DOCSLIB.ORG

F. Pellarin the ISOGENY THEOREM and the IRREDUCIBILITY THEOREM for ELLIPTIC CURVES: a SURVEY

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
F. Pellarin the ISOGENY THEOREM and the IRREDUCIBILITY THEOREM for ELLIPTIC CURVES: a SURVEY Rend. Sem. Mat. Univ. Poi. Torino Voi. 53, 4 (1995) Number Theory F. Pellarin THE ISOGENY THEOREM AND THE IRREDUCIBILITY THEOREM FOR ELLIPTIC CURVES: A SURVEY Abstract.The purpose of the present paper is to give a survey on the isogeny theorem and the irreducibility theorem for elliptic curves with regard to effectiveness. After some preliminaries, we present in Section 2 the most important questions on this subject throughout the work of Serre, making use of ultrametric techniques. Transcendental techniques are introduced in Section 3: these are advantageously used by Choodnovsky to get effective versions of the isogeny theorem. In Section 4 we present the work of Masser and Wustholz on polynomial bounds and applications of the effective isogeny theorem. Finally we show recent improvements of the previous results and applications to the effective irreducibility theorem. 1. Background In the following text, k is a number fìeld such that [k : Q] = d, k an algebraic closure of k in C, and G the Galois group Gal(fc|fc). Let E\, E2 be two elliptic curves defined over k (with Weierstrass models defined over k). In this text we will only deal with elliptic curves defined over a number fìeld k. DEFINITION. We say that E\ and E2 are isogenous if there exists a surjective morphism E\ —• Ei. We say that E\ and E2 are fc-isogenous if there exists a surjective morphism E\ —» E2 commuting with G. An isogeny (resp. k-isogeny) E\ —> E2 is a surjective morphism (resp. surjective morphism defined over k) whose kernel is a subgroup ofEi. r Write Hom(JEi,i?2) f° the /£-module of isogenies E\ —* E2 and Homjs(EiìE2) ' for the Z-module of &-isogenies E\ —> E2, with the Constant zero map as the identity. In the particular case E\ — E2 — E, the Z-modules Hom(i?i,i?2) and Hom^Ei,E2) are endowed with ring structures End(^) and Enók(E) respectively. Remember that End(i?) has rank at most 2 over Z, and that E has complex multiplication if the rank is 2 (in this case End(i?) is isomorphic to an order of an imaginary quadratic fìeld). Otherwise E has not complex multiplication, and the rank is 1. The Z-module Hom(^i, E2) has rank at most 1 if Ex has no complex multiplications 390 E Pellarin (see chapter III of [29]). We associate with E\ and E2 two lattices Ài and A2 of C such that there exist analytic isomorphisms C/Ai = #i(C) and C/A2 = E2(C). The curves Et and E2 are isogenous if and only if there exists a non-constant C-linear function a : C —*• C such that «Ai C A2 (see chapter VI of [29]). DEFINITION. The degree of an isogeny E\ —»• E2 is the cardinal of its kernel, that is, the finite index [A2 : a Ai]. Other useful tools will be described when necessary in the text, otherwise a reference will be given. 2. Tate modules We recali in brief the definition of the Tate module of an elliptic curve (see [8] chapter I and pages 90-103 of [29]). Let E be an elliptic curve. For a prime number £ and a positive integer ??., the groups E\tn\ form a projective system for the multiplication by £. DEFINITION. The Tate module Tt(E) of E at £ is the projective limit of the system E[£n\ Since G acts over each E[in], it acts over Tt(E), and Ti(E) is endowed with a structure of Qi[G]-module: this is a Galois Module. Let E\ and E2 be two elliptic curves and write Hom^^i),^.^)) for the TLf module of ^-linear functions and ìlomk(Ti(Ei),Ti(E2)) for the Z^-module of 2^-linear functions commuting with G. 2.1 The isogeny Theorem and the irreducibility Theorem. THEOREM 2.1. For ali prime numbers £ and ali positive integers n, the restriction n of the k-isogenies of Hom.k(Ei,E2) to the points of £ -torsion induces a canonical isomorphism Homk(EuE2) ®a 1Lt - Hom^T^Ey),Tt(E2)). This Theorem was first proved by J.-P. Serre in 1966 (see [3] and [4]) in the case Ei — E2 when the modular invariant of E\ is not an algebraic integer. His proof uses the "fìniteness" Theorem of Shafarevich (see [2]). Tate proved the Theorem in 1966 (see [5]) in the case of abelian varieties defìned over a finite field k, but in his article he remarks that the Theorem for elliptic curves is a consequence of results of Deuring in 1941 (see [1] and the bibliography of Serre's book [8]). The isogeny theorem and the irreducibility theorem for elliptic curves 391 The analogous result for abelian varieties defìned' over a number field remained a conjecture ("of Tate") until 1983, when it was proved by Faltings (see [20] and [21], see also [24]). The useful consequence for us is the following (see also p. IV-14 of [8]). THEOREM 2.2 (Isogeny Theorem). Let E\ and E2 be two elliptic curves defìned over k and fa a prime number £. Then E\ and E2 are k-isogenous if and only ifthe Galois modules Tt(Ei) and T^En) are isomorphic. This result, issued from [20], is stronger than the one in [8] because it deals with curves with integrai modular invariant. In chapter IV of this last reference there is another important result (see p. IV-9) for the case E\ — Ei — E., arising from the Theorem of Shafarevich [2] on the places of good reduction of E (see also [29] p. 265) that gives the following LEMMA 2.1. Let E be an elliptic curve over k. There are only finitely many k-isomorphism classes of elliptic curves k-isogenous to E. We say that Tt(E) is irreducible if G does not fix any linear subspace of dimension 1 of T£(E). THEOREM 2.3 (Irreducibility). Let E be an elliptic curve over k and let us suppose that E has no complex multiplications over k. Then Ti(E) is irreducible for ali prime numbers £ and E[£] is irreducible for £ > £Q where £Q is a Constant depending on E and k. The proof of the second property is a simple application of the Lemma 2.1. We recali in brief the proof in [29] p. 265. Let $i C E[i] be a non-trivial G-invariant subgroup. Then <3><» is cyclic of order £ and the quotient of E by $i is isomorphic to an elliptic curve Et over k which is k- isogenous to E by an isogeny 4>i with kernel $t. By varying £ and using the fact that Endk(E) — 7L, we obtain the result. In [12] Serre proved a more general result involving the ?2-torsion points E[n] for a rational integer n not necessarily a prime number in the case of no complex multiplication over k. The group G acts on E[n] for n rational integer and the image of the representation in Aut(i?[n]) has an index bounded by a Constant depending only on E and k . This implies LEMMA 2.2. Let E be an elliptic curve over k with no complex multiplications over k. Then there exist two constants c\ and c2 depending only on E and k such that the n An n 4n relative degree [k(E[£ ]) : k] satisfies cx£ < [k(E[£ ]) : k] < c2£ . Proof The main result of [12] implies that the cardinal of Gal(k(E[£n])\k) is An greater than Cl£ = ciCard(GL2(F^)). 392 E Pellarin n Define E(£) as the union of ali groups E[£ ]. Then pt(Ga\(k(Et)\k)) is a closed Lie subgroup of Enók(Tt(E)) and the cardinal of Ga\(k(E[f,n])\k) has order of growth at most £4n as n —» oo. REMARK. We may evaluate the degree of an isogeny by using Theprem 2.1. LEMMA 2.3. Let <j> be a k-isogeny front E\ to E2 and let fa be the image of $ by the naturai injection (see [29] p. 92) (2.1) Homk(EliE2)->Hpmk(Ti(E1)ìTl(E2)). Then deg <ft — ± det fa. Let Ti(/i) be the projective limit of the groups pcn of the P-th roots of unity with the structure of Galois module (see [8] p. 1-3). Let <j> : E\ —• E2 be a fc-isogeny. By the proposition 8.3 p. 99 of [29], there exists an alternating non-degenerate and G'-invariant bilinear pairing, called Weil pairing ei:Ti(Ei)xTt{Ei)->Tl(p), 1=1,2, such that <j> and its dual isogeny 0 (with the property that <j><j> = [deg(<^)], see [29] p. 84) are adjoint by e,;. Let fa be the image of </» in Hoirifc(T£(i?i), Ti(E2)), let us fix a 2^-basis vi, v2 of Tt(Ei) and a 2^-basis v[,v'2 ofTi(E2). Let ( , j be the matrix representing $£ in these bases. We have, by the properties of e; (see [29] p. 135): de ei(i/i,t;2) 8(*) = ci([deg(^)]t;i,t;2) = ei(<j>cfavi,v2) = e2(favi,fav2) — e2{av[ + cv'n, bv[ -f dv'0) Now we observe that the matrix of fa is the adjoint matrix of the matrix of fa. By det( the fact that deg^ = deg0 we also have e^v^.v'^^^ = ei(i>i, v2) ^. We obtain deg^ = ±det fa. REMARK - COMPLEX MULTIPLICATION. If E has complex multiplication over k, ali the problems sketched before are simpler: fìrstly, it is well known that the action of Ga\(k\k) over Tt{E) is abelian ([29] p.
Load more

Recommended publications
  • Abelian Varieties
    Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties .
    [Show full text]
  • Elliptic Curves, Isogenies, and Endomorphism Rings
    Elliptic curves, isogenies, and endomorphism rings Jana Sot´akov´a QuSoft/University of Amsterdam July 23, 2020 Abstract Protocols based on isogenies of elliptic curves are one of the hot topic in post-quantum cryptography, unique in their computational assumptions. This note strives to explain the beauty of the isogeny landscape to students in number theory using three different isogeny graphs - nice cycles and the Schreier graphs of group actions in the commutative isogeny-based cryptography, the beautiful isogeny volcanoes that we can walk up and down, and the Ramanujan graphs of SIDH. This is a written exposition 1 of the talk I gave at the ANTS summer school 2020, available online at https://youtu.be/hHD1tqFqjEQ?t=4. Contents 1 Introduction 1 2 Background 2 2.1 Elliptic curves . .2 2.2 Isogenies . .3 2.3 Endomorphisms . .4 2.4 From ideals to isogenies . .6 3 CM and commutative isogeny-based protocols 7 3.1 The main theorem of complex multiplication . .7 3.2 Diffie-Hellman using groups . .7 3.3 Diffie-Hellman using group actions . .8 3.4 Commutative isogeny-based cryptography . .8 3.5 Isogeny graphs . .9 4 Other `-isogenies 10 5 Supersingular isogeny graphs 13 5.1 Supersingular curves and isogenies . 13 5.2 SIDH . 15 6 Conclusions 16 1Please contact me with any comments or remarks at [email protected]. 1 1 Introduction There are three different aspects of isogenies in cryptography, roughly corresponding to three different isogeny graphs: unions of cycles as used in CSIDH, isogeny volcanoes as first studied by Kohel, and Ramanujan graphs upon which SIDH and SIKE are built.
    [Show full text]
  • Commutative Algebraic Groups up to Isogeny, in Which the Problems Raised by Imperfect fields Become Tractable; This Yields Rather Simple and Uniform Structure Results
    Commutative algebraic groups up to isogeny Michel Brion Abstract Consider the abelian category Ck of commutative group schemes of finite type over a field k. By results of Serre and Oort, Ck has homological dimension 1 (resp. 2) if k is algebraically closed of characteristic 0 (resp. positive). In this article, we explore the abelian category of commutative algebraic groups up to isogeny, defined as the quotient of Ck by the full subcategory Fk of finite k-group schemes. We show that Ck/Fk has homological dimension 1, and we determine its projective or injective objects. We also obtain structure results for Ck/Fk, which take a simpler form in positive characteristics. Contents 1 Introduction 2 2 Structure of algebraic groups 5 2.1 Preliminaryresults ............................... 5 2.2 Characteristiczero ............................... 8 2.3 Positive characteristics . 10 3 The isogeny category of algebraic groups 12 3.1 Definitionandfirstproperties . .. .. 12 3.2 Divisiblegroups................................. 17 3.3 Fieldextensions................................. 20 4 Tori, abelian varieties, and homological dimension 22 arXiv:1602.00222v2 [math.AG] 27 Sep 2016 4.1 Tori ....................................... 22 4.2 Abelianvarieties ................................ 24 4.3 Vanishingofextensiongroups . 28 5 Structure of isogeny categories 30 5.1 Vector extensions of abelian varieties . 30 5.2 Semi-abelian varieties . 35 5.3 Productdecompositions ............................ 36 1 1 Introduction There has been much recent progress on the structure of algebraic groups over an arbitrary field; in particular, on the classification of pseudo-reductive groups (see [CGP15, CP15]). Yet commutative algebraic groups over an imperfect field remain somewhat mysterious, e.g., extensions with unipotent quotients are largely unknown; see [To13] for interesting results, examples, and questions.
    [Show full text]
  • Lecture 2: Abelian Varieties the Subject of Abelian Varieties Is Vast
    Lecture 2: Abelian varieties The subject of abelian varieties is vast. In these notes we will hit some highlights of the theory, stressing examples and intuition rather than proofs (due to lack of time, among other reasons). We will note analogies with the more concrete case of elliptic curves (as in [Si]), as well as places where elliptic curves provide a poor guide for what to expect. In addition, we will use the complex-analytic theory and the example of Jacobians as sources of inspiration and examples to illustrate general results. For arithmetic purposes it is essential to allow the ground field to be of arbitrary characteristic (e.g., finite fields), and not algebraically closed. At the end, we explain Chabauty's clever argument that proves the Mordell conjecture for curves of genus g ≥ 2 whose Jacobian has Mordell{Weil rank less than g. In later lectures we will bootstrap from the case of individual abelian varieties to \families" of abelian varieties, or as we will call them, abelian schemes. This concept, along with related notions such as \good reduction" and \semistable reduction" will be an essential ingredient in Faltings' method. But for this lecture we restrict attention to the more concrete setting of a single fixed abelian variety over a field. Nonetheless, the later need for a mature theory of families over a parameter space of mixed-characteristic (even for the special case of modular curves over Z(p), required to make sense of \reduction mod p" for modular curves) forces us to allow arbitrary ground fields, including imperfect ones like Fp(t), as those always show up at generic points of special fibers when the parameter space has positive-dimensional fibers in each characteristic, as will happen in nearly all examples of moduli spaces that arise in Faltings' work and throughout arithmetic geometry.
    [Show full text]
  • Basic Theory of Abelian Varieties 1. Definitions
    ABELIAN VARIETIES J.S. MILNE Abstract. These are the notes for Math 731, taught at the University of Michigan in Fall 1991, somewhat revised from those handed out during the course. They are available at www.math.lsa.umich.edu/∼jmilne/. Please send comments and corrections to me at [email protected]. v1.1 (July 27, 1998). First version on the web. These notes are in more primitive form than my other notes — the reader should view them as an unfinished painting in which some areas are only sketched in. Contents Introduction 1 Part I: Basic Theory of Abelian Varieties 1. Definitions; Basic Properties. 7 2. Abelian Varieties over the Complex Numbers. 9 3. Rational Maps Into Abelian Varieties 15 4. The Theorem of the Cube. 19 5. Abelian Varieties are Projective 25 6. Isogenies 29 7. The Dual Abelian Variety. 32 8. The Dual Exact Sequence. 38 9. Endomorphisms 40 10. Polarizations and Invertible Sheaves 50 11. The Etale Cohomology of an Abelian Variety 51 12. Weil Pairings 52 13. The Rosati Involution 53 14. The Zeta Function of an Abelian Variety 54 15. Families of Abelian Varieties 57 16. Abelian Varieties over Finite Fields 60 17. Jacobian Varieties 64 18. Abel and Jacobi 67 Part II: Finiteness Theorems 19. Introduction 70 20. N´eron models; Semistable Reduction 76 21. The Tate Conjecture; Semisimplicity. 77 c 1998 J.S. Milne. You may make one copy of these notes for your own personal use. 1 0J.S.MILNE 22. Geometric Finiteness Theorems 82 23. Finiteness I implies Finiteness II. 87 24.
    [Show full text]
  • Duality of Abelian Varieties
    Duality of abelian varieties R. van Bommel Wednesday 16 November 2016 Abstract. In this talk the dual of an abelian variety will be defined. It will be shown that this dual is an abelian variety again. Using dual morphisms, we will finally show that the dual of the dual is canonically isomorphic to the orginal abelian variety. The author would like to thank Bas Edixhoven, Julian Lyczak and the other participants to the seminar for their valuable contributions. 0 Conventions and notation The letters A and B will be used to denote an abelian variety over a base scheme, S = Spec k, the spectrum of a field k. 1 Dual abelian variety Last week, Smit discussed the Picard functor. In our case he showed that PicA=S is a separated scheme, locally of finite type over S. It represents the relative Picard functor ∗ Sch=S ! Set : T 7! Pic(AT ) = prT Pic(T ): By construction it is a group scheme, whose identity point e 2 PicA=S(S) is given by the structure sheaf OA. Definition 1. The dual of A is _ 0 A := PicA=S; the connected component of PicA=S containing the identity e. Example 2. For an elliptic curve over S, i.e. an abelian variety of dimension 1, the dual is canonically isomorphic to itself, using for example Riemann-Roch. In the rest of this section, there will be an outline of a proof of the fact that A_ is again an abelian variety. 1 0 Lemma 3. Let X be a (proper) variety over S.
    [Show full text]
  • QUOTIENTS of ELLIPTIC CURVES OVER FINITE FIELDS 1. Introduction by Tate's Isogeny Theorem [14, P. 139], Two Elliptic Curves Ov
    QUOTIENTS OF ELLIPTIC CURVES OVER FINITE FIELDS JEFFREY D. ACHTER AND SIMAN WONG Abstract. Fix a prime `, and let Fq be a finite field with q ≡ 1 (mod `) elements. If 2 2 ` > 2 and q ` 1, we show that asymptotically (` − 1) =2` of the elliptic curves E=Fq with complete rational `-torsion are such that E=hP i does not have complete rational `-torsion for any point P 2 E(Fq) of order `. For ` = 2 the asymptotic density is 0 or 1=4, depending whether q ≡ 1 (mod 4) or 3 (mod 4). We also show that for any `, if E=Fq has an Fq-rational point R of order `2, then E=h`Ri always has complete rational `-torsion. 1. Introduction By Tate's isogeny theorem [14, p. 139], two elliptic curves over a finite field k are k- isogenous if and only if they have the same number of k-rational points. However, the isogeny need not preserve the group structure of the k-rational points of the respective 2 curves. For example, for E : y = x(x − 1)(x + 1) we have E(F5) ' Z=2 × Z=2. By explicit computations (cf. [15], [13, p. 70]), we find that E=h(0; 0)i is F5-isomorphic to E as elliptic curves (not via the quotient map, of course), but that the group of F5-rational points of both E=h(1; 0)i and E=h(−1; 0)i are cyclic. In this paper we investigate the problem of how often the quotient map by a k-rational torsion point preserves the group structure.
    [Show full text]
  • Abelian Varieties Over Finite Fields
    Abelian varieties over finite fields Frans Oort Mathematisch Instituut, P.O. Box. 80.010, NL - 3508 TA Utrecht The Netherlands e-mail: [email protected] Abstract. A. Weil proved that the geometric Frobenius π = F a of an abelian variety over a finite field with q = pa elements has absolute value √ q for every embedding. T. Honda and J. Tate showed that A 7→ πA gives a bijection between the set of isogeny classes of simple abelian varieties over Fq and the set of conjugacy classes of q-Weil numbers. Higher-dimensional varieties over finite fields, Summer school in G¨ottingen, June 2007 Introduction We could try to classify isomorphism classes of abelian varieties. The theory of moduli spaces of polarized abelian varieties answers this question completely. This is a geometric theory. However in this general, abstract theory it is often not easy to exhibit explicit examples, to construct abelian varieties with required properties. A coarser classification is that of studying isogeny classes of abelian varieties.A wonderful and powerful theorem, the Honda-Tate theory, gives a complete classification of isogeny classes of abelian varieties over a finite field, see Theorem 1.2. The basic idea starts with a theorem by A. Weil, a proof for the Weil conjec- ture for an abelian variety A over a finite field K = Fq, see 3.2: the geometric Frobenius π of A/K is an algebraic integer A √ which for every embedding ψ : Q(πA) → C has absolute value | ψ(πA) |= q. For an abelian variety A over K = Fq the assignment A 7→ πA associates to A its geometric Frobenius πA; the isogeny class of A gives the conjugacy class of the algebraic integer πA, and conversely an algebraic integer which is a Weil q-number determines an isogeny class, as T.
    [Show full text]
  • 6 Isogeny Kernels and Division Polynomials
    18.783 Elliptic Curves Spring 2017 Lecture #6 02/27/2017 6 Isogeny kernels and division polynomials In this lecture we continue our study of isogenies of elliptic curves. Recall that an isogeny is a surjective morphism that is also a group homomorphism, equivalently, a non-constant rational map that fixes the identity. In the previous lecture we showed that every nonzero 2 3 isogeny α: E1 ! E2 between elliptic curves in short Weierstrass form y = x + Ax + B can be written in the standard affine form u(x) s(x) α(x; y) = ; y ; v(x) t(x) where u ? v and s ? t are pairs of relatively prime polynomials in k[x].1 For any affine point (x0; y0) 2 E1(k¯), we have α(x0; y0) = 0 if and only if v(x0) = 0 (equivalently, if and only if t(x0) = 0; see Lemma 5.22 and Corollary 5.23). This follows from the fact that ker α is a subgroup, so if P = (x0; y0) 2 ker α then so is −P = (x0; −y0), and this accounts for every point in E(k¯) with x-coordinate x0. It follows that ker α = f(x0; y0) 2 E1(k¯): v(x0) = 0g [ f0g is determined by the polynomial v(x) (here 0 := (0 : 1 : 0) is the point at infinity). When α is the multiplication-by-n map P 7! nP = P + ··· + P (which is an isogeny because it is a group homomorphism defined by a non-constant rational map), the kernel of α is the n-torsion subgroup E[n] := fP 2 E(k¯): nP = 0g: Torsion subgroups play a key role in the theory of elliptic curves.
    [Show full text]
  • Abelian Varieties. Lecture 5
    Abelian Varieties. Lecture 5. Elisa Lorenzo Garc´ıa 29th February 2016 Contents 1 Basic definitions 1 1.1 Isogenies . .2 1.2 The Tate-module . .3 1.3 Tate-Honda Theorem . .3 1.4 Albert's classification . .4 1.5 The dual abelian variety . .4 2 Abelian Varieties over the complex numbers 4 3 Jacobians 5 4 Exercises 6 The main reference for today's lecture is paper [3]. 1 Basic definitions Let us fix k = Fq a finite field. An abelian variety over k is a subset of some projective n-space which 1. is defined by polynomials equations on the coordinates with coefficients in k. 2. is connected, and 3. has a group law which is algebraic (in the sense that the coordinates f the product of two points are rational functions of the coordinates of the factors). 1 In other words, an abelian variety A~k is an algebraic projective smooth and irreducible variety endow with a group law, that is, rational maps × ∶ A × A → A ∶ (P; Q) → P + Q and A → A ∶ P → −P . We usually denote the trivial element by O, that moreover, can be proved to be defined over the base field k. The first Theorem one proves is that abelian varieties are commutative, and this justifies why we denote the group law with the addition symbol. Example 1.1. Elliptic curves with the group law explained in last lectures, are abelian vari- eties of dimension 1. In particular, all dimension 1 abelian varieties are elliptic curves. The product of n elliptic curves E1 × ::: × En is an abelian variety of dimension n.
    [Show full text]
  • 18.783 S19 Elliptic Curves, Lecture 6: Isogeny Kernels and Division
    18.783 Elliptic Curves Spring 2019 Lecture #6 02/25/2019 6 Isogeny kernels and division polynomials In this lecture we continue our study of isogenies of elliptic curves. Recall that an isogeny is a surjective morphism that is also a group homomorphism, equivalently, a non-constant rational map that fixes the identity. In the previous lecture we showed that every nonzero 2 3 isogeny α: E1 ! E2 between elliptic curves in short Weierstrass form y = x + Ax + B can be written in the standard aÿne form u(x) s(x) α(x; y) = ; y ; v(x) t(x) 1 where u ? v and s ? t are pairs of relatively prime polynomials in k[x]. For any aÿne point (x0; y0) 2 E1(k¯), we have α(x0; y0) = 0 if and only if v(x0) = 0 (equivalently, if and only if t(x0) = 0; see Lemma 5.26 and Corollary 5.27). This follows from the fact that ker α is a subgroup, so if P = (x0; y0) 2 ker α then so is −P = (x0; −y0), and this accounts for every point in E(k¯) with x-coordinate x0. It follows that ker α = f(x0; y0) 2 E1(k¯): v(x0) = 0g [ f0g is determined by the polynomial v(x) (here 0 := (0 : 1 : 0) is the point at infinity). When α is the multiplication-by-n map P 7! nP = P + ··· + P (which is an isogeny because it is a group homomorphism defined by a non-constant rational map), the kernel of α is the n-torsion subgroup E[n] := fP 2 E(k¯) : nP = 0g: Torsion subgroups play a key role in the theory of elliptic curves.
    [Show full text]
  • Faltings's Proof of the Mordell Conjecture
    Faltings's Proof of the Mordell Conjecture Organized by Bhargav Bhatt and Andrew Snowden∗ Fall 2016 Abstract Our plan is to try to understand Faltings's proof of the Mordell conjecture. The focus will be on his first proof, which is more algebraic in nature, proves the Shafarevich and Tate conjectures, and also gives us a chance to learn about some nearby topics, such as the moduli space of abelian varieties or p-adic Hodge theory. The seminar will meet 4:10{5:30 every Thursday in 1360. Some relevant references are [CS86; Fal86; Tat66; Tat67], as well as notes from a seminar on this topic at Stanford recently: http://math.stanford.edu/~akshay/ntslearn.html Contents 1 September 15: Overview (Andrew Snowden) 3 1.1 Statements of Faltings's results . .3 1.2 Overview of the proof . .4 2 September 22: Endomorphisms of abelian varieties over finite fields (following Tate) (Emanuel Reinecke) 6 2.1 Motivation . .6 2.2 The isogeny category . .7 2.3 Initial reductions . .8 2.4 Proof of Tate's theorem . .9 2.5 Why does Hyp(k; A; `) hold if k = Fq?............................... 10 2.6 The weaker finiteness statement Hyp(k; A; d; `).......................... 11 3 September 29: Semiabelian varieties (Brandon Carter) 11 3.1 Motivation . 12 3.2 N´eronmodels . 12 3.3 Semiabelian varieties and semistable reduction . 13 3.4 The Hodge line bundle !A=S .................................... 14 3.5 A N´eronmapping property for semiabelian varieties . 14 3.6 Gabber's Lemma . 15 4 October 6: The Faltings height and basic finiteness theorems (Bhargav Bhatt) 17 4.1 Heights on Pn ...........................................
    [Show full text]