over k, of rank 2. We provide an explicit description of the Artin symbol in the division fields of psi, and discuss applications to problems focused on the splitting of primes of A in infinite families of division fields of ψ. This is joint work with Mihran Papikian. 5
6 Gerhard Frey (University of Duisburg-Essen) Galois Theory: the Key to Numbers and Cyphers. Very often Diophantine problems can be stated in an elementary way but it is notoriously hard to solve them. The most famous example for this phenomenon was Fermat s Last Theorem. The situation becomes better whenever one finds a mathematical structure behind the problem, and in many cases this structure is delivered by the action of the Galois group on geometric objects like torsion points of elliptic curves or, more generally, abelian varieties. Then the arithmetic of Galois representations plays a dominant role. A key role in this game is occupied by Jacobian varieties of modular curves. These varieties are very well understood, and connections to modular forms allow deep theoretical and practical insights. A typical and very interesting aspect is that there is an intense interplay between theoretical and algorithmic approaches. And there is an amazing and important additional bonus: Many of the used methods and obtained results can be applied to construct public key crypto systems and to discuss their security. In the lectures we shall try to explain both background and results as well as conjectures and applications of this in this exciting part of arithmetic geometry. Joachim von zur Gathen (Bonn-Aachen International Center for Information Technology) Computational complexity and cryptography. This lecture series exemplifies the influence of theoretical computer science, and of complexity theory in particular, on cryptography. The first example are reductions, coming from computability and the theory of P vs. NP, and now a standard tool in theoretical cryptography. The second one concerns structured models, called generic computations in cryptography. Here one does not consider a general model of computation like a Turing machine, but specialized calculations, such as arithmetic operations for arithmetic problems. In such a model, the discrete logarithm problem can actually be proved to be hard. The lectures end with an idiosyncratic view on ethics in science. 6
7 Thomas Mittelholzer (IBM Zurich Research Laboratory) Coding Techniques for Data Storage Systems. From a coding perspective, data storage devices can be considered as a communication channel in time, where data is stored at some instance in time and read back later. In addition to the underlying channel characteristics, coding for storage needs to take application specific requirements into account, namely, very low error rates, limited delay, and limited (decoding) complexity. After a brief introduction to algebraic coding theory, typical coding techniques for storage systems will be presented. A key aspect is the assessment of the code performance at very low error rates. For bounded distance decoding, the analysis is based on the minimum distance and weight enumerators of linear codes. For codes based on graphs - such as low-density parity-check codes, the analysis relies on graph-based objects such as stopping sets, which determine the low error-rate performance of graph-based decoders. Matthew Papanikolas (Texas A&M University) Hyperderivative power sums and log-algebraicity identities. Power sums of polynomials over finite fields are central in the theory of characteristic p zeta values and L-values. Thakur showed that these sums can be used to discover precise log-algebraicity identities on the Carlitz module. In investigating log-algebraicity on tensor powers of the Carlitz module, we find that power sums of polynomials and their hyperderivatives play a key role. We will discuss both vanishing criteria and explicit formulas for hyperderivative power sums, as well as demonstrate their connections with log-algebraicity and special values of L-functions. Federico Pellarin (University of Saint-Etienne) On the values of Goss zeta function. We can associate Goss zeta function to a ring A when it can be obtained as the integral closure of F q [T ] in a finite extension of F q (T ), yielding some analogies with Dedekind s zeta function. In this talk, however, we will essentially discuss the case A = F q [T ] itself. Then, a wide phenomenology (zeros, special values etc.) suggests a parallel with Riemann s zeta function. In particular, denoting this function associated to A = F q [T ] by ζ, we have the identity ζ(0) = 1, which is easily proved by the simplest combinatorial arguments. Since no known functional equation for this zeta exists, this identity cannot be deduced from analytic properties of some gamma factor. We will consider another class of (entire rigid analytic) functions, satisfying functional equations from which, for instance, the formula ζ(0) = 1 above can be deduced analytically. Other features of these functions will be also described. 7
8 Marc Palm (Göttingen University) Which factors can occur in an automorphic representations? Modular forms, Maass forms, and more generally irreducible automorphic representations over a global field admit a factorization into local irreducible representations. A local representation is a unitary representation of GL 2 (F ) for a local field F. There are uncountably many local representations, but only countably many automorphic representations. It is a natural question to ask, which combinations of factors can occur in an automorphic representation. I will talk about one tool box (trace formulas), its limitations and discuss my results in the context of GL(2). Hans-Georg Rück (University of Kassel) On uniformizable t-modules, an example. We study a family of t-modules and give sufficient and necessary conditions for their uniformizability. Michael Spiess (Bielefeld University) p-adic L-functions via group cohomology. We introduce a construction of the p-adic L-function of a totally real field via group cohomology and explain an application to the phenomena of trivial zeros. Lenny Taelman (Leiden University) Cyclotomic function fields and values of L-functions. Let A = F q [T ], let P A be monic irreducible. For a (non-trivial) character χ : (A/P A) F q one has the Goss L-value L(1, χ) = a χ(a) a F q ((1/T )) (with a ranging over the monic a A that are coprime with P ), as well as a P - adic version of this L-value. In this talk we will give arithmetic interpretations of these L-values. In particular, we will consider characteristic p analogues of the Mazur-Wiles theorem and the Kummer-Vandiver conjecture. This is joint work with Bruno Anglés. 8
9 Jan Tuitman (University of Oxford) Computing zeta functions with p-adic cohomology. To compute the number of rational points (or the zeta function) of an algebraic variety over a (large) finite field, one can try to compute its p-adic (also called rigid) cohomology spaces (with their action of the Frobenius map), and then use a Lefschetz trace formula. In this talk we will sketch the construction of p-adic cohomology, talk about how it behaves in families, and try to give examples. Douglas Ulmer (Georgia Institute of Technology) Arithmetic of Jacobians in Artin-Schreier extensions. I will discuss a construction of Jacobians over function fields which satisfy the BSD conjecture and have large Mordell-Weil rank in towers of Artin-Schreier extensions. Much of this is parallel to previous results on Kummer extensions, but one remarkable new phenomenon occurs: height pairings on one elliptic curve are related to point counts on another family of elliptic curves, somewhat reminiscent of mirror symmetry. This is joint work with Rachel Pries. Fu-Tsun Wei (National Tsing-Hua University) On Rankin triple product L-functions and function field analogue of Gross- Kudla formula. In this talk, I will discuss the analytic properties of triple product L-functions associated to Drinfeld type newforms whose levels are equal and square-free. Automorphic forms of Drinfeld type can be viewed as function field analogue of weight 2 modular forms. After a brief review of the basic facts about Drinfled type cusp forms, I will give the functional equation of these triple product L-functions. When the root number is positive, I present an analogue of Gross Kudla formula for the central critical values. Two examples will be shown at the end. Yuri Zarhin (Pennsylvania State University) Brauer-Grothendieck groups and Brauer-Manin sets. We discuss finiteness properties of Brauer-Grothendieck groups with special reference to abelian varieties and K3 surfaces. This is a report on a joint work with Alexei Skorobogatov. 9
Lenny Taelman s body of work on Drinfeld modules
Lenny Taelman s body of work on Drinfeld modules Seminar in the summer semester 2015 at Universität Heidelberg Prof Dr. Gebhard Böckle, Dr. Rudolph Perkins, Dr. Patrik Hubschmid 1 Introduction In the 1930
More informationAbstracts of papers. Amod Agashe
Abstracts of papers Amod Agashe In this document, I have assembled the abstracts of my work so far. All of the papers mentioned below are available at http://www.math.fsu.edu/~agashe/math.html 1) On invisible
More informationClass groups and Galois representations
and Galois representations UC Berkeley ENS February 15, 2008 For the J. Herbrand centennaire, I will revisit a subject that I studied when I first came to Paris as a mathematician, in 1975 1976. At the
More informationGalois Theory of Several Variables
On National Taiwan University August 24, 2009, Nankai Institute Algebraic relations We are interested in understanding transcendental invariants which arise naturally in mathematics. Satisfactory understanding
More informationArithmetic of elliptic curves over function fields
Arithmetic of elliptic curves over function fields Massimo Bertolini and Rodolfo Venerucci The goal of this seminar is to understand some of the main results on elliptic curves over function fields of
More informationIntroduction to Elliptic Curves
IAS/Park City Mathematics Series Volume XX, XXXX Introduction to Elliptic Curves Alice Silverberg Introduction Why study elliptic curves? Solving equations is a classical problem with a long history. Starting
More informationTranscendence theory in positive characteristic
Prof. Dr. Gebhard Böckle, Dr. Patrik Hubschmid Working group seminar WS 2012/13 Transcendence theory in positive characteristic Wednesdays from 9:15 to 10:45, INF 368, room 248 In this seminar we will
More information15 Elliptic curves and Fermat s last theorem
15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine
More informationElliptic curves over function fields 1
Elliptic curves over function fields 1 Douglas Ulmer and July 6, 2009 Goals for this lecture series: Explain old results of Tate and others on the BSD conjecture over function fields Show how certain classes
More information(Received: ) Notation
The Mathematics Student, Vol. 76, Nos. 1-4 (2007), 203-211 RECENT DEVELOPMENTS IN FUNCTION FIELD ARITHMETIC DINESH S. THAKUR (Received: 29-01-2008) Notation Z = {integers} Q = {rational numbers} R = {real
More informationCalculation and arithmetic significance of modular forms
Calculation and arithmetic significance of modular forms Gabor Wiese 07/11/2014 An elliptic curve Let us consider the elliptic curve given by the (affine) equation y 2 + y = x 3 x 2 10x 20 We show its
More informationt-motives: Hodge structures, transcendence and other motivic aspects
t-motives: Hodge structures, transcendence and other motivic aspects G. Böckle, D. Goss, U. Hartl, M. Papanikolas December 6, 2009 1 Overview of the field Drinfeld in 1974, in his seminal paper [10], revolutionized
More informationWhat is the Langlands program all about?
What is the Langlands program all about? Laurent Lafforgue November 13, 2013 Hua Loo-Keng Distinguished Lecture Academy of Mathematics and Systems Science, Chinese Academy of Sciences This talk is mainly
More informationProjects on elliptic curves and modular forms
Projects on elliptic curves and modular forms Math 480, Spring 2010 In the following are 11 projects for this course. Some of the projects are rather ambitious and may very well be the topic of a master
More informationWorkshop Automorphic Galois Representations, L-functions and Arithmetic Columbia June 17th-22nd, 2006 Saturday, June 17th 9:30 Welcome breakfast in
Workshop Automorphic Galois Representations, L-functions and Arithmetic Columbia June 17th-22nd, 2006 Saturday, June 17th 9:30 Welcome breakfast in the Lounge (5th floor) 10:00-11:30 Hida I 11:45-12:45
More informationAlberta Number Theory Days X th meeting (18w2226) May 11 13, 2018
Alberta Number Theory Days X th meeting (18w2226) May 11 13, 2018 May 12: Saturday Morning 9:00-9:10 Opening Remarks 9:10-10:00 Alice Silverberg Title: A leisurely tour through torus, abelian variety,
More informationForschungsseminar on Quaternion Algebras
Forschungsseminar on Quaternion Algebras Organisers: Gebhard Böckle, Juan Marcos Cerviño, Lassina Dembélé, Gerhard Frey, Gabor Wiese Sommersemester 2008 Abstract The goal of the seminar is to obtain a
More informationMod p Galois representations attached to modular forms
Mod p Galois representations attached to modular forms Ken Ribet UC Berkeley April 7, 2006 After Serre s article on elliptic curves was written in the early 1970s, his techniques were generalized and extended
More informationTwo Types of Equations. Babylonians. Solving Certain Cubic Equations: An Introduction to the Birch and Swinnerton-Dyer Conjecture
2 Solving Certain Cubic Equations: An Introduction to the Birch and Swinnerton-Dyer Conjecture Two Types of Equations Differential f '( x) = f( x) x 2 Algebraic 3x+ 2= 0 February 28, 2004 at Brown SUMS
More informationGalois Representations
9 Galois Representations This book has explained the idea that all elliptic curves over Q arise from modular forms. Chapters 1 and introduced elliptic curves and modular curves as Riemann surfaces, and
More informationPossibilities for Shafarevich-Tate Groups of Modular Abelian Varieties
Possibilities for Shafarevich-Tate Groups of Modular Abelian Varieties William Stein Harvard University August 22, 2003 for Microsoft Research Overview of Talk 1. Abelian Varieties 2. Shafarevich-Tate
More informationA p-adic Birch and Swinnerton-Dyer conjecture for modular abelian varieties
A p-adic Birch and Swinnerton-Dyer conjecture for modular abelian varieties Steffen Müller Universität Hamburg joint with Jennifer Balakrishnan and William Stein Rational points on curves: A p-adic and
More informationCongruent Number Problem and Elliptic curves
Congruent Number Problem and Elliptic curves December 12, 2010 Contents 1 Congruent Number problem 2 1.1 1 is not a congruent number.................................. 2 2 Certain Elliptic Curves 4 3 Using
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationTables of elliptic curves over number fields
Tables of elliptic curves over number fields John Cremona University of Warwick 10 March 2014 Overview 1 Why make tables? What is a table? 2 Simple enumeration 3 Using modularity 4 Curves with prescribed
More informationCONGRUENT NUMBERS AND ELLIPTIC CURVES
CONGRUENT NUMBERS AND ELLIPTIC CURVES JIM BROWN Abstract. In this short paper we consider congruent numbers and how they give rise to elliptic curves. We will begin with very basic notions before moving
More informationResidual modular Galois representations: images and applications
Residual modular Galois representations: images and applications Samuele Anni University of Warwick London Number Theory Seminar King s College London, 20 th May 2015 Mod l modular forms 1 Mod l modular
More informationZeta functions of buildings and Shimura varieties
Zeta functions of buildings and Shimura varieties Jerome William Hoffman January 6, 2008 0-0 Outline 1. Modular curves and graphs. 2. An example: X 0 (37). 3. Zeta functions for buildings? 4. Coxeter systems.
More informationComputations with Coleman integrals
Computations with Coleman integrals Jennifer Balakrishnan Harvard University, Department of Mathematics AWM 40 Years and Counting (Number Theory Session) Saturday, September 17, 2011 Outline 1 Introduction
More informationGalois groups with restricted ramification
Galois groups with restricted ramification Romyar Sharifi Harvard University 1 Unique factorization: Let K be a number field, a finite extension of the rational numbers Q. The ring of integers O K of K
More informationThe Arithmetic of Elliptic Curves
The Arithmetic of Elliptic Curves Sungkon Chang The Anne and Sigmund Hudson Mathematics and Computing Luncheon Colloquium Series OUTLINE Elliptic Curves as Diophantine Equations Group Laws and Mordell-Weil
More informationDiophantine equations and beyond
Diophantine equations and beyond lecture King Faisal prize 2014 Gerd Faltings Max Planck Institute for Mathematics 31.3.2014 G. Faltings (MPIM) Diophantine equations and beyond 31.3.2014 1 / 23 Introduction
More informationL-functions and Arithmetic. Conference Program. Title: On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication
Monday, June 13 9:15-9:30am Barry Mazur, Harvard University Welcoming remarks 9:30-10:30am John Coates, University of Cambridge Title: On the conjecture of Birch and Swinnerton-Dyer for elliptic curves
More informationModularity of Abelian Varieties
1 Modularity of Abelian Varieties This is page 1 Printer: Opaque this 1.1 Modularity Over Q Definition 1.1.1 (Modular Abelian Variety). Let A be an abelian variety over Q. Then A is modular if there exists
More informationOn values of Modular Forms at Algebraic Points
On values of Modular Forms at Algebraic Points Jing Yu National Taiwan University, Taipei, Taiwan August 14, 2010, 18th ICFIDCAA, Macau Hermite-Lindemann-Weierstrass In value distribution theory the exponential
More informationVerification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves
Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves William Stein University of California, San Diego http://modular.fas.harvard.edu/ Bremen: July 2005 1 This talk reports
More informationGenus 2 Curves of p-rank 1 via CM method
School of Mathematical Sciences University College Dublin Ireland and Claude Shannon Institute April 2009, GeoCrypt Joint work with Laura Hitt, Michael Naehrig, Marco Streng Introduction This talk is about
More informationOutline of the Seminar Topics on elliptic curves Saarbrücken,
Outline of the Seminar Topics on elliptic curves Saarbrücken, 11.09.2017 Contents A Number theory and algebraic geometry 2 B Elliptic curves 2 1 Rational points on elliptic curves (Mordell s Theorem) 5
More informationUniversity of Rochester Topology Seminar. Doug Ravenel
Beyond elliptic cohomology and TMF: where number theory and stable homotopy theory meet in mortal combat. University of Rochester Topology Seminar Doug Ravenel March 10, 2006 1 2 1. Introduction This talk
More informationElliptic Curves & Number Theory. R. Sujatha School of Mathematics TIFR
Elliptic Curves & Number Theory R. Sujatha School of Mathematics TIFR Aim: To explain the connection between a simple ancient problem in number theory and a deep sophisticated conjecture about Elliptic
More informationComputation of zeta and L-functions: feasibility and applications
Computation of zeta and L-functions: feasibility and applications Kiran S. Kedlaya Department of Mathematics, University of California, San Diego School of Mathematics, Institute for Advanced Study (2018
More informationIntroduction to Arithmetic Geometry
Introduction to Arithmetic Geometry 18.782 Andrew V. Sutherland September 5, 2013 What is arithmetic geometry? Arithmetic geometry applies the techniques of algebraic geometry to problems in number theory
More informationComputer methods for Hilbert modular forms
Computer methods for Hilbert modular forms John Voight University of Vermont Workshop on Computer Methods for L-functions and Automorphic Forms Centre de Récherche Mathématiques (CRM) 22 March 2010 Computer
More informationModern Number Theory: Rank of Elliptic Curves
Modern Number Theory: Rank of Elliptic Curves Department of Mathematics University of California, Irvine October 24, 2007 Rank of Outline 1 Introduction Basics Algebraic Structure 2 The Problem Relation
More informationElliptic Curves and Analogies Between Number Fields and Function Fields
Heegner Points and Rankin L-Series MSRI Publications Volume 49, 2004 Elliptic Curves and Analogies Between Number Fields and Function Fields DOUGLAS ULMER Abstract. Well-known analogies between number
More informationOn the equality case of the Ramanujan Conjecture for Hilbert modular forms
On the equality case of the Ramanujan Conjecture for Hilbert modular forms Liubomir Chiriac Abstract The generalized Ramanujan Conjecture for unitary cuspidal automorphic representations π on GL 2 posits
More informationOn the zeros of certain modular forms
On the zeros of certain modular forms Masanobu Kaneko Dedicated to Professor Yasutaka Ihara on the occasion of his 60th birthday. The aim of this short note is to list several families of modular forms
More informationVisibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one
Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one Amod Agashe February 20, 2009 Abstract Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e.,
More informationEndomorphism algebras of semistable abelian varieties over Q of GL(2)-type
of semistable abelian varieties over Q of GL(2)-type UC Berkeley Tatefest May 2, 2008 The abelian varieties in the title are now synonymous with certain types of modular forms. (This is true because we
More informationTitchmarsh divisor problem for abelian varieties of type I, II, III, and IV
Titchmarsh divisor problem for abelian varieties of type I, II, III, and IV Cristian Virdol Department of Mathematics Yonsei University cristian.virdol@gmail.com September 8, 05 Abstract We study Titchmarsh
More informationMA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26
MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26 1. Abelian Varieties of GL 2 -Type 1.1. Modularity Criteria. Here s what we ve shown so far: Fix a continuous residual representation : G Q GLV, where V is
More informationAre ζ-functions able to solve Diophantine equations?
Are ζ-functions able to solve Diophantine equations? An introduction to (non-commutative) Iwasawa theory Mathematical Institute University of Heidelberg CMS Winter 2007 Meeting Leibniz (1673) L-functions
More informationComputing coefficients of modular forms
Computing coefficients of modular forms (Work in progress; extension of results of Couveignes, Edixhoven et al.) Peter Bruin Mathematisch Instituut, Universiteit Leiden Théorie des nombres et applications
More informationGood reduction of the Brauer Manin obstruction
Good reduction of the Brauer Manin obstruction A joint work in progress with J-L. Colliot-Thélène Imperial College London Schloss Thurnau, July 2010 Notation: k is a number field, k v is the completion
More informationA Classical Introduction to Modern Number Theory
Kenneth Ireland Michael Rosen A Classical Introduction to Modern Number Theory Second Edition Springer Contents Preface to the Second Edition Preface v vii CHAPTER 1 Unique Factorization 1 1 Unique Factorization
More informationBSD and the Gross-Zagier Formula
BSD and the Gross-Zagier Formula Dylan Yott July 23, 2014 1 Birch and Swinnerton-Dyer Conjecture Consider E : y 2 x 3 +ax+b/q, an elliptic curve over Q. By the Mordell-Weil theorem, the group E(Q) is finitely
More informationON p-adic REPRESENTATIONS OF Gal(Q p /Q p ) WITH OPEN IMAGE
ON p-adic REPRESENTATIONS OF Gal(Q p /Q p ) WITH OPEN IMAGE KEENAN KIDWELL 1. Introduction Let p be a prime. Recently Greenberg has given a novel representation-theoretic criterion for an absolutely irreducible
More informationDiangle groups. by Gunther Cornelissen
Kinosaki talk, X 2000 Diangle groups by Gunther Cornelissen This is a report on joint work with Fumiharu Kato and Aristides Kontogeorgis which is to appear in Math. Ann., DOI 10.1007/s002080000183. Diangle
More informationFOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2
FOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2 CAMERON FRANC AND GEOFFREY MASON Abstract. We prove the following Theorem. Suppose that F = (f 1, f 2 ) is a 2-dimensional vector-valued
More informationComputing the image of Galois
Computing the image of Galois Andrew V. Sutherland Massachusetts Institute of Technology October 9, 2014 Andrew Sutherland (MIT) Computing the image of Galois 1 of 25 Elliptic curves Let E be an elliptic
More informationWiles theorem and the arithmetic of elliptic curves
Wiles theorem and the arithmetic of elliptic curves H. Darmon September 9, 2007 Contents 1 Prelude: plane conics, Fermat and Gauss 2 2 Elliptic curves and Wiles theorem 6 2.1 Wiles theorem and L(E/Q, s)..................
More informationNumber Theory/Representation Theory Notes Robbie Snellman ERD Spring 2011
Number Theory/Representation Theory Notes Robbie Snellman ERD Spring 2011 January 27 Speaker: Moshe Adrian Number Theorist Perspective: Number theorists are interested in studying Γ Q = Gal(Q/Q). One way
More informationHyperelliptic curves
1/40 Hyperelliptic curves Pierrick Gaudry Caramel LORIA CNRS, Université de Lorraine, Inria ECC Summer School 2013, Leuven 2/40 Plan What? Why? Group law: the Jacobian Cardinalities, torsion Hyperelliptic
More informationRank-one Twists of a Certain Elliptic Curve
Rank-one Twists of a Certain Elliptic Curve V. Vatsal University of Toronto 100 St. George Street Toronto M5S 1A1, Canada vatsal@math.toronto.edu June 18, 1999 Abstract The purpose of this note is to give
More informationCounting points on elliptic curves: Hasse s theorem and recent developments
Counting points on elliptic curves: Hasse s theorem and recent developments Igor Tolkov June 3, 009 Abstract We introduce the the elliptic curve and the problem of counting the number of points on the
More informationSome algebraic number theory and the reciprocity map
Some algebraic number theory and the reciprocity map Ervin Thiagalingam September 28, 2015 Motivation In Weinstein s paper, the main problem is to find a rule (reciprocity law) for when an irreducible
More informationSolving Cubic Equations: An Introduction to the Birch and Swinnerton-Dyer Conjecture
Solving Cubic Equations: An Introduction to the Birch and Swinnerton-Dyer Conjecture William Stein (http://modular.ucsd.edu/talks) December 1, 2005, UCLA Colloquium 1 The Pythagorean Theorem c a 2 + b
More informationL-Polynomials of Curves over Finite Fields
School of Mathematical Sciences University College Dublin Ireland July 2015 12th Finite Fields and their Applications Conference Introduction This talk is about when the L-polynomial of one curve divides
More informationIndependence of Heegner Points Joseph H. Silverman (Joint work with Michael Rosen)
Independence of Heegner Points Joseph H. Silverman (Joint work with Michael Rosen) Brown University Cambridge University Number Theory Seminar Thursday, February 22, 2007 0 Modular Curves and Heegner Points
More informationElliptic curves and modularity
Elliptic curves and modularity For background and (most) proofs, we refer to [1]. 1 Weierstrass models Let K be any field. For any a 1, a 2, a 3, a 4, a 6 K consider the plane projective curve C given
More informationMODULAR FORMS AND ALGEBRAIC K-THEORY
MODULAR FORMS AND ALGEBRAIC K-THEORY A. J. SCHOLL In this paper, which follows closely the talk given at the conference, I will sketch an example of a non-trivial element of K 2 of a certain threefold,
More informationLaval University, Québec September 2010
Conférence Québec-Maine Laval University, Québec September 2010 The Birch and Swinnerton-Dyer conjecture for Q-curves and Oda s period relations... Joint work in progress with Victor Rotger (Barcelona),
More informationTITLES & ABSTRACTS OF TALKS
TITLES & ABSTRACTS OF TALKS Speaker: Reinier Broker Title: Computing Fourier coefficients of theta series Abstract: In this talk we explain Patterson s method to effectively compute Fourier coefficients
More informationA RIEMANN HYPOTHESIS FOR CHARACTERISTIC p L-FUNCTIONS
A RIEMANN HYPOTHESIS FOR CHARACTERISTIC p L-FUNCTIONS DAVID GOSS Abstract. We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic
More informationQUADRATIC CONGRUENCES FOR COHEN - EISENSTEIN SERIES.
QUADRATIC CONGRUENCES FOR COHEN - EISENSTEIN SERIES. P. GUERZHOY The notion of quadratic congruences was introduced in the recently appeared paper [1]. In this note we present another, somewhat more conceptual
More informationthis to include the explicit maps, please do so!
Contents 1. Introduction 1 2. Warmup: descent on A 2 + B 3 = N 2 3. A 2 + B 3 = N: enriched descent 3 4. The Faltings height 5 5. Isogeny and heights 6 6. The core of the proof that the height doesn t
More informationCOMPLEX MULTIPLICATION: LECTURE 15
COMPLEX MULTIPLICATION: LECTURE 15 Proposition 01 Let φ : E 1 E 2 be a non-constant isogeny, then #φ 1 (0) = deg s φ where deg s is the separable degree of φ Proof Silverman III 410 Exercise: i) Consider
More informationThe complexity of Diophantine equations
The complexity of Diophantine equations Colloquium McMaster University Hamilton, Ontario April 2005 The basic question A Diophantine equation is a polynomial equation f(x 1,..., x n ) = 0 with integer
More informationp-divisible Groups and the Chromatic Filtration
p-divisible Groups and the Chromatic Filtration January 20, 2010 1 Chromatic Homotopy Theory Some problems in homotopy theory involve studying the interaction between generalized cohomology theories. This
More informationExplicit Complex Multiplication
Explicit Complex Multiplication Benjamin Smith INRIA Saclay Île-de-France & Laboratoire d Informatique de l École polytechnique (LIX) Eindhoven, September 2008 Smith (INRIA & LIX) Explicit CM Eindhoven,
More informationElliptic Curves Spring 2019 Problem Set #7 Due: 04/08/2019
18.783 Elliptic Curves Spring 2019 Problem Set #7 Due: 04/08/2019 Description These problems are related to the material covered in Lectures 13-14. Instructions: Solve problem 1 and then solve one of Problems
More informationOverview of the proof
of the proof UC Berkeley CIRM 16 juillet 2007 Saturday: Berkeley CDG Sunday: CDG MRS Gare Saint Charles CIRM Monday: Jet lag Jet lag = Slides Basic setup and notation G = Gal(Q/Q) We deal with 2-dimensional
More informationNon CM p-adic analytic families of modular forms
Non CM p-adic analytic families of modular forms Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. The author is partially supported by the NSF grant: DMS 1464106. Abstract:
More informationConstructing genus 2 curves over finite fields
Constructing genus 2 curves over finite fields Kirsten Eisenträger The Pennsylvania State University Fq12, Saratoga Springs July 15, 2015 1 / 34 Curves and cryptography RSA: most widely used public key
More informationHecke Operators for Arithmetic Groups via Cell Complexes. Mark McConnell. Center for Communications Research, Princeton
Hecke Operators for Arithmetic Groups via Cell Complexes 1 Hecke Operators for Arithmetic Groups via Cell Complexes Mark McConnell Center for Communications Research, Princeton Hecke Operators for Arithmetic
More informationThe Arithmetic of Noncongruence Modular Forms. Winnie Li. Pennsylvania State University, U.S.A. and National Center for Theoretical Sciences, Taiwan
The Arithmetic of Noncongruence Modular Forms Winnie Li Pennsylvania State University, U.S.A. and National Center for Theoretical Sciences, Taiwan 1 Modular forms A modular form is a holomorphic function
More informationIsogeny invariance of the BSD conjecture
Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p
More informationLecture 4: Examples of automorphic forms on the unitary group U(3)
Lecture 4: Examples of automorphic forms on the unitary group U(3) Lassina Dembélé Department of Mathematics University of Calgary August 9, 2006 Motivation The main goal of this talk is to show how one
More informationEquations for Hilbert modular surfaces
Equations for Hilbert modular surfaces Abhinav Kumar MIT April 24, 2013 Introduction Outline of talk Elliptic curves, moduli spaces, abelian varieties 2/31 Introduction Outline of talk Elliptic curves,
More informationl-adic Representations
l-adic Representations S. M.-C. 26 October 2016 Our goal today is to understand l-adic Galois representations a bit better, mostly by relating them to representations appearing in geometry. First we ll
More informationHONDA-TATE THEOREM FOR ELLIPTIC CURVES
HONDA-TATE THEOREM FOR ELLIPTIC CURVES MIHRAN PAPIKIAN 1. Introduction These are the notes from a reading seminar for graduate students that I organised at Penn State during the 2011-12 academic year.
More informationFundamental groups, polylogarithms, and Diophantine
Fundamental groups, polylogarithms, and Diophantine geometry 1 X: smooth variety over Q. So X defined by equations with rational coefficients. Topology Arithmetic of X Geometry 3 Serious aspects of the
More informationNumber Theory Seminar Spring, 2018: Modularity
Number Theory Seminar Spring, 2018: Modularity Motivation The main topic of the seminar is the classical theory of modularity à la Wiles, Taylor Wiles, Diamond, Conrad, Breuil, Kisin,.... Modularity grew
More information1 Absolute values and discrete valuations
18.785 Number theory I Lecture #1 Fall 2015 09/10/2015 1 Absolute values and discrete valuations 1.1 Introduction At its core, number theory is the study of the ring Z and its fraction field Q. Many questions
More informationCYCLOTOMIC FIELDS CARL ERICKSON
CYCLOTOMIC FIELDS CARL ERICKSON Cyclotomic fields are an interesting laboratory for algebraic number theory because they are connected to fundamental problems - Fermat s Last Theorem for example - and
More informationThe Birch & Swinnerton-Dyer conjecture. Karl Rubin MSRI, January
The Birch & Swinnerton-Dyer conjecture Karl Rubin MSRI, January 18 2006 Outline Statement of the conjectures Definitions Results Methods Birch & Swinnerton-Dyer conjecture Suppose that A is an abelian
More informationRIMS. Ibukiyama Zhuravlev. B.Heim
RIMS ( ) 13:30-14:30 ( ) Title: Generalized Maass relations and lifts. Abstract: (1) Duke-Imamoglu-Ikeda Eichler-Zagier- Ibukiyama Zhuravlev L- L- (2) L- L- L B.Heim 14:45-15:45 ( ) Title: Kaneko-Zagier
More informationCAN A DRINFELD MODULE BE MODULAR?
CAN A DRINFELD MODULE BE MODULAR? DAVID GOSS Abstract. Let k be a global function field with field of constants F r, r = p m, and let be a fixed place of k. In his habilitation thesis [Boc2], Gebhard Böckle
More informationGalois Representations
Galois Representations Samir Siksek 12 July 2016 Representations of Elliptic Curves Crash Course E/Q elliptic curve; G Q = Gal(Q/Q); p prime. Fact: There is a τ H such that E(C) = C Z + τz = R Z R Z. Easy
More informationOn Universal Elliptic Curves Over Igusa Curves
On Universal Elliptic Curves Over Igusa Curves Douglas L. Ulmer Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 To my parents in their 50 th year The purpose of this
More information