ABSTRACT. the characteristic polynomial of the Frobenius endomorphism. We show how this

Transcription

1 ABSTRACT Title of dissertation: COMMUTATIVE ENDOMORPHISM RINGS OF SIMPLE ABELIAN VARIETIES OVER FINITE FIELDS Jeremy Bradford, Doctor of Philosohy, 01 Dissertation directed by: Professor Lawrence C. Washington Deartment of Mathematics In this thesis we look at simle abelian varieties defined over a finite field k = F n with End k (A) commutative. We derive a formula that connects the - rank r(a) with the slitting behavior of in E = Q(π), where π is a root of the characteristic olynomial of the Frobenius endomorhism. We show how this formula can be used to exlicitly list all ossible slitting behaviors of in O E, and we do so for abelian varieties of dimension less than or equal to four defined over F. We then look for when divides [O E : Z[π, π]]. This allows us to rove that the endomorhism ring of an absolutely simle abelian surface is maximal at when 3. We also derive a condition that guarantees that divides [O E : Z[π, π]]. Last, we exlicitly describe the structure of some intermediate subrings of -ower index between Z[π, π] and O E when A is an abelian 3-fold with r(a) = 1.

2 COMMUTATIVE ENDOMORPHISM RINGS OF SIMPLE ABELIAN VARIETIES OVER FINITE FIELDS by Jeremy Bradford Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in artial fulfillment of the requirements for the degree of Doctor of Philosohy 01 Advisory Committee: Professor Lawrence C. Washington, Chair/Advisor Professor Patrick Brosnan Professor William Gasarch Professor Niranjan Ramachandran Professor Harry Tamvakis

3 Table of Contents 1 Introduction 1 Background 5.1 Abelian Varieties Endomorhism Rings Subrings of Q End k (A) Ellitic Curves The Theorems of Waterhouse and Nakamura Newton Polygons Definitions and Proerties Newton Polygons and Local Invariants Classification of Weil olynomials Ellitic Curves Abelian Surfaces Abelian 3-folds Abelian 4-folds The slitting of and the -rank 9 4 The Index [O E : Z[π, π]] Ellitic Curves Abelian Surfaces Higher dimensional abelian varieties Subrings of CM Fields of Degree 6 Corresonding to Abelian 3-folds of -rank 1 54 A Slitting of folds folds Bibliograhy 74 ii

4 Chater 1 Introduction The driving force behind this thesis is the roblem of comuting the endomorhism ring End k (A) of a given abelian variety A defined over a finite field k. Comuting the endomorhism ring of A gives information about the isomorhism class of A within its isogeny class because isomorhic varieties must have isomorhic endomorhism rings. Comuting endomorhism rings also has otential alications to crytograhy. In hyerellitic curve crytograhy, one needs to construct a hyerellitic curve over a finite field such that its Jacobian has a secified number of oints. In the genus two case, Eisenträger and Lauter develoed a method for constructing such curves by using the Chinese Remainder Theorem [1]. Their CRT method requires checking whether or not the endomorhism ring of a Jacobian of a genus curve is the full ring of integers in a CM field. A follow-u aer by Freeman and Lauter gave a robabilistic algorithm for determining whether the endomorhism ring of a Jacobian is the full ring of integers []. A general algorithm for comuting the endomorhism ring of an abelian variety over a finite would be a natural extension of their work. David Kohel exlored this roblem in [3] when A is an ordinary ellitic curve by using the grah of l-isogenies between ellitic curves in the same isogeny class as A. He discovered that the grah of l-isogenies has the general shae of a volcano, 1

5 where the ellitic curves with maximal endomorhism ring form the rim at the to and the ellitic curves with minimal endomorhism ring form the base. Kohel was able to develo an algorithm that comuted the endomorhism ring by exloiting the structure of this grah. This grah is often referred to as the isogeny volcano and it has found a number of alications beyond simly comuting the endomorhism ring. For examle, Sutherland was able to use isogeny volcanos to comute secializations of modular olynomials [4]. More recently Bisson and Sutherland have develoed other algorithms for calculating the endomorhism ring of A when A is an ordinary ellitic curve [5]. Their algorithms search for relations in the ideal class grou to comute the conductor of End k (A). Thus the algorithms of Bisson and Sutherland, like Kohel s algorithm, rely on the fact that End k (A) is some order in the quadratic imaginary field Q End k (A). Orders in quadratic imaginary fields are comletely determined by their conductor, or equivalently, by their index in the ring of integers. The algorithms of Kohel, Bisson, and Sutherland calculate this index and thus determine End k (A). For higher dimensional abelian varieties, we will restrict ourselves to the case where Q End k (A) is a field. In general, the index of End k (A) in the ring of integers does not necessarily determine the order End k (A). This means that an algorithm that comutes the index of End k (A) in the ring of integers will not be sufficient to in down the endomorhism ring of A, a difficulty not encountered in the case of ordinary ellitic curves. We do not resolve this difficulty in this thesis but rather attemt to narrow down the ossible orders that End k (A) can be inside the ring of integers. It is known [6, 3.5] that End k (A) must contain Z[π, π] and so we will

6 focus on the index of Z[π, π] in the ring of integers. Secifically we will look at the question of the maximality or non-maximality of Z[π, π] at. We will show that the maximality or non-maximality of Z[π, π] at is connected with the -rank of the abelian variety and the slitting behavior of in the CM field Q End k (A). We rove a theorem that comletely describes the relationshi between the slitting behavior of and the -rank and then use this theorem to work out all ossible slitting behaviors in the cases of abelian varieties defined over F u to dimension four. We also note that recent work of Zaytsev has significant overla with this ortion of the thesis. Zaytsev has been looking at the more general question of the isomorhism tye of A[] as a finite grou scheme. He is able to connect the slitting behavior of in E with the first truncated Barsotti-Tate grou scheme A[] for abelian varieties u to dimension three [7]. We use the classification of the ossible slitting behaviors of to examine when divides the index [O E : Z[π, π]]. If A is an absolutely simle abelian surface, then we rove that does not divide this index for 17. This allows us to rove that End k (A) is maximal at for 3. For higher dimensional abelian varieties, we find sufficient conditions on the -rank and the slitting behavior of to guarantee that Z[π, π] is non-maximal at. We also briefly exlore the roblem of exlicitly describing intermediate rings R between Z[π, π] and the maximal order such that [R : Z[π, π]] is divisible by. It is not comletely clear which of these intermediate rings can arise as the endomorhism ring of an abelian variety. Waterhouse and Nakamura have rovided some results that guarantee the maximality of the endomorhism ring at under certain 3

7 hyotheses [6, 5.3], [8]. However, they both make the rather strong assumtion that End k (A) contains the maximal order of the totally real subfield of index two. If a general algorithm for comuting the endomorhism ring of an abelian variety over a finite field is to be develoed, then it seems that we need to know more about the ossible intermediate subrings. In this thesis we give the exlicit structure of some intermediate subrings for abelian 3-folds with -rank r(a) = 1. Under a certain hyothesis relating the coefficients of the characteristic olynomial of Frobenius, we show that there is always an intermediate subring with index. Using numerical evidence, we also conjecture that this intermediate subring is unique. 4

8 Chater Background The goals of this background chater are (1) to introduce the notation used throughout the thesis, and () state definitions and results with the aim of making this thesis reasonably self-contained. We also hoe that this background section will be illuminating by clearly showing how the the new results that we rove fit into the growing body of theory on endomorhism rings of abelian varieties over finite fields. Throughout this chater, known results are stated without roof but citations are always rovided..1 Abelian Varieties We begin with a short review of the theory of abelian varieties over finite fields (see [9] for a general reference). Let A be an abelian variety of dimension g defined over the finite field k := F q where q = n. We will often assume that A is absolutely simle, which just means that A is simle and when we extend scalars to the algebraic closure k we have that A k k is also simle. If k is an extension of k contained in k, then A(k ) denotes the set of oints on A with coordinates in k. Since A is an abelian variety there is an addition morhism A A A defined over k that makes A(k ) into an abelian grou for every extension k of k. For any ositive integer m, A(k )[m] will denote the m-torsion elements of the abelian grou 5

9 A(k ). If m N is relatively rime to, then A( k)[m] (Z/mZ) g. Thus for any rime l we have A( k)[l d ] (Z/l d Z) g. Multilication by l induces a ma A( k)[l d ] A( k)[l d 1 ]. The Tate Module is the inverse limit T l (A) := lim A( k)[l d ]. T l (A) is therefore a free Z l -module of rank g. On the other hand, A( k)[] (Z/Z) r(a) for some 0 r(a) g. The integer r(a) is the -rank of A. Every integer from 0 to g arises as the -rank of some abelian variety over k of dimension g. If r(a) = g then A is said to be ordinary. A morhism of abelian varieties ϕ : A 1 A is a morhism of varieties which is defined over k and is also comatible with the addition mas of A 1 and A. In articular ϕ : A 1 (k ) A (k ) is a homomorhism of grous. If dim(a 1 ) = dim(a ), then an isogeny ϕ : A 1 A is a morhism of abelian varieties with finite kernel. An endomorhism of A is a morhism of abelian varieties from A to A. For examle, for every integer m > 0 there is a multilication-by-m endomorhism [m] : A A which takes a oint P A( k) to [m](p ) := P +P + +P, the m-fold sum. Another examle is the ma [ 1](P ) := P which sends P to its additive inverse. Comosing these two endomorhisms allows us to treat the integers Z as endomorhisms of A. Since A is defined over the finite field k = F q, there is also a distinguished endomorhism F : A A induced by the q-th ower Frobenius automorhism of the field k. We call F the Frobenius endomorhism of A. It is known that an endomorhism ϕ of A defined over k is actually defined over k if and only if F ϕ = ϕf. We denote the set of endomorhism of A defined over k by End k (A). Under comosition and oint-wise addition, End k (A) becomes a ring which we call the endomorhism ring of A. 6

10 If l is a rime, then F induces a linear transformation on the Tate module T l (A). One way to define the characteristic olynomial of F is to take the characteristic olynomial of the linear transformation that F induces on T l (A). However, it is then far from clear that this is a olynomial with integer coefficients indeendent of the rime l. Thus we will instead use an alternative definition of the characteristic olynomial due to Weil. In order to describe this alternative, we need to give some more definitions. For an abelian variety A over k, the function field k(a) is the field of rational functions from A to P 1 (k). Let α : A 1 A be a morhism of abelian varieties and let ψ k(a ). Then we can define α (ψ) := ψ α k(a 1 ). Thus α : k(a ) k(a 1 ) allows us to identify k(a ) with a subfield of k(a 1 ). If α 0 and [k(a 1 ) : α k(a )] is finite, then the degree of α is [k(a 1 ) : α k(a )]. By convention, we define the degree of the zero morhism to be 0. For examle, deg[n] = n g and deg F = q (see [9] I.7. and II.1.). To define the characteristic olynomial of an endomorhism we use the following theorem: Theorem.1 ([9] Theorem I.10.9). Let α End k (A). There is a unique monic olynomial P α Z[x] of degree g such that P α (r) = deg(α r) for all r Z. The characteristic olynomial of the endomorhism α is the monic olynomial P α Z[x]. In this thesis we let f Z[x] be the characteristic olynomial of the Frobenius endomorhism F. We will often assume that f is irreducible. In such cases we will identify a root π of f with the Frobenius endomorhism F and then by abuse of language will call the root π the Frobenius endomorhism. 7

11 For any isogeny α : A B there is an isogeny ᾱ : B A called the dual isogeny such that ᾱα = [deg α] A and αᾱ = [deg α] B. Thus the relation A is isogenous to B is an equivalence relation on abelian varieties over k. The isogeny class of A is the set of abelian varieties over k that are isogenous to A. Tate discovered a remarkable connection between the characteristic olynomial f of the Frobenius endomorhism and the isogeny class of the abelian variety: Theorem. ([10] Theorem 1). Let A and B be abelian varieties over a finite field k, and let f A and f B be the characteristic olynomials of their Frobenius endomorhisms relative to k. Then (b) The following are equivalent: (b1) B is k-isogenous to an abelian subvariety of A defined over k. (b3) f B divides f A. (c) The following are equivalent: (c1) A and B are k-isogenous. (c) f A = f B. (c4) A(k ) = B(k ) for every finite extension k of k. Remark.3. To kee this introductory section brief we have only stated some of the arts of the original theorem in [10] but have enumerated these selections using the original enumeration given in [10]. For the sake of comleteness of this introduction we include the following well-known theorem: 8

12 Theorem.4 ([9] Theorem II.1.1). Let A be an abelian variety over the finite field k, let f be the characteristic olynomial of the Frobenius endomorhism, and let k m be the field extension of k of degree m. Write f(x) = g i=1 (x a i) for a i C. Then (a) #A(k m ) = g i=1 (1 am i ) for all m 1, and (b) (Riemann hyothesis) a i = q 1. A Weil q-number is a comlex number π C such that if ϕ : Q(π) C is an embedding of fields, then ϕ(π) = q 1. A Weil olynomial is a olynomial in Z[x] whose roots are all Weil q-numbers. Two Weil q-numbers π 1 and π are considered equivalent if they have the same minimal olynomial, or equivalently if there is an isomorhism ϕ : Q(π 1 ) Q(π ) such that ϕ(π 1 ) = π. By the Riemann hyothesis, the characteristic olynomial of Frobenius is a Weil olynomial. The converse is also true, due to Tate and Honda: Theorem.5 ([11] Theorem 1). There is a bijection between the set of isogeny classes of simle abelian varieties over k and the equivalence classes of Weil q- numbers. The bijection is given by associating to a simle abelian variety A over k a root of the characteristic olynomial of Frobenius.. Endomorhism Rings We begin by stating some fundamental results in the theory of endomorhism rings of abelian varieties. Theorem.6 ([9] I.10.6, I.10.15). For abelian varieties A and B over k, the set 9

13 Hom k (A, B) of morhisms of abelian varieties from A to B is a finitely generated torsion-free Z-module with rank at most 4 dim(a) dim(b). In articular End k (A) is a finitely generated torsion-free Z-module with rank at most 4 dim(a). Theorem.7 ([10] Theorem ). Let A be an abelian variety of dimension g over a finite field k. Let F be the Frobenius endomorhism of A relative to k and f its characteristic olynomial. (a) The algebra E := Q[F ] is the center of the semisimle algebra S = Q End k (A). (b) We have g [S : Q] (g) (c) The following are equivalent: (c1) [S : Q] = g. (c) f has no multile root. (c3) S = E. (c4) S is commutative. (d) The following are equivalent: (d1) [S : Q] = (g). (d) f is a ower of a linear olynomial. (d3) E = Q. 10

14 (d4) S is isomorhic to the algebra of g by g matrices over the quaternion algebra D over Q which is ramified only at and. (d5) A is k-isogenous to the g-th ower of a suersingular ellitic curve, all of whose endomorhisms are defined over k. (e) A is k-isogenous to a ower of a k-simle abelian variety if and only if f is a ower of a Q-irreducible olynomial P. When this is the case S is a central simle algebra over E which slits at all finite rimes v of E not dividing = char(k), but does not slit at any real rime of E. We will usually be dealing with simle abelian varieties. In these cases Theorem.7(e) will aly and thus f = h e for some irreducible olynomial h Z[x]. The olynomial h will be a Weil olynomial (Theorem.4) with root π. Furthermore, [S : E] = e (see [11,.14] and also [1, Ch. IV]) hence the endomorhism ring of a simle abelian variety is commutative if and only if e = 1. Theorem.5 says that a root π of a Weil olynomial h determines an isogeny class of simle abelian varieties. But isogenous abelian varieties have isomorhic endomorhism rings after tensoring with Q, so we would like to be able to determine this ring S from the given root π. We do this by first calculating the local invariants of the central simle algebra S over the field E = Q(π). Theorem.7(e) states that we get a local invariant of 1 at every real lace of E and that S slits at every finite rime other than those that lie over. If v is a lace of E over, then Tate roved ([11, Thm. 1]) inv v (S) v(π) v(q) [E v : Q ] = v(π) f v n 11 (mod 1) (.1)

15 where f v is the degree of the residue field at v. The exonent e is the least common multile of the denominators of the local invariants of the central simle algebra S. Since we are assuming that our abelian varieties are simle, every nonzero endomorhism of A is an isogeny. Recall that every isogeny α has a dual ᾱ End k (A) such that αᾱ = [deg α] Z. Thus when we formally extend scalars to form S = Q End k (A) we see that any nonzero endomorhism α has inverse [deg α] 1 ᾱ. Thus we get that S is a central division algebra over E. This means that the local invariants uniquely determine S ([1] VIII.4.), hence we have recovered S from π. We now sketch the structure of Weil q-numbers as described in detail in [6, Ch. ]. First suose that there is a real rime of E. Then π = ± q. If n is even then π Z, E = Q, S is the quaternion algebra over Q ramified only at and, and A is a suersingular ellitic curve with all endomorhisms defined over k. If n is odd then E = Q( ), S is the quaternion algebra over Q( ) ramified only at the two infinite laces, and A is a simle abelian surface. If E does not have a real rime, then let β = π + π = π + q. Then K = Q(β) is a totally real subfield and π π satisfies X βx + q, hence E is a quadratic imaginary extension of K. That is, E is a CM field with totally real subfield K = Q(π + q/π). In this thesis we will deal with this latter case almost exclusively..3 Subrings of Q End k (A) This section will be dealing with the question of which subrings of Q End k (A) arise as endomorhism rings of abelian varieties. We will always assume that A is 1

16 simle. We do not yet require that End k (A) be commutative. Thus Q End k (A) will be a central division algebra over E = Q(π), and End k (A) is commutative if and only if E = Q End k (A)..3.1 Ellitic Curves When dealing with an ellitic curve C we know that π End k (C). Furthermore the characteristic olynomial of Frobenius is X βx +q and β = +1 #E(k), hence β Z. In articular, π = β π Z[π]. Therefore any subring of Q End k (C) that arises as the endomorhism ring of an ellitic curve contains at a minimum Z[π] = Z[π, π]. Waterhouse fully analyzed which subrings between Z[π] and Q End k (C) arise as the endomorhism rings of ellitic curves: Theorem.8 ([6] 4.). Let S be the endomorhism algebra of an isogeny class of ellitic curves. The orders in S which are endomorhism rings of curves in the isogeny class are as follows: (a) If the curves are suersingular with all endomorhisms defined over k, the maximal orders. (b) If the curves are not suersingular, all orders containing Z[π]. (c) If the curves are suersingular with not all endomorhisms defined over k, the orders which contain Z[π] and are maximal at. 13

17 .3. The Theorems of Waterhouse and Nakamura Now let A be a simle abelian variety of dimension g > 1 with commutative endomorhism ring. By Theorem.7(c) we have that E = Q(π) = Q End k (A). It is still true that π End k (A), but in general β = π + π will not be an integer. As a consequence, we should exect Z[π, π] to strictly contain Z[π]. This means that Z[π, π] End k (A) will have to serve as our lower bound for ossible endomorhism rings instead of the simler ring Z[π]. The task now is to analyze which subrings between Z[π, π] and the ring of integers O E may be the endomorhism ring of A. One way to begin is to choose a rime l and then localize to get R l := Z l R contained in E l := Q l E. In the roof of Theorem.8, Waterhouse found the following result: Porism.9 ([6] 4.3). Let E be the endomorhism algebra of an isogeny class of simle abelian varieties and assume E is commutative. Let R be any order in E containing Z[π, π]. Then there is a variety A in the isogeny class with End k (A) l = R l for all rimes l. This just leaves us to deal with subrings between Z[π, π] and O E whose index in O E is divisible by. By analyzing invariant sublattices of the Dieudonné module T (A) and their corresonding orders, Waterhouse was able to rove the following theorem: Theorem.10 ([6] 5.3). Let A be a simle variety with E commutative. Let K be the totally real subfield of index in E, and assume that slits comletely in K. Assume also that R = End k (A) O E contains the ring of integers O K of K. Then 14

18 R is maximal at. This result was strengthened in a follow-u aer by Nakamura [8, Thm. 1]. Nakamura s theorem also assumes that End k (A) contains O K, but he weakens the hyothesis that slits comletely in K. The hyothesis that he assumes in its lace is a bit technical and so we do not reroduce the theorem here but instead refer the interested reader to the original article [8]..4 Newton Polygons.4.1 Definitions and Proerties Let h(x) = x d + a d 1 x d a 1 x + a 0 Z[x]. Let l be a rime and consider the set S h := {(0, ord l a 0 ), (1, ord l a 1 ),..., (d 1, ord l a d 1 ), (d, 0)} of oints in the lane. The Newton Polygon N l (h) is the lower convex hull of the oints in S h. That is, N l (h) is the highest convex sequence of connected line segments connecting (0, ord l a 0 ) to (d, 0) such that all the oints in S h lie on or above this sequence of line segments. N l (h) may be constructed as follows: start with a vertical line l drawn through (0, ord l a 0 ) and rotate l about this oint counterclockwise until l touches another oint in S h. In fact, l may now touch several oints in S h, so we draw the line segment joining (0, ord l a 0 ) to the last such oint (i 1, ord l a i1 ) in S h that l currently touches. This line segment is the first segment of N l (h). Next we rotate l further about (i 1, ord l a i1 ) until l hits a further oint in S h. As before, l may be touching more than one oint in S h, so we draw the segment joining (i 1, ord l a i1 ) to the last such oint (i, ord l a i ) in S h that l currently touches. This line segment is 15

19 the second segment of N l (h). We reeat this rocess of rotating l about the oint (i j, ord l a ij ) counterclockwise until we hit another oint in S h and then drawing a line segment from (i j, ord l a ij ) to the furthest oint (i j+1, ord l a ij+1 ) that l currently touches to get the next line segment of N l (h). For examle, let f(x) = x 6 + 3x 5 + 5x 4 + 5x 3 + 5x + 75x Then the Newton olygon N 5 (f) is We have the following standard result: Lemma.11 ([13], IV Lemma 4). Let F be the slitting field of h in C and let L be a rime of O F lying over l. Let h(x) = (x α 1 )(x α ) (x α d ) be the factorization of h in F. Let v be the extension of ord l to F induced by the rime L and let λ i = v(α i ). If λ is the sloe of a segment of N l (h) having horizontal length m, then recisely m of the λ i are equal to λ. When the characteristic olynomial of Frobenius f is irreducible, we have designated π to be some chosen root of f. When we need to enumerate the comlete set of roots of f in C we will use both {α 1,..., α g } and {π 1, π 1,..., π g, π g }. We 16

20 then let β i := π i + π i and likewise β := π + π. Thus β i and β are totally real and K = Q(β). We let E be the slitting field of f in C and let K be Q(β 1, β,..., β g ), the Galois closure of K in C. With this notation we have: Theorem.1 ([14] Proosition 3.1). (a) r(a) is the sum of the multilicities of the non-zero roots of the (mod )- reduced characteristic olynomial f. (b) r(a) = #{α i / P 1 i g} where P is a rime ideal over in the ring of integers of E. (c) r(a) = #{β i / P 1 i g} where P is a rime ideal over in the ring of integers of K. Putting these last two results together easily yields: Corollary.13. r(a) is the length of the zero-sloe segment of N (f). this thesis: We next state a result which will be exanded into a more general theorem in Theorem.14 ([14] Proosition 3.). Let A/F q be an F q -simle abelian variety. Then: (a) A is ordinary (i.e. r(a) = g) if and only if the ideals (π) and ( π) (equivalently the ideals (π + π), ()) are relatively rime in E. (b) r(a) = 0 if and only if every rime P () divides (π) in E (equivalently, divides (π + π)). 17

21 (c) A is k-isogenous to a ower of a suersingular ellitic curve if and only if (π) = ( π). If A is an abelian variety and f the characteristic olynomial of Frobenius, then A is suersingular if N (f) is just a single line segment. An abelian variety is suersingular if and only if A k k falls under case (d5) of Theorem.7 over k [15, 4.]. If A is an ellitic curve or abelian surface, then suersingular is equivalent to r(a) = 0. When q =, this can be seen by examining the ossible Newton olygons. The characteristic olynomial of Frobenius f is a Weil olynomial, and thus it must be of the form f(x) = x + a 1 x + if A is an ellitic curve or f(x) = x 4 +a 1 x 3 +a x +a 1 x+ if A is an abelian surface (see Section.5). It is then easy to see that, if N (f) has a vertex, then it also has a zero-sloe segment. However, when g 3, such Newton Polygons are ossible. For examle consider the Weil olynomial f(x) = x 6 5x x 4 35x x 15x + 15 for = 5. We see that N 5 (f) has no zero-sloe segment, yet it is not a single line segment because of the vertex at (3, 1). Thus for g 3, suersingular is not equivalent to r(a) = Newton Polygons and Local Invariants The Newton olygon can be helful when calculating the local invariants. Let f be an irreducible Weil olynomial, π a root of f, and E = Q(π). Let be a rime in O E over, and let v be the corresonding valuation. Recall from equation (.1) that the local invariant i is given by i v (π) v (q) [E : Q ] (mod 1) 18

22 3 Figure.1: Newton Polygon for f This formula can be rewritten if we introduce some new notation. Let Q be a fixed algebraic closure of Q and let α 1,..., α g be the roots of f in Q. Each comleted field E can be embedded into Q in d := [E : Q ] different ways because E is searable over Q. Each embedding of E into Q sends π to one of the roots α i of f. It is known that the valuation v on Q extends uniquely to a valuation on Q. If ϕ i : E Q is an embedding, then we take v to be normalized so that it agrees with the ullback of v via ϕ i. That is, for all x E, we have that v (x) = v (ϕ i (x)). In articular we have that v (π) = v (ϕ i (π)). may define If α i1,..., α id are all the images of π under all embeddings of E, then we f (x) := d j=1 (x α ij ). Each f corresonds to a Gal(Q /Q )-conjugacy class of roots of f [16, Thm. II.], f is irreducible in Z [x], and f = f. Let π be a reresentative of the Galois 19

23 conjugacy class of roots of f. All the roots of f have the same -adic valuation, hence i v (π) v (q) [E : Q ] = v (π )d v (q) = v (f (0)) n (mod 1) (.) In this new equation, we get that i v(π)d n (mod 1). We can now see clearly the connection with Newton olygons. The negatives of the sloes of the segments of the Newton olygon will corresond to the numbers v (π ) in the numerators of the local invariants. For examle, consider the irreducible Weil olynomial f(x) = x + x + 4. This olynomial has no real roots, hence the only non-integer local invariants must come from rimes over. The Newton olygon N (f) has two segments with sloes 3 and 1. These two segments each have horizontal length one, so therefore f = (x α 1 )(x α ) where x α 1 and x α are the irreducible factors of f in Z [x]. In articular, we get two rimes 1 and over. Without loss of generality we may assume that f 1 (x) = x α 1, f (x) = x α, v (α 1 ) = 3, and v (α ) = 1. Alying equation (.) we get that i 1 = 3 and i 4 = 1. The 4 least common multile of the denominators of the local invariants is 4, hence the olynomial (x +x+ 4 ) 4 is the characteristic olynomial of a simle abelian variety of dimension 4 defined over F 4 with non-commutative endomorhism ring..5 Classification of Weil olynomials Weil olynomials corresonding to the characteristic olynomial of Frobenius of a simle abelian variety A have been classified when A is an ellitic curve, an abelian surface, or an abelian 3- or 4-fold. Classification of Weil olynomials is usually a two- 0

24 ste rocess. The first ste is determining the Weil olynomials of a given degree. Such a olynomial h then has an exonent e which is comletely determined by h (see Section.). The second ste is to find conditions on the coefficients of h that cause e to have the roerty that deg h e = g, where g is the dimension of the class of abelian varieties under investigation (in our case g = 1,, 3 or 4). The resulting olynomial h will then have the roerty that h e is the characteristic olynomial of Frobenius for an isogeny class of simle abelian varieties of dimension g. In this thesis we will only be dealing with the case where End k (A) is commutative, which corresonds to e = 1. A reliminary observation to the classification of Weil olynomials is to recall Theorem.4, which says that the characteristic olynomial of Frobenius f is always a Weil olynomial. In articular, if f has no real root, then over the real numbers we get a factorization g f(x) = (x β i x + q). i=1 If we exand this roduct, we see that the coefficients of f have a symmetry. In articular, we get that f is of the form f(x) = x g + a 1 x g 1 + a x g + + a g x g + a g 1 qx g 1 + a g q x g + + q g..5.1 Ellitic Curves The classification of Weil olynomials corresonding to ellitic curves is due to Waterhouse: Theorem.15 ([6] 4.1). Let k = F q where q = n. The isogeny classes of elli- 1

25 tic curves defined over k are in one-to-one corresondence with rational integers β having β q and satisfying one of the following conditions: (a) (β, ) = 1; (b) If n is even : β = ± q; (c) If n is even and 1 (mod 3) : β = ± q; (d) If n is odd and = or 3 : β = ± n+1 ; (e) If either (i) n is odd or (ii) n is even and 1 (mod 4) : β = 0; The first of these are not suersingular; the second are suersingular and have all their endomorhisms defined over k; the rest are suersingular but do not have all their endomorhisms defined over k. Remark.16. The corresonding characteristic olynomial of Frobenius is always h e = x βx + q. Cases (a) and (c)-(e) have h irreducible and e = 1 while in case (b) we get h(x) = x ± q and e =. The condition β q ensures that h is a Weil olynomial while the conditions (a) and (c)-(e) ensure that the local invariants determined by h yield e = Abelian Surfaces The classification of Weil olynomials of abelian surfaces is rimarily due to Rück. He roved the following theorem: Theorem.17 ([17] 1.1). The set of irreducible Weil olynomials f(x) of degree four that corresond to simle abelian surfaces is the set of olynomials f(x) =

26 X 4 + a 1 X 3 + a X + a 1 qx + q where the integers a 1 and a satisfy the following conditions: (a) a 1 < 4 q and a 1 q q < a < a 1/4 + q, (b) a 1 4a + 8q is not a square in Z, (c) one of the following conditions is satisfied: (i) v (a 1 ) = 0, v (a ) n/ and (a + q) 4qa 1 is not a square in Z. (ii) v (a ) = 0. (iii) v (a 1 ) n/, v (a ) n and h(x) has no root in Z. Remark.18. Condition (a) ensures that f is a Weil olynomial, condition (b) makes f irreducible, and condition (c) ensures that e = 1. The three cases (i), (ii), and (iii) in (c) corresond to surfaces with -rank 1,, and 0, resectively. We now look at the ossibility that f = h or f = h 4 for h an irreducible Weil olynomial. If f = h 4, then h is linear and thus h has a real root. The case when h has a real root has already been discussed in the closing aragrah of Section of this chater. This leaves the case where f = h for h an irreducible Weil quadratic olynomial with no real root. Let h(x) = x βx + q, let π be a root, and let E = Q(π). In order to guarantee that h has no real root we need β < q. We also need to get e =, where e is the least common multile of the denominators of the local invariants. In articular, we need the local invariants for the rimes of O E over to be equal to 1. If the Newton olygon N (h) has a vertex at (1, v (β)), then v (β) < n. Examining the sloes of the Newton olygon we get that 3

27 the local invariants are v(β) n and n v(β) n. However, the condition that v (β) < n means that v(β) n cannot be 1, a contradiction. Therefore, (1, v (β)) cannot be a vertex of N (h). But this means that N (h) is just a single line segment, hence h corresonds to a suersingular abelian variety. However, no suersingular abelian variety is absolutely simle [15, 4.], so we ignore this case..5.3 Abelian 3-folds The classification of Weil olynomials of abelian 3-folds is rimarily due to Haloui. First we remark that if f = h e is the characteristic olynomial of Frobenius corresonding to an isogeny class of simle abelian 3-folds, then f does not have a real root. This is due to the analysis of Weil q-numbers given at the end of section.. Such Weil q-numbers always corresond to either a suersingular ellitic curve or an abelian surface. Thus if f has a real root, the corresonding abelian 3-fold is not simle. This allows us to eliminate the cases e = and e = 6, so either e = 1 and f is irreducible or else f = h 3. The first theorem in the classification of abelian 3-folds will simly assume that the characteristic olynomial of Frobenius f does not have a real root. Thus the roots of f occur as airs of comlex conjugate Weil q-numbers and so f must be of the form f(x) = x 6 + a 1 x 5 + a x 4 + a 3 x 3 + a qx + a 1 q x + q 3. We have the following result due to Haloui: Theorem.19 ([18] 1.1). Let f(x) = x 6 + a 1 x 5 + a x 4 + a 3 x 3 + a qx + a 1 q x + q 3. Then f is a Weil olynomial with no real root if and only if the following conditions 4

28 hold: (a) a 1 < 6 q, (b) 4 q a 1 9q < a a q, (c) a a 1a 3 + qa 1 7 (a 1 3a + 9q) 3/ a 3 a a 1a 3 + qa (a 1 3a + 9q) 3/, (d) qa 1 qa q q < a 3 < qa 1 + qa + q q. Next we state conditions for when such an f is irreducible: Proosition.0 ([18] 1.3). Set r = a a 3q, s = a3 1 7 a 1a 3 qa 1 + a 3, = s 4 7 r3, u = s+. Then f(x) is irreducible over Q if and only if 0 and u is not a cube in Q( ). Last we give conditions that ensure e = 1: Theorem.1 ([18] 1.4). Let f(x) = x 6 + a 1 x 5 + a x 4 + a 3 x 3 + a qx + a 1 q x + q 3 be an irreducible Weil q-olynomial. Then f is the characteristic olynomial of an abelian 3-fold if and only if one of the following conditions holds: (a) v (a 3 ) = 0, (b) v (a ) = 0, v (a 3 ) n/, and f has no root of valuation n/ in Q, (c) v (a 1 ) = 0, v (a ) n/, v (a 3 ) n, and f has no root of valuation n/ in Q, 5

29 (d) v (a 1 ) n/3, v (a ) n/3, v (a 3 ) = n and f has no root in Q, (e) v (a 1 ) n/, v (a ) n, v (a 3 ) 3n/, and f has no root in Q nor factor of degree three in Z [x]. The -ranks of abelian varieties in (a)-(e) are resectively 3,,1,0, and 0. The abelian varieties in case (e) are suersingular. The case where e = 3 is dealt with in [19]. We will not be dealing with this case so we omit the statement of the theorem..5.4 Abelian 4-folds The classification of abelian 4-folds is due to Haloui, Singh and Xing. However, there are errors in the draft of the aer of Haloui and Singh ([0]) that was laced in the arxiv. Thus the statement of the theorems in this thesis will not exactly match those found in [0]. Since we are rimarily interested in Weil olynomials corresonding to simle abelian varieties, we may again restrict to Weil olynomials that do not have any real roots. We already know that the general form for the characteristic olynomial of Frobenius is f(x) = x 8 + a 1 x 7 + a x 6 + a 3 x 5 + a 4 x 4 + a 3 qx 3 + a q x + a 1 q 3 x + q 4. We have the following theorem: Theorem. ([0] 1.1). Let f(x) = x 8 + a 1 x 7 + a x 6 + a 3 x 5 + a 4 x 4 + a 3 qx 3 + 6

30 a q x + a 1 q 3 x + q 4 be a olynomial with integer coefficients. Set r = 3a a 4q r 3 = a3 1 8 qa 1 a 1a + a 3 r 4 = 3a qa 1 + a 1 a 16 a 1a 3 4 qa + q j = e πi 3 ( ω = 1 4 8r 6 540rr r3 4 + i9 r 3 ( r3 8 7 r3 ) 3/) 1/3 where ω is some third root. Now let S := {ω + ω + r 6, jω + jω + r 6, jω + j ω + r 6 }. S is a set of three real numbers so let γ 1 γ γ 3 be the three elements of S arranged from least to greatest. Then f is a Weil olynomial with no real root if and only if the following conditions hold: (a) a 1 < 8 q, (b) 6 q a 1 0q < a 3a q, (c) 9qa 1 4 qa 16q q < a 3 < 9qa qa + 16q q, (d) a a 1a + qa 1 ( 3 r ) 3/ a 3 a a 1a + qa 1 + ( 3 r ) 3/, (e) q qa 1 + a 3 qa q < a 4, (f) γ 1 r 4 a 4 γ r 4. The next theorem assumes that f is irreducible and then finds conditions that force e = 1: Theorem.3 ([0] 1.). Let f(x) = x 8 + a 1 x 7 + a x 6 + a 3 x 5 + a 4 x 4 + a 3 qx 3 + a q x + a 1 q 3 x + q 4 be an irreducible Weil olynomial. Then f is the characteristic olynomial of an abelian 4-fold if and only if one of the following conditions holds: 7

31 (a) v (a 4 ) = 0, (b) v (a 3 ) = 0, v (a 4 ) n/, and f has no root of valuation n/ in Q, (c) v (a ) = 0, v (a 3 ) n/, v (a 4 ) n, and f has no root of valuation n/ in Q, (d) v (a 1 ) = 0, v (a ) n/, v (a 3 ) n, v (a 4 ) n, and f has no root of valuation n/ nor factor of degree three in Q, (e) v (a 1 ) = 0, v (a ) n/3, v (a 3 ) n/3, v (a 4 ) = n, and f has no root of valuation n/3 or n/3 in Q, (f) v (a 1 ) n/3, v (a ) n/3, v (a 3 ) = n, v (a 4 ) 3n/, and f has no root in Q, (g) v (a 1 ) n/4, v (a ) n/, v (a 3 ) = 3n/4, v (a 4 ) = n, and f has no root in Q nor factor of degree or 3 in Q, (h) v (a 1 ) n/, v (a ) n, v (a 3 ) = 3n/, v (a 4 ) n, and f has no root in Q nor factor of degree 3 in Q. The -ranks of abelian varieties in cases (a)-(h) are 4,3,,1,1,0,0, and 0. The abelian varieties in case (h) are suersingular. The other ossibilities are e = and e = 4, both of which are worked out in [19]. We will not be dealing with these cases so we omit the statement of the theorem. 8

32 Chater 3 The slitting of and the -rank Proosition 3.1. Let f Z[x] be a monic irreducible olynomial of degree d with roots α 1, α,..., α d. Let E = Q(α 1,..., α d ) and let Z be a rime. Let α be a fixed root of f, let P 0 be a fixed rime over in E, let a = ord P0 (α), and let N be the number of roots of f with P 0 -adic valuation equal to a. Then ( ) #{P : P divides and ordp (α) = a} N = d #{P : P divides } (3.1) where P runs through the rimes of E. Proof. Let G = Gal(E /Q), H G the stabilizer of α, and D G the stabilizer of P 0. G acts transitively on the roots of f so we may ick elements τ i G with the roerty that τ i (α i ) = α. Suose that τ Hτ i Hτ j. Then τ = hτ i for some h H and thus τ(α i ) = h(τ i (α i )) = h(α) = α, and likewise τ(α j ) = α. But τ is an isomorhism and f has distinct roots, so i = j. Thus {Hτ i : 1 i d} are the right cosets of H. By relabeling if necessary we may assume without loss of generality that (i) ord P0 (α 1 ) = ord P0 (α ) = = ord P0 (α N ) = a (ii) ord P0 (α i ) a if i > N. 9

33 Thus we now get #{P : P, ord P (α) = a} #{P : P } = #{τ(p 0) : ord τ(p0 )(α) = a, τ G} [G : D] = #{τ : ord τ(p 0 )(α) = a}/ D [G : D] = #{τ : ord P 0 (τ 1 (α)) = a} G = #{τ τ(α i) = α for some 1 i N} G ( N ) # i=1 Hτ i = G = N H G = N 1 [G : H] = N d Remark 3.. The Newton olygon N (f) of f in the above theorem will have a segment of length N with sloe a/e where e is the ramification index of in E. Normally we think of the Newton olygon as a local object due to the choice of a rime P 0 over. However, the right hand side of (3.1) is determined with the global data. Thus this theorem rovides a kind of dual way to view the Newton olygon. Instead of fixing a articular rime P 0, we may instead fix the root α and comute the right hand side of (3.1). Theorem 3.3. Let A be a simle abelian variety over k with commutative endomorhism ring, r(a) the -rank of A, f the characteristic olynomial of Frobenius, 30

34 π a root of f in C, and β = π + π. Let E = Q(π) and K = Q(β). Let E be the Galois closure of E in C and let K be the Galois closure of K in C. Then we have ( ) #{P : P divides and π / P} r(a) = g #{P : P divides } (3.) where P runs through the rimes of E, and ( ) #{P : P divides and β / P } r(a) = g. (3.3) #{P : P divides } where P runs through the rimes of K. Proof. Equation (3.) follows immediately from Corollary.13 and the above remark. Equation (3.3) will follow from (3.). First we observe that K and E are both Galois over Q and so the slitting behavior of the rime P in E is indeendent of the choice P. In other words, if one rime P over in K is slit/ramified/inert in E, then all rimes over in K are slit/ramified/inert in E. Suose that P is ramified or inert in E and P is the rime in E over P. This is equivalent to having P c = P. We have that π π = q P and P is rime, so either π P or π P. If π P, then taking conjugates gives π P c = P. Thus we must have that π, π P, and therefore we also must have β P. As shown above, this behavior is indeendent of the choice of rime P, so we get that π P for all rimes P over in E and that β P for all rimes P over in K. In articular, #{P : P divides and π / P} = #{P : P divides and β / P } = 0. Thus (3.3) agrees with (3.3) in this case, namely r(a) = 0. 31

35 Suose that P is a rime in K which does not contain β. Then the argument in the revious aragrah shows that we must have that P slits in E, and consequently all rimes in K over slit in E. Let P and P c be the distinct rimes in E that lie over P. We know that π must lie in at least one of {P, P c }, so without loss of generality we may assume that π P and by taking conjugates we also get that π P c. As β = π + π and β / P by assumtion, it follows that π / P c. Thus #{P : P divides P and π / P} = 1. Now suose that there is a rime P in E such that π / P. But π π = q P, hence π P. Taking conjugates gives π P c. Let P = P O K. Then we have that β / P and #{P : P divides P and π / P} = 1. Putting this together with the revious aragrah gives #{P : P divides and π / P} = #{P : P divides and β / P }. Since every rime in K slits in E, the denominator in (3.3) is half the value of the denominator in (3.). Thus the formula in (3.3) follows from (3.). Proosition 3.4. Let f, E, and α be as in Proosition 3.1. Let E = Q(α) and let P 1,..., P s be the rimes of E over. For each P i let e i be the ramification index of P i over and f i the degree of the extension of residue fields for P i over. Let e be the ramification index of in the Galois extension E and let a/e be a sloe of 3

36 a segment of N (f) with length N. Let S a = {i : ord Pi (α) = ae i, 1 i s}. Then e N = i S a e i f i. (3.4) Proof. We already have that E is the Galois closure of E in C. Let g i denote the number of rimes in E that lie over P i. Since E is Galois over Q, it is also Galois over E. Thus we can let a i be the ramification index of P i in E and let b i be the degree of the extension of residue fields with resect to P i. As E is Galois over E we get that [E : E] = a 1 b 1 g 1 = a b g = = a s b s g s. (3.5) Now let m be the degree of the extension of residue fields with resect to in E. We then have that a i e i = e and b i f i = m for all 1 i s. Thus dividing (3.5) through by em we get g 1 e 1 f 1 = g e f = = g s e s f s. We now turn to Proosition 3.1. With our notation, #{P : P } = s i=1 g i. Furthermore, if P lies over P i, then ord P (α) = a if and only if ord Pi (α) = a a i = ae i e. 33

37 If we define S i := {P : P P i }, then g i = S i. Putting it all together we get ( ) #{P : P, ordp (α) = a} N = d #{P : P } ( ) #{P : P divides Pi for some i S a } = d #{P : P } ( ) i S = d a S i s j=1 g j = d ( ) S i s i S a j=1 g j = d ( ) g i i S a s j=1 e j f j e i f i g i = d ( ) e i f i s i S a j=1 e jf j = d ( ) ei f i d i S a = i S a e i f i Remark 3.5. This gives a refinement of the relation d = s i=1 e if i. Theorem 3.6. Let A be a simle abelian variety of dimension g defined over the finite field k = F q for q = n. Let r(a) denote the -rank of the abelian grou A( k)[]. Let f Z[x] be the characteristic olynomial of the Frobenius endomorhism of A and suose that f is irreducible. Let π be a root of f, β = π + π, E = Q(π), K = Q(β). For each rime P in K over and P in E over, let e(p ) and e(p) be the ramification index of P and P over, resectively, and let f(p ) and f(p) be degree of the extension of Z/Z corresonding the the rimes P and P, resectively. 34

38 Then r(a) = π / P e(p)f(p) (3.6) where the sum ranges over the rimes in E over not containing π, and r(a) = β / P e(p )f(p ) (3.7) where the sum ranges over the rimes in K over not containing β. Proof. (3.6) follows from Proosition 3.4 by letting α = π and a = 0. The roof of (3.7) follows from the fact that any rime P of K that does not contain β must slit in E as P and P c. But E is quadratic over K, so if P slits then we get that e(p ) = e(p) = e(p c ) and f(p ) = f(p) = f(p c ). Exactly one of {P, P c } does not contain π and thus every summand in (3.7) aears in (3.6). Conversely, if π / P for some P, then P c P and π P c. Furthermore, if we let P = P O K, then e(p ) = e(p) and f(p ) = f(p) because E is quadratic over K and P slits in E. Since β = π + π, π / P, and π P, we see that β / P. Thus every summand in (3.6) aears in (3.7), hence (3.6) and (3.7) have the same summands. Remark 3.7. (a) Equation (3.7) is easily seen to be qualitatively correct because g = [K : Q] = P e(p )f(p ). Thus r(a) must lie between 0 and g, as required. (b) This theorem extends Theorem.14, which was only able to determine the extremes, either r(a) = 0 or r(a) = g. Examle 3.8. (Ellitic Curves) Let C be an ellitic curve over k with Q End k (C) commutative. Then E = Q(π) is a quadratic imaginary extension of Q. Suose 35

39 that in O E we get a factorization () = PP c. Then we must have that (π) = P i (P c ) n i. The two local invariants are therefore i/n and (n i)/n. In order for these to be integers, we must have i {0, n}. Thus, only one of {P, P c } contains π and so equation (3.6) gives r(c) = 1. If is inert or ramifies in E, then there is only one rime P in E over and it must contain π. Thus (3.6) gives r(c) = 0. Examle 3.9. (Abelian Surfaces) This examle is contained in the roof of [14, Thm. 3.7] but we resent it here in light of Theorem 3.6 which gives the calculations a slightly different flavor than that found in [14]. Let A/k be an absolutely simle abelian variety of dimension. Then the factorization of () in E can only be one of the following cases: a) () = P 1(P c 1) ( ramifies in K) b) () = P 1 P c 1 ( is inert in K) c) () = P 1 P c 1P s, 1 s ( slits comletely in K but not in E) d) () = P 1 P c 1P P c ( slits comletely in E) Note that cases like () = P 4 are not ossible. If () = P 4, then (π) = P and ( π) = P. Thus (π) = ( π), hence A is suersingular, contradicting the hyothesis that A is absolutely simle. In case a), the factorization of the ideal (π) in E is P i 1(P c 1) n i for some 0 i n. The local invariants i/n and (n i)/n are integers if and only if i {0, n, n}. The case i = n is not ossible because then (π) = ( π) and A would be suersingular hence not absolutely simle. Thus (π) is either P n 1 or (P c 1) n. Without loss of generality we may assume that (π) = (P 1 ) n. The sum in (3.6) only contains one summand corresonding to P c 1. For P c 1 we have e 1 = and f 1 = 1 36

40 and thus r(a) = by Theorem 3.6. In case b), in order for the local invariants integers, it must be that (π) is P n 1 or (P c 1) n. Without loss of generality assume (π) = (P 1 ) n. Since is inert in K it follows that e 1 = 1 and f 1 = for the rime P c 1 and therefore r(a) =. In case c) we have that (π) is P n 1 P sn/ or (P c 1) n P sn/. Without loss of generality assume that (π) = (P 1 ) n P sn/. Thus P c 1 is the only rime over in E that does not contain π. We have that e 1 = 1 and f 1 = 1 for P c 1 and so r(a) = 1. In case d) the ideal (π) is P n 1 P n, P n 1 (P c ) n, (P c 1) n P n, or (P c 1) n (P c ) n. For all rimes in E over we have that e i = f i = 1. In every one of the four cases for the factorization of (π) we see that π is not contained in exactly two rimes of the rimes of E over and thus r(a) =. Examle (Abelian 3-folds) The case of abelian 3-folds can be handled similarly to that of abelian surfaces if one is atient enough to enumerate all ossible slitting behaviors of in K and E. We will not do this exhaustively here, but we will hit uon some highlights. For an examle of such an analysis, suose that A is defined over k = F. Suose that () = P 1 P in K and suose that P 1 slits in E into P 1 and P1. c Suose also that P is inert in E, and let P be the unique rime in E lying over P. Then it must be that (π) is P 1 P or P1P c. Without loss of generality assume that it is the first case. Then β / P 1 and β P. Therefore by Theorem 3.6 we get that r(a) = 1. An examle of an abelian 3-fold with this behavior is the isogeny class corresonding to the Weil olynomial f(x) = x 6 3x 5 +10x 3 75x+15. One difference between abelian surfaces and abelian 3-folds is that the slitting 37