Lattices and Rational Points
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Sheet 8 Solutions Are Due on 08.06.18
Dr. Severin Bunk Algebra und Zahlentheorie Fachbereich Mathematik Algebraic Topology Universit¨atHamburg Summer 2018 Sheet 8 Solutions are due on 08.06.18. Problem 8.1 (a) Show that for each abelian group D, the tensor product ( ) D is right-exact. That is, α β − ⊗ show that for every short exact sequence 0 A B C 0 of abelian groups, there → → → → is an exact sequence of abelian groups α id β idD A D ⊗ D B D ⊗ C D 0 . ⊗ ⊗ ⊗ β (b) Prove that if 0 A α B C 0 is split exact, then for any abelian group D the → → → → induced sequence α id β idD 0 A D ⊗ D B D ⊗ C D 0 . ⊗ ⊗ ⊗ is exact. (c) Is the abelian group (Q, +) free? Show that ( ) Q is exact. − ⊗ Hint: Observe that for any abelian group A and field k, the tensor product A k is a ⊗ k-vector space. For the second part, start by working out the kernel of the map qA : A A Q, a a 1. → ⊗ 7→ ⊗ (d) The rank of an abelian group A can be defined as rk(A) := dim (A Q) . Q ⊗ n Show that if A is finitely generated, i.e. A = Z Zq1 Zqm , then rk(A) = n. Moreover, α β ∼ ⊕ ⊕· · ·⊕ show that if 0 A B C 0 is any short exact sequence of finitely generated abelian → → → → groups, then rk(B) = rk(A) + rk(C) . 1 Problem 8.2 Let X be a a finite CW complex. The Euler characteristic of X is defined as n χ(X) := ( 1) rk Hn(X, Z) Z . -
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. -
JETIR Research Journal
© 2018 JETIR July 2018, Volume 5, Issue 7 www.jetir.org (ISSN-2349-5162) A STUDY ON FREE ABELIAN GROUPS AND ITS PROPERTIES Meera Rose Joseph1 1Assistant Professor, Nirmala College, Muvattupuzha Abstract: An Abelian group, also called a commutative group is a group in which the result of applying the group operation to two group elements does not depend on their order. In this paper we discuss about Free abelian groups and its properties, where Free abelian groups are special cases of Free modules as abelian groups are nothing but modules over the ring. we are taking into account how a Free abelian group is related to Free group and Torsion free abelian group. Also we consider one of the important application of Free abelian groups that is “The proof of Fundamental Theorem of Finitely Generated Abelian Groups” Keywords: Free abelian group, Free modules, Free group, Torsion free abelian group 1. Introduction A group is an ordered pair (G,*), where G is a nonempty set and * is a binary operation on G such that the following properties hold: (1) For all a, b, c ∊ G, a * (b * c) = (a * b) * c. (associative law) (2) There exists e ∊ G such that, for any a ∊ G , a*e = a = e*a. (Existence of an identity) (3) For each a ∊ G, there exits b ∊ G, there exists b ∊ G such that a * b = e = b * a. (existence of an inverse). An Abelian group is a group with the property that a * b = b * a. Again a Free Abelian group is an Abelian group with a basis where a basis is a subset of the elements such that every group element can be found by adding or subtracting a finite number of basis elements and such that every element, its expression as a linear combination of basis elements is unique. -
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. -
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 -
Arxiv:1803.02069V1 [Math.NT] 6 Mar 2018 Ayelpi Uvsdfie Yeutoso H Form the of Equations by Defined Curves Elliptic Many Be Have Respectively
SEQUENCES OF CONSECUTIVE SQUARES ON QUARTIC ELLIPTIC CURVES MOHAMED KAMEL AND MOHAMMAD SADEK Abstract. Let C : y2 = ax4 + bx2 + c, be an elliptic curve defined over Q. A set of rational points (xi,yi) ∈ C(Q), i = 1, 2, ··· , is said to be a sequence 2 of consecutive squares if xi = (u + i) , i = 1, 2, ··· , for some u ∈ Q. Using ideas of Mestre, we construct infinitely many elliptic curves C with sequences of consecutive squares of length at least 6. It turns out that these 6 rational points are independent. We then strengthen this result by proving that for a fixed 6- term sequence of consecutive squares, there are infinitely many elliptic curves C with the latter sequence forming the x-coordinates of six rational points in C(Q). 1. Introduction In [3], Bremner started discussing the existence of long sequences on elliptic curves. He produced an infinite family of elliptic curves with arithmetic progres- sion sequences of length 8. Several authors displayed infinite families of elliptic curves with long arithmetic progression sequences, see [5, 7, 10]. A geometric progression sequence is another type of sequence that has been studied on elliptic and hyperelliptic curves. Infinitely many (hyper)elliptic curves with 5-term and 8-term geometric progression sequences have been introduced in [4] and [1] respectively. arXiv:1803.02069v1 [math.NT] 6 Mar 2018 In [6], sequences of consecutive squares on elliptic curves were studied. Infinitely many elliptic curves defined by equations of the form E : y2 = ax3 + bx + c, a, b, c ∈ Q, with 5-term sequences of consecutive squares were presented. -
Subgroups of Finitely Presented Metabelian Groups of Finite Rank
J. Austral. Math. Soc. 22 (Series A) (1976), 501-508. SUBGROUPS OF FINITELY PRESENTED METABELIAN GROUPS OF FINITE RANK JAMES BOLER (Received 6 January 1975; revised 9 April 1976) Abstract Let G be a finitely generated metabehan group whose derived group G' has finite rank. It is shown that G can be embedded in a finitely presented metabelian group H with H' of finite rank. Introduction Baumslag (1973) has shown that each finitely generated metabelian group can be embedded in a finitely presented metabelian group. Our objective here is to prove THEOREM 4.1. Let G be a finitely generated metabelian group whose derived group G' has finite rank. Then there exists a finitely presented metabelian group H with H' of finite rank, and an embedding of G into H. The proof of Theorem 4.1 makes use of the well-known connection between finitely generated metabelian groups and modules over finitely generated commutative rings. This relationship is mentioned in Section 1, along with some basic facts about the notion of rank. In Section 2 a residual property of certain modules is established. This property is used in Section 3 and Section 4 to prove Theorem 4.1. 1. Preliminaries We denote the ring of integers by Z and the field of rational numbers by Q. If x and y are elements of a group, we use the standard notation xy = y~lxy;[x, y] = x~'y~'xy. 501 Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.42, on 26 Sep 2021 at 01:04:11, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. -
On Rational Points in CFT Moduli Spaces
CALT-TH 2020-050, IPMU20-0117 On Rational Points in CFT Moduli Spaces Nathan Benjamin,a Christoph A. Keller,b Hirosi Ooguri,c;d and Ida G. Zadehe a Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA b Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089, USA c Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA d Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, 277-8583, Japan e International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy nathanb at princeton.edu, cakeller at math.arizona.edu, ooguri at caltech.edu, zadeh at ictp.it ABSTRACT Motivated by the search for rational points in moduli spaces of two-dimensional conformal arXiv:2011.07062v1 [hep-th] 13 Nov 2020 field theories, we investigate how points with enhanced symmetry algebras are distributed there. We first study the bosonic sigma-model with S1 target space in detail and uncover hitherto unknown features. We find for instance that the vanishing of the twist gap, though true for the S1 example, does not automatically follow from enhanced symmetry points being dense in the moduli space. We then explore the supersymmetric sigma- model on K3 by perturbing away from the torus orbifold locus. Though we do not reach a definite conclusion on the distribution of enhanced symmetry points in the K3 moduli space, we make several observations on how chiral currents can emerge and disappear under conformal perturbation theory. November 13, 2020 Contents 1 Introduction 1 2 Warm-up: S1 3 2.1 Spectrum . -
(Linear) Algebraic Groups 1 Basic Definitions and Main Examples
Study Group: (Linear) Algebraic Groups 1 Basic Denitions and Main Examples (Matt) Denition 1.1. Let (where is some eld) then n is an ane algebraic I C K[x] K VI = fP 2 A : f(P ) = 08f 2 Ig set. If I is prime, then VI is an ane algebraic variety. Denition 1.2. A linear algebraic group, G, is a variety V=K with a group structure such that the group operations are morphisms and V is ane. K-rational point e 2 G and K-morphisms µ : G × G ! G (µ(x; y) = xy) and i : G ! G (i(x) = x−1) K[G] is the K algebra of regular functions in V . ∆ : K[G] ! K[G] ⊗K K[G] the comultiplication, ι : K[G] ! K[G] is the coinverse We get the following axioms: (∆ ⊗ 1) ◦ ∆ = (1 ⊗ ∆) ◦ ∆ Example. : where , 1. , ∼ . n2 ∼ . Ga µ(x; y) = x+y a; y 2 A K[x] ∆ : x 7! x⊗1+1⊗y 2 K[x]⊗K[y] = K[x; y] Ga = Mn(K) : ∗ 2 dened by . Then . So Gm K ! V = (x; y) 2 A jxy = 1 t 7! (t; 1=t) (x1; y1) · (x2; y2) = (x1x2; y1y2) ∆ : x 7! x ⊗ x; y 7! y ⊗ y. Note that =∼ GL (K) p p Gm p 1 p p p Let , , −1 x1−y1 2 . Let K = Q( 2)=Q (x1 + y1 2)(x2 + y2 2) = (x1x2 + 2y1y2) + (x1y2 + x2y1) 2 (x1 + y1 2) = 2 2 x1−2y1 2 2, then we can use . , . g = x1−2y1 V (g 6= 0) ∆ : x1 7! x1⊗x2+2y1⊗y2 y1 7! x1⊗y2+x2⊗y1 ι : x1 7! x1=g; y1 7! −y1=g n .