DISSERTATION CONJUGACY CLASSES OF MATRIX GROUPS OVER LOCAL RINGS AND AN APPLICATION TO THE ENUMERATION OF ABELIAN VARIETIES Submitted by Cassandra L Williams Department of Mathematics In partial fulfillment of the requirements For the Degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Summer 2012 Doctoral Committee: Advisor: Jeffrey Achter Richard Eykholt Alexander Hulpke Tim Penttila Copyright by Cassandra Lee Williams 2012 All Rights Reserved ABSTRACT CONJUGACY CLASSES OF MATRIX GROUPS OVER LOCAL RINGS AND AN APPLICATION TO THE ENUMERATION OF ABELIAN VARIETIES The Frobenius endomorphism of an abelian variety over a finite field F of dimen- q sion g can be considered as an element of the finite matrix group GSp (Z/(cid:96)r). The 2g characteristic polynomial of such a matrix defines a union of conjugacy classes in the group, as well as a totally imaginary number field K of degree 2g over Q. Suppose g = 1 or 2. We compute the proportion of matrices with a fixed characteristic polyno- mial by first computing the sizes of conjugacy classes in GL (Z/(cid:96)r) and GSp (Z/(cid:96)r). 2 4 Then we use an equidistribution assumption to show that this proportion is related to the number of abelian varieties over a finite field with complex multiplication by the maximal order of K via a theorem of Everett Howe. ii ACKNOWLEDGEMENTS I would like to thank Jeff Achter for all of his guidance, and Hilary Smallwood for her help in completing Appendix B. iii TABLE OF CONTENTS 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Background: Number Theory and Algebraic Geometry . . . . . . . . . . . 3 2.1 Splitting of primes in Galois extensions of Q . . . . . . . . . . . . . . 3 2.2 Dirichlet characters, L-series, and the Dedekind zeta function . . . . . 8 2.3 Class numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Abelian varieties and isogeny . . . . . . . . . . . . . . . . . . . . . . 13 3. Motivation and Overview: A theorem of Gekeler . . . . . . . . . . . . . . . 19 4. The Conjugacy Classes of GL (Z/(cid:96)r) . . . . . . . . . . . . . . . . . . . . . 25 2 4.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 A classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2.1 An overview of Section 2 of Avni, et al . . . . . . . . . . . . . 29 4.2.2 Enumerating similarity and conjugacy classes . . . . . . . . . 31 4.3 The order of a conjugacy class of GL (Z/(cid:96)r) . . . . . . . . . . . . . . 35 2 4.3.1 A number theoretic approach . . . . . . . . . . . . . . . . . . 38 4.3.2 An algebraic geometric approach: Smoothness . . . . . . . . . 42 5. The number of elliptic curves with complex multiplication by the maximal order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.1 Primes, conjugacy classes, and characters . . . . . . . . . . . . . . . . 50 5.2 The proportion of Frobenius elements and elliptic curves with complex multiplication by O . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 K 6. The Conjugacy Classes of GSp (Z/(cid:96)r) . . . . . . . . . . . . . . . . . . . . . 56 4 6.1 A brief introduction to the symplectic groups . . . . . . . . . . . . . 56 6.2 The conjugacy classes of GSp (F ) . . . . . . . . . . . . . . . . . . . . 58 4 (cid:96) 6.3 Centralizer orders: Regular semisimple classes . . . . . . . . . . . . . 68 6.4 Centralizer orders: Non-regular classes . . . . . . . . . . . . . . . . . 70 7. Primes, conjugacy classes, and characters in imaginary quartic extensions . 79 7.1 Complex conjugation in O and eigenspaces of γ . . . . . . . . . . . 80 K 7.2 Cyclic quartic extensions . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.3 Biquadratic quartic extensions . . . . . . . . . . . . . . . . . . . . . . 86 7.4 The constant ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8. The number of abelian surfaces with complex multiplication by the maximal order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9. Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 97 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 iv Appendices 103 A. A failure of matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 B. The Sato-Tate conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 v 1. INTRODUCTION Abelian varieties are geometric objects defined as projective zero sets of polynomials which have a (commutative) group structure on their points. For example, elliptic curves are abelian varieties of genus one. An isogeny between two such objects is a particular type of map, and it induces an equivalence relation on the set of all abelian varieties of a particular dimension. Over fields of characteristic zero, isogeny classes of abelian varieties with complex multiplication can be represented by the totally imaginary field in which the complex multiplicationlies. Whiletheseclassesmaybeinfinite, thenumberofabelianvarieties with complex multiplication by the maximal order in an imaginary field (i.e., the ring of integers) is finite. In fact, the number of such varieties is closely related to the class number of the imaginary field. Alternatively, over finite fields Tate’s theorem [22] says that isogeny classes of abelian varieties are determined by the characteristic polynomial of the Frobenius endomorphism. TheseFrobeniusmapsarenaturallyelementsofGL (Z/n)forn ∈ N, 2g and since every abelian variety admits a polarization we can instead represent them in GSp (Z/n). In this group, we can compute the size of the conjugacy class of such 2g a matrix. An interesting question one could ask is how likely is it that a random abelian variety over a finite field F has a Frobenius endomorphism with a given character- q istic polynomial f. Gekeler formulated this question for elliptic curves [9] and, after assuming the Frobenius elements are equidistributed in the matrix group, he showed that the answer has to do with the local factors in the Euler product expansion of an 1 L-series. Note that this equidistribution assumption cannot possibly be true, since there are only finitely many abelian varieties of a given dimension over F . q In this paper, we have two main goals. First, we use number theory, algebraic geometry, and group theory to determine representatives for and sizes of conjugacy classes of the matrix groups GL (Z/(cid:96)r) and GSp (Z/(cid:96)r). In particular, we prove 2 4 a smoothness result (Theorem 4.3.9) on centralizers in to find a formula for the centralizer order of any γ ∈ GL (Z/(cid:96)r). We use this conjugacy class data to efficiently 2 reproduce Gekeler’s result for GL (Z/(cid:96)r). 2 Gekeler noted that the probability he computed was equal to an Euler factor of an L-series. We interpret this relationship slightly differently; consider this L-series as one that occurs in the Dedekind zeta function of a certain totally imaginary field. Then the L-series is (up to a real constant) related to the class number of the field, and a theorem of Howe (as communicated in [10]) gives a formula for the size of the set of abelian varieties over a finite field with complex multiplication by the maximal order of the field in terms of the class number. Our second goal is to use this theorem to interpret Gekeler’s result in terms of the size of an isogeny class of elliptic curves, and extend the computation to abelian surfaces using our results about the conjugacy classes of GSp (Z/(cid:96)r). 4 LetC(γ)denotetheconjugacyclassofγ. Ourmainresultisthefollowingtheorem. Theorem 1.0.1 (Main Theorem). Suppose f(T) ∈ Z[T] is such that K = Split(f) is a totally imaginary quartic field. Let I be the set of isomorphism classes of K principally polarized abelian surfaces over a finite field with complex multiplication by the maximal order of K. Then for an explicit real constant ξ, 1 (cid:89) #{cyclic γ ∈ GSp (F )|C(γ) ↔ f mod (cid:96)} #I = 4 (cid:96) . K ξ (cid:96)−2 #Sp (F ) (cid:96)∈Z prime 4 (cid:96) 2 2. BACKGROUND: NUMBER THEORY AND ALGEBRAIC GEOMETRY We will make use of many standard results from algebraic number theory and alge- braic geometry, and for the sake of convenience we collect them here. In addition, we will set some notation. 2.1 Splitting of primes in Galois extensions of Q Although many results are cited with specific references, much of the material in this section comes from classic graduate texts in number theory, perhaps most frequently from Marcus [16], Neukirch [18], and Serre [19]. Recall that a Dedekind domain is an integral domain such that every ideal is finitely generated, every nonzero prime ideal is maximal, and such that it is inte- grally closed. An important attribute of Dedekind domains is that they have unique factorization of ideals. Let A be a Dedekind domain, and K its field of fractions. Let L be a finite Galois extension of K and let B be the integral closure of A in L. L ⊃ B ⊃ q (cid:54) ∪ K ⊃ A ⊃ p The ring B is also a Dedekind domain, so we have unique factorization of prime ideals in B. For any prime p ⊂ A, pB ⊂ B and so we can ask how the ideal p factors in B. 3 Suppose r (cid:89) pB = qe1qe2...qer = qe(qi/p) 1 2 r i i=1 with the q ⊂ B prime ideals and e(q /p) ∈ Z the inertial degree of q over p. We i i + i collect here some results about primes in such an extension. Lemma 2.1.1. [18, Section I.9] The ring B is stable under Gal(L/K). For any ∼ σ ∈ Gal(L/K) and q ⊂ B prime, σ(q) is also prime; in fact, B/σ(q) = B/q. Lemma 2.1.2. [16, Theorem 23] The group Gal(L/K) acts transitively on the set of primes of B lying over a prime p ∈ A. Since B is a Dedekind domain, any nonzero prime ideal q ⊂ B is maximal and so B/q is a field; we denote it by κ(q) = B/q and call it the residue field of q. If q∩A = p then κ(q) is a finite extension of κ(p), and we call f(q/p) = [κ(q) : κ(p)] the residue degree of q over p. Corollary 2.1.3. [16, Corollary to Theorem 23] If L/K is a finite Galois extension where q and q(cid:48) lie over p, then f(q/p) = f(q(cid:48)/p) and e(q/p) = e(q(cid:48)/p). We will denote f(q/p) and e(q/p) as f = f(p) and e = e(p) respectively. Proposition 2.1.4. [18, Proposition I.8.2] Suppose (cid:89) (cid:89) pB = qe(qi/p) = qei i i | | qi p qi p and let f = f(q /p). Then i i (cid:88) e f = [L : K] = n. i i i 4