How Many Rational Points Does a Random Curve Have?
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Algebraic Curves and Surfaces
Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces . -
Arxiv:2002.02220V2 [Math.AG] 8 Jun 2020 in the Early 80’S, V
Contemporary Mathematics Toward good families of codes from towers of surfaces Alain Couvreur, Philippe Lebacque, and Marc Perret To our late teacher, colleague and friend Gilles Lachaud Abstract. We introduce in this article a new method to estimate the mini- mum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be \liftable" under finite morphisms, paving the way toward the construction of good codes from towers of surfaces. In the same direction, we establish a criterion for a surface with a fixed finite set of closed points P to have an infinite tower of `{´etalecovers in which P splits totally. We conclude by stating several open problems. In par- ticular, we relate the existence of asymptotically good codes from general type surfaces with a very ample canonical class to the behaviour of their number of rational points with respect to their K2 and coherent Euler characteristic. Contents Introduction 1 1. Codes from surfaces 3 2. Infinite ´etaletowers of surfaces 15 3. Open problems 26 Acknowledgements 29 References 29 Introduction arXiv:2002.02220v2 [math.AG] 8 Jun 2020 In the early 80's, V. D. Goppa proposed a construction of codes from algebraic curves [21]. A pleasant feature of these codes from curves is that they benefit from an elementary but rather sharp lower bound for the minimum distance, the so{called Goppa designed distance. In addition, an easy lower bound for the dimension can be derived from Riemann-Roch theorem. The very simple nature of these lower bounds for the parameters permits to formulate a concise criterion for a sequence of curves to provide asymptotically good codes : roughly speaking, the curves should have a large number of rational points compared to their genus. -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
Proceedings of the Conference on Promoting Undergraduate Research in Mathematics Proceedings of the Conference on Promoting Undergraduate Research in Mathematics
In the span of twenty years, research in Proceedings mathematics by undergraduates has gone from rare to commonplace. To examine Promoting Undergraduate Research in Mathematics Gallian, Editor of the and promote this trend, the AMS and Proceedings NSA sponsored a three-day conference Conference in September 2006 to bring together of the mathematicians who were involving on Promoting undergraduates across the country in Conference on mathematics research to exchange ideas, Undergraduate discuss issues of common concern, establish contacts, and gather information Research in that would be of value to others interested Promoting in engaging undergraduates in mathematics Mathematics research. The conference featured plenary talks, panel discussions, and small group sessions. The topics included summer Undergraduate research programs, academic year research Joseph A. Gallian opportunities, diversity issues, assessment Editor methods, and perspectives from alumni Research in of research programs. This volume is the proceedings of that conference. Mathematics Joseph A. Gallian Editor AMS on the Web PURM/GALLIAN2 www.ams.org AMS 472 pages • Spine Width: 7/8 inch • Two Color Cover: This plate PMS 4515 (Beige) This plate PMS 323 (Teal) Proceedings of the Conference on Promoting Undergraduate Research in Mathematics Proceedings of the Conference on Promoting Undergraduate Research in Mathematics Joseph A. Gallian Editor American Mathematical Society Providence, Rhode Island Cover photography courtesy of Cindy Wyels, CSU Channel Islands, Camarillo, CA, and Darren Narayan, Rochester Institute of Technology, Rochester, NY. This material is based upon work supported by National Security Agency under grant H98230-06-1-0095. ISBN-13: 978-0-8218-4321-5 ISBN-10: 0-8218-4321-4 Copyright c 2007 American Mathematical Society Printed in the United States of America. -
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 the Density of Rational Points on Elliptic Fibrations
On the density of rational points on elliptic fibrations F. A. Bogomolov Courant Institute of Mathematical Sciences, N.Y.U. 251 Mercer str. New York, (NY) 10012, U.S.A. e-mail: [email protected] and Yu. Tschinkel Dept. of Mathematics, U.I.C. 851 South Morgan str. Chicago, (IL) 60607-7045, U.S.A. e-mail: [email protected] arXiv:math/9811043v1 [math.AG] 7 Nov 1998 1 Introduction Let X be an algebraic variety defined over a number field F . We will say that rational points are potentially dense if there exists a finite extension K/F such that the set of K-rational points X(K) is Zariski dense in X. The main problem is to relate this property to geometric invariants of X. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]). Varieties which are not of general type can be thought of as triple fibra- tions X → Y → Z, where the generic fiber of X → Y is rationally connected, Y → Z is a Kodaira fibration with generic fiber of Kodaira dimension 0 and the base has Kodaira dimension ≤ 0 (cf. [3]). In this paper we study mostly varieties of dimension 2 and 3. In dimension 2 the picture is as follows: Rationally connected surfaces are rational over some finite extension of F and therefore the problem has an easy solution in this case. -
Rational Points and Obstructions to Their Existence 2015 Arizona Winter School Problem Set Extended Version
RATIONAL POINTS AND OBSTRUCTIONS TO THEIR EXISTENCE 2015 ARIZONA WINTER SCHOOL PROBLEM SET EXTENDED VERSION KĘSTUTIS ČESNAVIČIUS The primary goal of the problems below is to build up familiarity with some useful lemmas and examples that are related to the theme of the Winter School. Notation. For a field k, we denote by k a choice of its algebraic closure, and by ks Ă k the resulting separable closure. If k is a number field and v is its place, we write kv for the corresponding completion. If k “ Q, we write p ¤ 8 to emphasize that p is allowed be the infinite place; for this particular p, we write Qp to mean R. For a base scheme S and S-schemes X and Y , we write XpY q for the set of S-morphisms Y Ñ X. When dealing with affine schemes we sometimes omit Spec for brevity: for instance, we write Br R in place of BrpSpec Rq. A ‘torsor’ always means a ‘right torsor.’ Acknowledgements. I thank the organizers of the Arizona Winter School 2015 for the opportunity to design this problem set. I thank Alena Pirutka for helpful comments. 1. Rational points In this section, k is a field and X is a k-scheme. A rational point of X is an element x P Xpkq, i.e., a section x: Spec k Ñ X of the structure map X Ñ Spec k. 1.1. Suppose that X “ Spec krT1;:::;Tns . Find a natural bijection pf1;:::;fmq n Xpkq ÐÑ tpx1; : : : ; xnq P k such that fipx1; : : : ; xnq “ 0 for every i “ 1; : : : ; mu: krT1;:::;Tns Hint. -
IV.5 Arithmetic Geometry Jordan S
i 372 IV. Branches of Mathematics where the aj,i1,...,in are indeterminates. If we write with many nice pictures and reproductions. A Scrap- g1f1 + ··· + gmfm as a polynomial in the variables book of Complex Curve Theory (American Mathemat- x1,...,xn, then all the coefficients must vanish, save ical Society, Providence, RI, 2003), by C. H. Clemens, the constant term which must equal 1. Thus we get and Complex Algebraic Curves (Cambridge University a system of linear equations in the indeterminates Press, Cambridge, 1992), by F. Kirwan, also start at an easily accessible level, but then delve more quickly into aj,i1,...,in . The solvability of systems of linear equations is well-known (with good computer implementations). advanced subjects. Thus we can decide if there is a solution with deg gj The best introduction to the techniques of algebraic 100. Of course it is possible that 100 was too small geometry is Undergraduate Algebraic Geometry (Cam- a guess, and we may have to repeat the process with bridge University Press, Cambridge, 1988), by M. Reid. larger and larger degree bounds. Will this ever end? For those wishing for a general overview, An Invitation The answer is given by the following result, which was to Algebraic Geometry (Springer, New York, 2000), by proved only recently. K. E. Smith, L. Kahanpää, P. Kekäläinen, and W. Traves, is a good choice, while Algebraic Geometry (Springer, New Effective Nullstellensatz. Let f1,...,fm be polyno- York, 1995), by J. Harris, and Basic Algebraic Geometry, mials of degree less than or equal to d in n variables, volumes I and II (Springer, New York, 1994), by I. -
Elliptic Curves Over Q∞ Are Modular
Elliptic curves over Q1 are modular Jack A. Thorne∗ July 11, 2016 Abstract We show that if p is a prime, then all elliptic curves defined over the cyclotomic Zp-extension of Q are modular. 1 Introduction Our goal in this paper is to prove the following theorem: Theorem 1. Let p be a prime, and let F be a number field which is contained in the cyclotomic Zp-extension of Q. Let E be an elliptic curve over F . Then E is modular. By definition, the cyclotomic Zp-extension of Q is the unique subfield Q1 of Q(ζp1 ) with Galois ∼ group Gal(Q1=Q) = Zp. It is totally real. An elliptic curve E over a number field F is said to be modular if there is a regular algebraic automorphic representation π of GL2(AF ) which has the same L-function as E. This is one of several equivalent formulations; if F is totally real, then π will be generated by vectors which can be interpreted as Hilbert modular forms of parallel weight 2. The modularity of a given elliptic curve E has many useful consequences. It implies that the L-function of E has an analytic continuation to the whole complex plane, allowing one to formulate the Birch{Swinnerton-Dyer conjecture for E unconditionally. When the order of vanishing of the L-function at the point s = 1 is at most 1, this conjecture is known, in its weak form, in many cases [Zha01]. The modularity of all elliptic curves over Q has been known since work of Wiles and Breuil, Conrad, Diamond, and Taylor [Wil95], [TW95], [CDT99], [BCDT01]. -
Arxiv:1806.01691V2 [Math.NT] 22 Aug 2018 Ubro Primes
INDIVISIBILITY OF HEEGNER POINTS AND ARITHMETIC APPLICATIONS ASHAY A. BURUNGALE, FRANCESC CASTELLA, AND CHAN-HO KIM Abstract. We upgrade Howard’s divisibility towards Perrin-Riou’s Heegner point main con- jecture to the predicted equality. Contrary to previous works in this direction, our main result allows for the classical Heegner hypothesis and non-squarefree conductors. The main ingredi- ents we exploit are W. Zhang’s proof of Kolyvagin’s conjecture, Kolyvagin’s structure theorem for Shafarevich–Tate groups, and the explicit reciprocity law for Heegner points. Contents 1. Introduction 1 2. Selmer structures 4 3. Heegner point Kolyvagin systems 5 4. Equivalent main conjectures 7 5. Equivalent special value formulas 7 6. Skinner–Urban lifting lemma 9 7. Proof of the main results 9 References 10 1. Introduction Let E/Q be an elliptic curve of conductor N and let K be an imaginary quadratic field of discriminant DK with (DK ,N) = 1. Then K determines a factorization N = N +N − with N + (resp. N −) divisible only by primes which are split (resp. inert) in K. Throughout this paper, the following hypothesis will be in force: arXiv:1806.01691v2 [math.NT] 22 Aug 2018 Assumption 1.1 (Generalized Heegner hypothesis). N − is the square-free product of an even number of primes. Let p > 3 be a good ordinary prime for E with (p, DK ) = 1, and let K∞ be the anti- cyclotomic Zp-extension of K. Under Assumption 1.1, the theory of complex multiplication + provides a collection of CM points on a Shimura curve with “Γ0(N )-level structure” attached to the quaternion algebra B/Q of discriminant N − defined over ring class extensions of K. -
Rational Points on Curves
Rational points on curves December 6, 2013 Contents 1 Curves of genus 0 2 1.1 Rational points.......................................2 1.2 Representation by quadratic forms............................4 1.3 Local representatbility...................................5 2 Curves of genus 1 6 2.1 Basic algebraic geometry of curves............................6 2.2 Curves of genus 1......................................8 2.3 Elliptic curves over C or finite fields........................... 10 3 Arithmetic curves 10 3.1 Divisors and line bundles................................. 10 3.2 Riemann{Roch....................................... 13 3.3 Applications......................................... 13 4 Mordell{Weil theorem 15 4.1 Mordell{Weil theorem................................... 15 4.2 Weak Mordell{Weil theorem................................ 16 5 Hermitian line bundles 17 5.1 Divisors, line bundles, and metrics............................ 17 5.2 Arithmetic line bundles and arithmetic divisors..................... 18 5.3 Heights........................................... 19 6 Heights without hermitian line bundles 20 6.1 Behavior under morphisms................................ 21 6.2 Nothcott property..................................... 23 6.3 Neron{Tate height on elliptic curves........................... 24 7 Birch and Swinnerton-Dyer conjectures 25 7.1 Tate{Shafarevich groups.................................. 25 7.2 L-functions......................................... 26 7.3 Geometric analog...................................... 28 1 8 Constant -
Coregular Spaces and Genus One Curves
Cambridge Journal of Mathematics Volume 4, Number 1, 1–119, 2016 Coregular spaces and genus one curves Manjul Bhargava∗ and Wei Ho† A coregular space is a representation of an algebraic group for which the ring of polynomial invariants is freely generated. We show that the orbits of many coregular irreducible representations of algebraic groups where the number of generating invariants is at least two, over a (not necessarily algebraically closed) field k, correspond to genus one curves over k together with line bundles, vector bundles, and/or points on their Jacobian curves. In particu- lar, we give explicit descriptions of certain moduli spaces of genus one curves with extra structure as quotients by algebraic groups of open subsets of affine spaces. In most cases, we also describe how the generators of the in- variant rings are geometrically manifested, often as coefficients for the Jacobian families of elliptic curves for the genus one curves. We also show how many of the correspondences, including their proofs, are special cases of two general constructions related to Hermitian matrices and cubic Jordan algebras. In forthcoming work, we use these orbit parametrizations to determine the average sizes of Selmer groups for various families of elliptic curves. 1 Introduction 3 2 Main results I: Genus one curves and 2×2×2×2 hypercubes 9 2.1 Preliminaries on 2 × 2 × 2 cubes 10 2.2 On 2 × 2 × 2 × 2 hypercubes 11 2.3 Genus one curves from hypercubes 13 2.4 Orbit classification of 2 × 2 × 2 × 2 hypercubes 17 2.5 Symmetrization 18 2.6 Triple symmetrization 19 ∗Supported by a Packard Foundation Fellowship and NSF grant DMS-1001828.