Field Extensions Generated by Kernels of Isogenies by Jonathan Love A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Mathematics University of Toronto c Copyright 2016 by Jonathan Love Abstract Field Extensions Generated by Kernels of Isogenies Jonathan Love Master of Science Graduate Department of Mathematics University of Toronto 2016 Given an odd prime p, A technique due to Jean-Fran¸coisMestre allows one to construct infinitely many quadratic fields for which the ideal class group has p{rank at least 2, using a degree p isogeny between elliptic curves such that the kernel has a rational point. This technique only works for primes p ≤ 7; we attempt to generalize the construction for larger primes. One line of approach uses higher degree isogenies (which have no rational point in the kernel), from which we obtain higher-degree number fields with p{rank at least two. In the process, we collect data on the number fields generated by the points in the kernel of an isogeny, and make a series of conjectures based on the data. We also discuss the possibilities and limitations of replacing the elliptic curves in Mestre's technique with more general abelian varieties. ii Contents 1 Motivation: Ideal Class Groups1 1.1 Results and Open Problems...................................2 2 Elliptic Curves and Isogenies5 2.0.1 The Projective Plane...................................5 2.1 Elliptic Curves..........................................6 2.2 Isogenies..............................................7 2.3 Generating Quadratic Fields...................................8 2.3.1 Preimages of Quotient Maps..............................9 3 Isogenies with Irrational Kernel 11 3.1 Data on Kernel Fields...................................... 12 3.2 Observations from the Data................................... 12 3.2.1 p = 11; 19; 43; 67..................................... 13 3.2.2 p = 37........................................... 15 3.2.3 p = 17........................................... 16 3.2.4 p = 13........................................... 16 3.3 Preimages of points........................................ 17 3.3.1 Points with X 2 Q .................................... 17 Examples......................................... 18 Ramification....................................... 19 3.3.2 Points with X 2 K .................................... 19 4 Higher Genus Curves 21 4.1 Background............................................ 21 4.2 Hyperelliptic curves....................................... 22 4.3 Z=pZ covers by Descent..................................... 22 4.3.1 A family of hyperelliptic curves............................. 22 4.4 Proportion of Points on a Curve for which p j h(Q(P )).................... 24 5 Appendix: Description of Code 28 5.1 Computing Ideal Class Groups................................. 28 5.1.1 Classes of Binary Quadratic Forms........................... 28 5.1.2 The Class Group of Forms................................ 30 iii Operating on Form Orbits................................ 33 5.2 Points P on a curve with p j h(Q(P )).............................. 34 5.3 Kernel Fields of Isogenies.................................... 35 Bibliography 37 iv List of Tables 3.1 Extensions generated by kernels of isogenies.......................... 14 v List of Figures 4.1 Proportion of points (X; Y ) 2 X0(11) with X 2 Q and 5 j h(Q(P ))............. 24 4.2 Proportion of points (X; Y ) 2 X0(23) with X 2 Q and 11 j h(Q(P ))............ 24 4.3 y2 = x6 + 2x5 + 5x4 + 2x3 + 2x2 + 1 (has 11{torsion), p = 11................ 25 4.4 y2 = x6 + 2x5 + 5x4 + 2x3 + 2x2 + 2 (no 11{torsion), p = 11................. 25 4.5 y2 = x6 + 4x4 + 10x3 + 4x2 − 4x + 1 (has 13{torsion), p = 13................ 26 4.6 y2 = x6 + x5 + x4 + x3 + x2 + 3x + 1 (no 13{torsion), p = 13................ 26 4.7 y2 = (9x2 + 2x + 1)(32x3 + 81x2 − 6x + 1) (has 17{torsion), p = 17............. 27 4.8 y2 = (10 − x)(3x + 2)(72x4 + 96x3 + 45x2 − 38x + 5) (has 17{torsion), p = 17....... 27 vi Chapter 1 Motivation: Ideal Class Groups Consider a field K that is a finite extension of Q, and let OK be its ring of integers. In many ways, this ring behaves a lot like the integers. For instance, there is a well-defined notion of size, and elements can always be factored into a finite product of smaller irreducible elements. Studying these number systems can often give us insights into the structure of the integers themselves (as an example, the question of which integers can be expressed as a sum of two squares can be completely solved by considering factorization in Z[i]; see for instance section 2.2 of [12]). There is one very significant way in which these rings may fail to be like the integers, however: there is not always a unique way to factor an element into irreducible elements. This has considerable implications in number theory; for instance, if factorization into irreducibles were always unique, then we might have had a relatively simple proof of Fermat's last theorem as early as the 1800s ([7]). As a result, it would be useful to measure, in some sense, the extent to which unique factorization fails. Fortunately, a convenient tool for doing so exists, using the ideals of the ring. We note the following proposition: Theorem 1.0.1. Let OK be the ring of integers of a finite extension K=Q. Then elements of OK can be factored uniquely into irreducible elements if and only if every ideal of OK is a principal ideal. (see e.g. [22], Proposition 3.18 and preceding discussion). In particular, factorization of ideals of OK into prime ideals is always unique, and in a principal ideal domain, this factorization corresponds with the factorization of an element into irreducible elements. Thus, to study the failure of unique factorization, we need to study the ways in which an ideal can fail to be a principal ideal. The set of nonzero ideals IK of a ring of integers OK form a monoid under the operation of multi- plication. We define an equivalence relation ∼ on IK: given I;J 2 IK, we say I ∼ J if (a)I = (b)J for some a; b 2 OK . The set of equivalence classes is called Cl(K). One can check that multiplication is well defined on equivalence classes, and in fact makes Cl(K) into a group, called the ideal class group of K. The collection of all principal ideals is one of the equivalence classes, and corresponds to the identity element of Cl(K); thus, OK has unique factorization if and only if Cl(K) is the trivial group. So to better understand failure of unique factorization, Cl(K) is the structure we would like to understand. One of the most basic results about Cl(K) is the following: Theorem 1.0.2 ([22], Theorem 4.3). Let D be the discriminant of K, n = [K : Q], and 2r the number 1 Chapter 1. Motivation: Ideal Class Groups 2 of non-real embeddings K,! C. Then every equivalence class of ideals has an ideal I with 4 r n! N(I) ≤ pjDj : π nn Since there are only finitely many ideals of any given norm, this allows us to prove an important corollary: Theorem 1.0.3. Cl(K) is a finite abelian group. In fact, we can directly compute Cl(K) simply by listing all ideals with norm no greater than the Minkowski bound and checking how they multiply together - though this algorithm is impractical for all but the simplest field extensions. A more efficient algorithm is discussed in the appendix, section 5.1. Instead of taking a field and computing its class group, one can ask the converse; given a finite abelian group A, which fields K have Cl(K) = A? Or, more generally, have A ≤ Cl(K)? Very little is known about this sort of question in general, but considerable progress has been made in the case of quadratic fields. 1.1 Results and Open Problems For the remainder of this section, D 2 will denote a fundamental discriminant (either D ≡ 1 (mod 4) Z p is squarefree, or D ≡ 2; 3 (mod 4) is squarefree), and K will be a quadratic field: K = ( D) for some 4 p Q fundamental discriminant D. Note that every quadratic field K = Q( M) can be written in this way; if s2 is the largest perfect square dividing M (so that M=s2 is squarefree), then ( 2 2 M; M=s2 ≡ 2; 3 (mod 4) D = s 1 2 2 s M; M=s ≡ 1 (mod 4) p p is a fundamental discriminant with Q( M) = Q( D). We define the class number h(K) := jCl(K)j. Though Gauss conjectured the following in 1801 ([14], Section V, Article 303), this fact was only proven in 1934 by Heilbronn ([16]): p Theorem 1.1.1. For each n 2 Z, there are only finitely many D < 0 for which h(Q( D)) = n. In fact, we have very strict bounds on the growth of the class number: Theorem 1.1.2 (Siegel, 1935 [26]). For all " > 0, p 1=2−" 1=2+" jDj h(Q( D)) jDj : (Here f(D) g(D) means that for all C 2 R, we have Cf(D) < g(D) for sufficiently large D). However, though this tells us a lot about the size of the class group, it fails to give us much more information about the structure of the group itself. However, not much progress has been made in this direction. We do have the following: Theorem 1.1.3 (Ankeny, 1955 [1]). For any positive integer n, there are infinitely many D < 0 such p that h(Q( D)) is divisible by n. Chapter 1. Motivation: Ideal Class Groups 3 By Cauchy's theorem, this tells us that whenever A is a product of distinct prime cyclic groups (i.e. when jAj is squarefree), A appears as a subgroup of the ideal class group for infinitely many imaginary quadratic fields.