Supersingular Zeros of Divisor Polynomials of Elliptic Curves of Prime Conductor
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Arxiv:2003.01675V1 [Math.NT] 3 Mar 2020 of Their Locations
DIVISORS OF MODULAR PARAMETRIZATIONS OF ELLIPTIC CURVES MICHAEL GRIFFIN AND JONATHAN HALES Abstract. The modularity theorem implies that for every elliptic curve E=Q there exist rational maps from the modular curve X0(N) to E, where N is the conductor of E. These maps may be expressed in terms of pairs of modular functions X(z) and Y (z) where X(z) and Y (z) satisfy the Weierstrass equation for E as well as a certain differential equation. Using these two relations, a recursive algorithm can be used to calculate the q - expansions of these parametrizations at any cusp. Using these functions, we determine the divisor of the parametrization and the preimage of rational points on E. We give a sufficient condition for when these preimages correspond to CM points on X0(N). We also examine a connection between the al- gebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the q-expansions of these objects. 1. Introduction and statement of results The modularity theorem [2, 12] guarantees that for every elliptic curve E of con- ductor N there exists a weight 2 newform fE of level N with Fourier coefficients in Z. The Eichler integral of fE (see (3)) and the Weierstrass }-function together give a rational map from the modular curve X0(N) to the coordinates of some model of E: This parametrization has singularities wherever the value of the Eichler integral is in the period lattice. Kodgis [6] showed computationally that many of the zeros of the Eichler integral occur at CM points. -
A Glimpse of the Laureate's Work
A glimpse of the Laureate’s work Alex Bellos Fermat’s Last Theorem – the problem that captured planets moved along their elliptical paths. By the beginning Andrew Wiles’ imagination as a boy, and that he proved of the nineteenth century, however, they were of interest three decades later – states that: for their own properties, and the subject of work by Niels Henrik Abel among others. There are no whole number solutions to the Modular forms are a much more abstract kind of equation xn + yn = zn when n is greater than 2. mathematical object. They are a certain type of mapping on a certain type of graph that exhibit an extremely high The theorem got its name because the French amateur number of symmetries. mathematician Pierre de Fermat wrote these words in Elliptic curves and modular forms had no apparent the margin of a book around 1637, together with the connection with each other. They were different fields, words: “I have a truly marvelous demonstration of this arising from different questions, studied by different people proposition which this margin is too narrow to contain.” who used different terminology and techniques. Yet in the The tantalizing suggestion of a proof was fantastic bait to 1950s two Japanese mathematicians, Yutaka Taniyama the many generations of mathematicians who tried and and Goro Shimura, had an idea that seemed to come out failed to find one. By the time Wiles was a boy Fermat’s of the blue: that on a deep level the fields were equivalent. Last Theorem had become the most famous unsolved The Japanese suggested that every elliptic curve could be problem in mathematics, and proving it was considered, associated with its own modular form, a claim known as by consensus, well beyond the reaches of available the Taniyama-Shimura conjecture, a surprising and radical conceptual tools. -
The Arbeitsgemeinschaft 2017: Higher Gross-Zagier Formulas Notes By
The Arbeitsgemeinschaft 2017: Higher Gross-Zagier Formulas Notes by Tony Feng1 Oberwolfach April 2-8, 2017 1version of April 11, 2017 Contents Note to the reader 5 Part 1. Day One 7 1. An overview of the Gross-Zagier and Waldspurger formulas (Yunqing Tang) 8 2. The stacks Bunn and Hecke (Timo Richarz) 13 3. (Moduli of Shtukas I) (Doug Ulmer) 17 4. Moduli of Shtukas II (Brian Smithling) 21 Part 2. Day Two 25 5. Automorphic forms over function fields (Ye Tian) 26 6. The work of Drinfeld (Arthur Cesar le Bras) 30 7. Analytic RTF: Geometric Side (Jingwei Xiao) 36 8. Analytic RTF: Spectral Side (Ilya Khayutin) 41 Part 3. Day Three 47 9. Geometric Interpretation of Orbital Integrals (Yihang Zhu) 48 10. Definition and properties of Md (Jochen Heinloth) 52 Part 4. Day Four 57 11. Intersection theory on stacks (Michael Rapoport) 58 12. LTF for Cohomological Correspondences (Davesh Maulik) 64 µ 13. Definition and description of HkM;d: expressing Ir(hD) as a trace (Liang Xiao) 71 3 4 CONTENTS 14. Alternative calculation of Ir(hD) (Yakov Varshavsky) 77 Part 5. Day Five 83 15. Comparison of Md and Nd; the weight factors (Ana Caraiani) 84 16. Horocycles (Lizao Ye) 89 17. Cohomological spectral decomposition and finishing the proof (Chao Li) 93 NOTE TO THE READER 5 Note to the reader This document consists of notes I live-TEXed during the Arbeitsgemeinschaft at Oberwolfach in April 2017. They should not be taken as a faithful transcription of the actual lectures; they represent only my personal perception of the talks. -
Andrew Wiles's Marvelous Proof
Andrew Wiles’s Marvelous Proof Henri Darmon ermat famously claimed to have discovered “a truly marvelous proof” of his Last Theorem, which the margin of his copy of Diophantus’s Arithmetica was too narrow to contain. While this proof (if it ever existed) is lost to posterity, AndrewF Wiles’s marvelous proof has been public for over two decades and has now earned him the Abel Prize. According to the prize citation, Wiles merits this recogni- tion “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.” Few can remain insensitive to the allure of Fermat’s Last Theorem, a riddle with roots in the mathematics of ancient Greece, simple enough to be understood It is also a centerpiece and appreciated Wiles giving his first lecture in Princeton about his by a novice of the “Langlands approach to proving the Modularity Conjecture in (like the ten- early 1994. year-old Andrew program,” the Wiles browsing of Theorem 2 of Karl Rubin’s contribution in this volume). the shelves of imposing, ambitious It is also a centerpiece of the “Langlands program,” the his local pub- imposing, ambitious edifice of results and conjectures lic library), yet edifice of results and which has come to dominate the number theorist’s view eluding the con- conjectures which of the world. This program has been described as a “grand certed efforts of unified theory” of mathematics. Taking a Norwegian per- the most brilliant has come to dominate spective, it connects the objects that occur in the works minds for well of Niels Hendrik Abel, such as elliptic curves and their over three cen- the number theorist’s associated Abelian integrals and Galois representations, turies, becoming with (frequently infinite-dimensional) linear representa- over its long his- view of the world. -
Fermat's Last Theorem
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions.[1] The proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death.[2] After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995; it was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016.[3] It also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.[4] Contents The 1670 edition of Diophantus's Arithmetica includes Fermat's Overview commentary, referred to as his "Last Pythagorean origins Theorem" (Observatio Domini Petri Subsequent developments and solution de Fermat), posthumously published Equivalent statements of the theorem by his son. -
Arxiv:2003.08242V1 [Math.HO] 18 Mar 2020
VIRTUES OF PRIORITY MICHAEL HARRIS In memory of Serge Lang INTRODUCTION:ORIGINALITY AND OTHER VIRTUES If hiring committees are arbiters of mathematical virtue, then letters of recom- mendation should give a good sense of the virtues most appreciated by mathemati- cians. You will not see “proves true theorems” among them. That’s merely part of the job description, and drawing attention to it would be analogous to saying an electrician won’t burn your house down, or a banker won’t steal from your ac- count. I don’t know how electricians or bankers recommend themselves to one another, but I have read a lot of letters for jobs and prizes in mathematics, and their language is revealing in its repetitiveness. Words like “innovative” or “original” are good, “influential” or “transformative” are better, and “breakthrough” or “deci- sive” carry more weight than “one of the best.” Best, of course, is “the best,” but it is only convincing when accompanied by some evidence of innovation or influence or decisiveness. When we try to answer the questions: what is being innovated or decided? who is being influenced? – we conclude that the virtues highlighted in reference letters point to mathematics as an undertaking relative to and within a community. This is hardly surprising, because those who write and read these letters do so in their capacity as representative members of this very community. Or I should say: mem- bers of overlapping communities, because the virtues of a branch of mathematics whose aims are defined by precisely formulated conjectures (like much of my own field of algebraic number theory) are very different from the virtues of an area that grows largely by exploring new phenomena in the hope of discovering simple This article was originally written in response to an invitation by a group of philosophers as part arXiv:2003.08242v1 [math.HO] 18 Mar 2020 of “a proposal for a special issue of the philosophy journal Synthese` on virtues and mathematics.” The invitation read, “We would be delighted to be able to list you as a prospective contributor. -
Elliptic Curves, Modularity, and Fermat's Last Theorem
Elliptic curves, modularity, and Fermat's Last Theorem For background and (most) proofs, we refer to [1]. 1 Weierstrass models Let K be any field. For any a1; a2; a3; a4; a6 2 K consider the plane projective curve C given by the equation 2 2 3 2 2 3 y z + a1xyz + a3yz = x + a2x z + a4xz + a6z : (1) An equation as above is called a Weierstrass equation. We also say that (1) is a Weierstrass model for C. L-Rational points For any field extension L=K we can consider the L-rational points on C, i.e. the points on C with coordinates in L: 2 C(L) := f(x : y : z) 2 PL : equation (1) is satisfied g: The point at infinity The point O := (0 : 1 : 0) 2 C(K) is the only K-rational point on C with z = 0. It is always smooth. Most of the time we shall instead of a homogeneous Weierstrass equation write an affine Weierstrass equation: 2 3 2 C : y + a1xy + a3y = x + a2x + a4x + a6 (2) which is understood to define a plane projective curve. The discriminant Define 2 b2 := a1 + 4a2; b4 := 2a4 + a1a3; 2 b6 := a3 + 4a6; 2 2 2 b8 := a1a6 + 4a2a6 − a1a3a4 + a2a3 − a4: For any Weierstrass equation we define its discriminant 2 3 2 ∆ := −b2b8 − 8b4 − 27b6 + 9b2b4b6: 1 Note that if char(K) 6= 2, then we can perform the coordinate transformation 2 3 2 y 7! (y − a1x − a3)=2 to arrive at an equation y = 4x + b2x + 2b4x + b6. -
The Modularity Theorem
R. van Dobben de Bruyn The Modularity Theorem Bachelor's thesis, June 21, 2011 Supervisors: Dr R.M. van Luijk, Dr C. Salgado Mathematisch Instituut, Universiteit Leiden Contents Introduction4 1 Elliptic Curves6 1.1 Definitions and Examples......................6 1.2 Minimal Weierstrass Form...................... 11 1.3 Reduction Modulo Primes...................... 14 1.4 The Frey Curve and Fermat's Last Theorem............ 15 2 Modular Forms 18 2.1 Definitions............................... 18 2.2 Eisenstein Series and the Discriminant............... 21 2.3 The Ring of Modular Forms..................... 25 2.4 Congruence Subgroups........................ 28 3 The Modularity Theorem 36 3.1 Statement of the Theorem...................... 36 3.2 Fermat's Last Theorem....................... 36 References 39 2 Introduction One of the longest standing open problems in mathematics was Fermat's Last Theorem, asserting that the equation an + bn = cn does not have any nontrivial (i.e. with abc 6= 0) integral solutions when n is larger than 2. The proof, which was completed in 1995 by Wiles and Taylor, relied heavily on the Modularity Theorem, relating elliptic curves over Q to modular forms. The Modularity Theorem has many different forms, some of which are stated in an analytic way using Riemann surfaces, while others are stated in a more algebraic way, using for instance L-series or Galois representations. This text will present an elegant, elementary formulation of the theorem, using nothing more than some basic vocabulary of both elliptic curves and modular forms. For elliptic curves E over Q, we will examine the reduction E~ of E modulo any prime p, thus introducing the quantity ~ ap(E) = p + 1 − #E(Fp): We will also give an almost complete description of the conductor NE associated to an elliptic curve E, and compute it for the curve used in the proof of Fermat's Last Theorem. -
The Modularity Theorem Jerry Shurman Y + Xy + Y = X − X − X − 14 −1, 0, −2, 4, 0, −2, 1, −4, 4, 6, 4,
The Modularity Theorem Jerry Shurman y2 + xy + y = x3 − x2 − x − 14 −1, 0, −2, 4, 0, −2, 1, −4, 4, 6, 4, . This talk discusses a result called the Modu- larity Theorem: All rational elliptic curves arise from modular forms. Taniyama first suggested in the 1950’s that a statement along these lines might be true, and a precise conjecture was formulated by Shimura. A 1967 paper of Weil provides strong theoretical evidence for the conjecture. The theorem was proved in the mid-1990’s for a large class of elliptic curves by Wiles with a key ingredient supplied by joint work with Taylor, completing the proof of Fermat’s Last The- orem after some 350 years. The Modularity Theorem was proved completely around 2000 by Breuil, Conrad, Diamond, and Taylor. I. A Motivating Example For any d ∈ Z, d 6= 0, consider a quadratic equation, Q : x2 − dy2 = 1. For any prime p not dividing 2d, let Q(Fp) de- note the solutions (x,y) of Q working over the e field Fp of p elements, i.e., working modulo p. Since we expect roughly p solutions, define a normalized solution-count ap(Q)= p −|Q(Fp)|. There is a bijective correspondencee 1 2 P (Fp) \{t : dt = 1} ←→ Q(Fp). Specifically (exercise), the map ine one direc- tion is 1+ dt2 2t t 7→ , , ∞ 7→ (−1, 0) 1 − dt2 1 − dt2! and the map in the other direction is y (x,y) 7→ , (−1, 0) 7→ ∞. x + 1 1 2 But the set P (Fp) \ {t : dt = 1} contains p − 1 elements or p + 1 elements depending on whether d is a square modulo p or not. -
Arxiv:2107.13991V2 [Math.AG] 1 Sep 2021 TBEVCO UDE NAHYPER-K a on BUNDLES VECTOR STABLE ..Telvsbpc Fa Betin Object an of Subspace LLV the 6.3
STABLE VECTOR BUNDLES ON A HYPER-KAHLER¨ MANIFOLD WITH A RANK 1 OBSTRUCTION MAP ARE MODULAR EYAL MARKMAN Abstract. Let X be an irreducible 2n-dimensional holomorphic symplectic manifold. A re- flexive sheaf F is modular, if its Azumaya algebra End(F ) deforms with X to every K¨ahler deformation of X. We show that if F is a slope-stable reflexive sheaf of positive rank and the obstruction map HH2(X) → Ext2(F, F ) has rank 1, then F is modular. We associate to such a sheaf a vector in the Looijenga-Lunts-Verbitsky lattice of rank b2(X) + 2. Three sources of examples of such modular sheaves emerge. The first source consists of slope-stable reflexive sheaves F of positive rank which are isomorphic to the image Φ(OX ) of the structure sheaf via an equivalence Φ : Db(X) → Db(Y ) of the derived categories of two irreducible holomorphic symplectic manifolds. The second source consists of such F , which are isomorphic to the image of a sky-scraper sheaf via a derived equivalence. The third source consists of images Φ(L) of torsion sheaves L supported as line bundles on holomorphic lagrangian submanifolds Z, such that Z deforms with X in co-dimension one in moduli and L is a rational power of the canonical line bundle of Z. An example of the first source is constructed using a stable and rigid vector bundle G on a K3 surface X to get the modular vector bundle F on the Hilbert scheme X[n], associated to the equivariant vector bundle G ⊠ · · · ⊠ G on Xn via the BKR-correspondence. -
Lectures on Modular Forms 1St Edition Free Download
FREE LECTURES ON MODULAR FORMS 1ST EDITION PDF Joseph J Lehner | 9780486821405 | | | | | Lectures on Modular Forms - Robert C. Gunning - Google книги In mathematicsa modular form is a complex analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular groupand also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topologysphere packingand string theory. Instead, modular functions are meromorphic that is, they are almost holomorphic except for a set of isolated points. Modular form theory is a special case of the more general theory of automorphic formsand therefore can now be seen as just the most concrete part of a rich theory of discrete groups. Modular forms can also be interpreted as sections of a specific line bundles on modular varieties. The dimensions of these spaces of modular forms can be computed using the Riemann—Roch theorem [2]. A Lectures on Modular Forms 1st edition form of weight k for the modular group. A modular form can equivalently be defined as a function F from the set of lattices in C to the set of complex numbers which satisfies certain conditions:. The simplest examples from this point of view are the Eisenstein series. Then E k is a modular form of weight k. An even unimodular lattice L in R n is a lattice generated by n Lectures on Modular Forms 1st edition forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in L is an even integer. -
On the Modularity of Elliptic Curves
Uniwersytet Warszawski Wydziaª Matematyki, Informatyki i Mechaniki Przemysªaw Chojecki Nr albumu: 262926 On the modularity of elliptic curves Praca licencjacka na kierunku MATEMATYKA Praca wykonana pod kierunkiem dra hab. Adriana Langera Instytut Matematyki Czerwiec 2009 O±wiadczenie kieruj¡cego prac¡ Potwierdzam, »e niniejsza praca zostaªa przygotowana pod moim kierunkiem i kwa- lifikuje si¦ do przedstawienia jej w post¦powaniu o nadanie tytuªu zawodowego. Data Podpis kieruj¡cego prac¡ O±wiadczenie autora (autorów) pracy Świadom odpowiedzialno±ci prawnej o±wiadczam, »e niniejsza praca dyplomowa zostaªa napisana przeze mnie samodzielnie i nie zawiera tre±ci uzyskanych w sposób niezgodny z obowi¡zuj¡cymi przepisami. O±wiadczam równie», »e przedstawiona praca nie byªa wcze±niej przedmiotem pro- cedur zwi¡zanych z uzyskaniem tytuªu zawodowego w wy»szej uczelni. O±wiadczam ponadto, »e niniejsza wersja pracy jest identyczna z zaª¡czon¡ wersj¡ elektroniczn¡. Data Podpis autora (autorów) pracy Streszczenie Praca ta jest przegla,dem podstawowych definicji i twierdze«, które sa, potrzebne do zrozu- mienia wyniku znanego obecnie jako twierdzenie o modularno±ci. Szczególny przypadek tego twierdzenia, wystarczaja,cy do otrzymania Wielkiego Twierdzenia Fermata jako wniosku, zo- staª udowodniony w 1995 roku przez Andrew Wilesa (z pomoca, Richarda Taylora), a w caªej ogólności w 2000 roku przez Breuila, Conrada, Diamonda i Taylora. Dawniej twierdzenie o modularno±ci byªo znane jako hipoteza Taniyamy-Shimury-Weila. In this work I make a survey of basic definitions and theorems needed to understand a result called the Modularity Theorem. It was proved in '95 by Andrew Wiles (with a help of Richard Taylor) in the case when an elliptic curve in question is semistable and later in '00, it was proved in full generality by Breuil-Conrad-Diamond-Taylor.