Mordell-Weil Groups of Large Rank in Towers

Transcription

1 Mordell-Weil Groups of Large Rank in Towers Item Type text; Electronic Dissertation Authors Occhipinti, Thomas Publisher The University of Arizona. Rights Copyright is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 21/04/ :45:47 Link to Item

2 Mordell-Weil Groups of Large Rank in Towers by Thomas Occhipinti Copyright c Thomas C. Occhipinti 2010 A Dissertation Submitted to the Faculty of the Department of Mathematics In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy In the Graduate College The University of Arizona

3 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Thomas Occhipinti entitled Mordell-Weil Groups of Large Rank in Towers and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Romyar Sharifi Ana-Maria Castravet William McCallum Pham H Tiep Date: 4/7/10 Date: 4/7/10 Date: 4/7/10 Date: 4/7/10 Final approval and acceptance of this dissertation is contingent upon the candidate s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Dissertation Director: Douglas Ulmer Date: 4/7/10

4 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder. SIGNED: Thomas Occhipinti

5 4 Acknowledgements I have been fortunate to have many great friends throughout my academic career. I will not attempt to name them all, but without their aid I could never have made it where I am today. Professionally, I am particularly indebted to John Dillon and Douglas Ulmer for working with me closely and providing me with interesting things to think about. The quality of this dissertation has been greatly improved by comments given by Douglas Ulmer, Romyar Sharifi, Pham Tiep, Ana-Maria Castravet, William McCallum, and Lisa Berger. Their time and effort in this regard is much appreciated. My parents, Theresa and Robert, have given me unwavering support throughout my life. Finally, I would like to thank Tristan for holding me together through this stressful year.

6 5 Table of Contents List of Figures Abstract Introduction Notations and Preliminaries Berger s Construction The Curves C d The geometry of the curves C d Endomorphism algebras of Jacobians of the curves C d The Special Cases f = g and g = 1/f An Observation about Endomorphism Algebras The f = g Case Comparison to Previous Results The m > 2 Case General Discussion A family of cases with m = Future Directions References

7 6 List of Figures Figure 2.1. Definition of S d Figure 9.1. Pairs (a, b) for which the polynomial F factors for a non-zero value of t

8 7 Abstract Let k be the algebraic closure of the field with q elements. We build upon recent work of Ulmer and Berger to give examples of elliptic curves and higher dimensional abelian varieties over the field K = k(t) with the property that their ranks become arbitrarily large when d th roots of the variable t are adjoined to K for d varying across the integers relatively prime to q. We also give a first example of an elliptic curve whose rank under such extensions grows linearly in d, for those d prime to q.

9 8 1. Introduction It is a classical question to determine which non-negative integers occur as the ranks of elliptic curves over the rational numbers. In particular, the largest rank known to occur as of this writing is 28, and it is unknown whether arbitrarily large ranks occur. Computations seem to suggest that asymptotically half of elliptic curves over Q have rank zero and half have rank one, making larger ranks over Q a rare occurrence [Sil99]. Analogous questions can be asked by replacing the rationals with a general number field and elliptic curves with abelian varieties of a fixed dimension and in these cases even less is known. It is of great interest that the picture over function fields in positive characteristic is drastically different. Indeed, constructions of elliptic curves of arbitrarily large rank over fields of the form F q (X) (the field of rational functions on a projective smooth curve X defined over F q ) have been known since The first such examples are due to Shafarevich and Tate [ST67] who give examples of isotrivial 1 elliptic curves over F q (t) with arbitrarily large ranks. In his paper [Ulm07], Ulmer shows that a wide class of elliptic curves over F q (t) attain large analytic ranks in the tower attained by replacing t with t d. The main 1 An elliptic curve over F q (t) is isotrivial if its j-invariant lies in the field F q. This is equivalent to the condition that it becomes isomorphic to a constant curve after a base change.

10 9 theorem is as follows. Theorem 1 (Ulmer). Let K = F q (t), and let E/K denote a non-isotrivial elliptic curve. Let n denote the part of the conductor of E/K prime to 0 and. Suppose that the degree of n is odd and that p 5. Then for d = q n + 1, ord s=1 L(E/K(t 1/d ), s) d 2n c, where c is a constant independent of n. For some specific curves E, he is able to demonstrate that the conclusion of the conjecture of Birch Swinnerton-Dyer 2 is true and this implies that those curves actually have large rank. For example this is known to be true of curves for which an affine equation with four monomials can be given. As a particular example, one can take the curve given affinely by the equation y 2 + xy = x 3 + t over the field F q (t). One very general class of elliptic curves over function fields for which BSD is known to hold is the class of such curves for which the associated surface is dominated by a product of curves (see below). In her thesis [Ber08], Berger demonstrates a very general method by which one can generate towers of elliptic surfaces for which all of the surfaces in the tower are dominated by a product of curves. Hence the conclusion of BSD is true for the associated elliptic curves, and the results of Ulmer in [Ulm07] 2 The conjecture of Birch and Swinnerton-Dyer, hereafter abbreviated as BSD, asserts that the analytic rank of the abelian variety A/K, that is ord s=1 L(A/K, s), is equal to the rank of the abelian group of K-rational points of A.

11 10 can in many cases be used to show that the large analytic ranks in fact correspond to large algebraic ranks. A recent paper of Ulmer [Ulm10] studies the geometry of Berger s construction and gives examples of elliptic curves of large rank over F q (t) together with explicit closed form formulas for generators of the Mordell-Weil groups of the curves. The curves given in this paper are arranged in towers of the form f(x, y, t d ) = 0 and attain large ranks when d divides q n + 1 for some n. This paper also gives a formula for the ranks of the elliptic curves E d which is independent of the previous L-function result. This formula is equation 3.4 and will play a key role in the proofs of the theorems in this paper. In the present work, Theorem 5, we give a one parameter family of examples of elliptic curves E d as above for which the rank of E d over F q (t) is always at least d. This phenomenon is not explained by Ulmer s previous L-function result [Ulm07] and hence seems to suggest a new source of rank. In Section 9.2, we will also use this point of view to construct a two parameter family of towers of four-dimensional abelian varieties which attain large rank.

12 11 2. Notations and Preliminaries We will throughout this paper always use the letter k to denote the perfect ground field over which we are operating. By a curve over a field K we will mean a one dimensional connected smooth scheme over Spec(K), and by a surface over K we will mean a two dimensional connected smooth scheme over Spec(K). We may associate to any given a surface S together with a generically smooth dominant morphism S P 1 k a curve over the field K := k(t) by taking the generic fiber of the morphism. By a tower of curves (respectively surfaces) we will mean a set of curves (respectively surfaces) X d indexed by the set of positive integers d which are relatively prime to the characteristic of the ground field k such that there is an element t k(x 1 ) such that k(x d ) is isomorphic to k(x 1 )(t 1/d ) and such that for every pair of positive integers e and d with e dividing d, there is a morphism X d X e which induces the inclusion of function fields k(x 1 )(t 1/e ) k(x 1 )(t 1/d ). Recall that a surface S over K is said to be dominated by a product of curves if there exist curves C and D over K such that there is a dominant rational map C D S. Let S be a surface over k together with a dominant morphism π : S P 1. Let t be a coordinate on P 1. Define the morphism π d : P 1 P 1 by the map t t d. We

13 12 will often use the notation S d to denote a surface, associated to the surface S and the morphism π, which is a smooth proper model of the fiber product S P 1 P 1 with respect to the maps π and π d given above. The existence of S d follows from a theorem of Lipman. See [LE06], Theorem This is summarized in the diagram below. S d S P 1 P 1 S P 1 t t d P 1 Figure 2.1. Definition of S d Note that this does not define S d uniquely, as there will be many smooth proper models of a given surface. When a more precise construction of the surfaces S d is necessary, it will be provided. We will say, following Berger, that a smooth surface S over k together with a surjective morphism π : S P 1 is dominated by products of curves in towers if S itself is dominated by a product of curves and the surfaces S d defined above are dominated by products of curves for every d prime to the characteristic of the ground field. Note that this is well defined even though S d is not. This is because being dominated by a product of curves is a birational invariant, and all possible choices for S d are birational to S P 1 P 1. For a more detailed discussion, see [Ber08]. We recall that if A and B are abelian varieties defined over a field K then a homomorphism φ from A to B is a morphism of varieties which is simultaneously a

14 13 homomorphism of groups. We denote by Hom K (A, B) the abelian group of homomorphisms from A to B defined over the field K. We denote by End K (A) the (not necessarily commutative) ring of homomorphisms from A to itself which are defined over the field K. If the field K is clear from context, we will sometimes omit it from the notation. We will further define Hom 0 K(A, B) to be Hom K (A, B) Z Q and End 0 K(A) to be End K (A) Z Q. Note that the isogenies in End K (A) become invertible in End 0 K(A). We will say that an abelian variety A over a field K is simple if A does not have any non-trivial abelian subvarieties defined over K. Note that by extending the ground field, a simple abelian variety may become non-simple. It is easy to show that if A is a simple abelian variety over K then End 0 K(A) is a division ring. See [Mil] for details. The theory of Honda and Tate classifies simple abelian varieties over a given finite field F q up to isogeny and also provides a complete description of the algebras End 0 F q (A) associated to these abelian varieties. In this case the abelian variety has a distinguished endomorphism, namely the Frobenius endomorphism relative to the field F q. For a given abelian variety A over F q, we will denote this endomorphism by π A. Suppose that A is simple. It can be shown, see [Tat68], that the span of π A and Q in the division ring End 0 k(a) is a number field. We may thus associate to π A a Galois conjugacy class of algebraic numbers, namely the set of images of π A under all embeddings of this number field into Q.

15 14 We recall the main theorem of Honda-Tate theory as it will be used later. The reader wishing a more detailed discussion of this theorem should see [Tat68]. Theorem 2 (Honda-Tate). 1. The map A π A induces a bijection between isogeny classes of simple abelian varieties over the field F q and Galois conjugacy classes of q-weil numbers E := End 0 F q (A) is a central simple algebra with center F := Q(π A ) dim(a) = [E : F ] 1/2 [F : Q] 1 A q-weil number is an algebraic integer α such that every embedding ι : Q(α) C satisfies ι(α) = q 1 2

16 15 3. Berger s Construction In [Ber08], Berger constructs a tower of surfaces S d over a field k indexed by the positive integers prime to the characteristic of k together with maps to P 1 k such that every surface in the tower is dominated by a product of curves. When k is finite, this implies that the conclusion of the BSD conjecture is true for the Jacobians of the generic fibers of these maps, i.e., that the analytic rank of the generic fiber of the morphism S d P 1 k is equal to the algebraic rank. Let C and D be proper geometrically irreducible smooth curves over the field k, and let f and g be separable non-constant rational functions on C and D respectively. Consider the rational map π : C D P 1 given by (P, Q) [f(p ) : g(q)]. Note that this rational map is undefined at those points (P, Q) where f(p ) = g(q) = 0 or f(p ) = g(q) =. Let S denote a surface obtained from C D by using blow-ups to resolve the map π to a morphism. Let S and π be such that the morphism π : S P 1 is a smooth proper relatively minimal model of the morphism π : S P 1. Then, we have a theorem of Berger: Theorem 3 (Berger). The S constructed above, together with the morphism π : S P 1, is dominated by products of curves in towers.

17 16 Remark: Let t be a coordinate on P 1. It is illustrative to note that the surface S is birational to the zero set of f tg in C D P 1, and hence the surface S d is birational to the zero set of f t d g in C D P 1. Let C d be the smooth projective curve given affinely by z d = f(x), and let D d be the smooth projective curve given affinely by w d = g(y). There is a rational map C d D d S d given on an affine patch by ((x, z), (y, w)) (x, y, t = z ), and hence we see the domination by a w product of curves explicitly. We will, in particular, be interested in studying the generic fibers of the morphisms S d P 1 constructed above for d relatively prime to the characteristic of k. Under mild hypotheses, to be discussed below, this generic fiber will be a curve over k(t), the possible issue being smoothness. In either case, we will denote the generic fiber of the morphism S d P 1 by X d. Recall that the inputs to the construction of S d, and hence X d, are two curves C and D over the field k and non-zero rational functions f and g on C and D respectively. Let m and n be the degrees of f and g, respectively, as morphisms to P 1. We will further assume, for simplicity, that the zeroes and poles of f and g are k-rational. Denote the zeroes of f by P 1, P 2,..., P M and the zeroes of g by Q 1, Q 2,..., Q N. Similarly, denote the poles of f by P 1, P 2,..., P M, and the poles of g by Q 1, Q 2,..., Q N. We will denote the multiplicity of P i by a i, the multiplicity of P i by a i, the multiplicity of Q i by b i and the multiplicity of Q i by b i. Hence, recalling the assumption that all P i, P i, Q i, and Q i are k-rational, we have that

18 17 m = M M a i = a i and n = i=1 i=1 N N b i = b i. (3.1) i=1 i=1 With this notation, we have the following theorem, also due to Berger. Theorem 4. If gcd(a 1,..., a M, a 1,..., a M, b 1,..., b N, b 1,..., b N ) = 1 and for all i, j, i, j the characteristic of k divides neither gcd(a i, b j ) nor gcd(a i, b j ), then X d is a smooth, proper curve of genus g(x d ) = mg(d) + ng(c) + (m 1)(n 1) i,j δ(a i, b j ) δ(a i, b j ), (3.2) i,j where g(c) and g(d) denote the genera of C and D respectively, and for any two positive integers a, b, δ(a, b) = (ab a b + gcd(a, b))/2. Remark: Note, in particular, that if f and g have distinct zeroes and poles (e.g., if all the a i, b i, a i, and b i are equal to 1), then the formula becomes much simpler, and one simply has g(x d ) = mg(d) + ng(c) + (m 1)(n 1). If we further assume, as we often will later, that C and D are both isomorphic to P 1, then the formula simplifies to g(x d ) = (m 1)(n 1). (3.3)

19 18 In his paper [Ulm10], Ulmer demonstrates a formula for the algebraic rank of the Jacobian of X d when d is relatively prime to the characteristic of k. The formula is as follows when k is algebraically closed. rank(jac(x d )) = dim Q (Hom 0 k(jac(c d ), Jac(D d )) µ d ) c 1 (d) + c 2 (d) (3.4) The formula requires a bit of explanation. Recall that C d and D d are the smooth projective models of the curves with affine models given by z d = f and w d = g respectively. Note that there is a natural action of µ d, the group of d th roots of unity (which we take to be in k for simplicity), on C d and D d which induces an action of µ d on Jac(C d ) and Jac(D d ) which is defined over k(µ d ). The action of µ d on C d is given affinely by µ d C d C d : (g, (x, z)) (x, gz), and the action on D d is similar. Note that if ζ µ d and φ Hom 0 k(jac(c d ), Jac(D d )), then both ζ φ and φ ζ are again elements of Hom 0 k(jac(c d ), Jac(D d )). However, there is no reason to expect them to be equal. If, however, ζ φ = φ ζ for all ζ µ d, then we will say that φ commutes with the action of µ d. The exponent of µ d on the term (Hom 0 k(jac(c d ), Jac(D d ))) µ d in the formula denotes the subalgebra of Hom 0 k(jac(c d ), Jac(D d )) consisting of elements which commute with the action of µ d. The dim Q denotes the Q-vector space dimension. The function c 1 (d) is non-negative and bounded above by a linear function of the form c d for some constant c determined by geometric data. The function c 2 (d) is periodic and hence

20 19 bounded. In order to give concise formulas for the functions c 1 and c 2, we will assume the field k is algebraically closed. For the general case, see [Ulm10]. We have ( ) c 1 (d) = (f v 1) d (3.5) v 0, where the sum is over the closed points v of P 1 k other than 0 and and f v denotes the number of irreducible components in the fiber of the map S P 1 above the point v. have For d relatively prime to the multiplicities of the zeros and poles of f and g we c 2 (d) = (M 1)(N 1) + (M 1)(N 1) (3.6) where, as above, M and N represent the number of zeros of the functions f and g respectively and M and N denote the number of poles of the functions f and g respectively. We will not use the formula in the case when d is not relatively prime to the multiplicities of the zeroes and poles of f and g, but it can be found in [Ulm10]. For the purposes of this paper, we will consider the above construction only with C = D = P 1 k, and we will choose f and g so that C d = D d.

21 20 4. The Curves C d 4.1. The geometry of the curves C d Let us fix a rational function f k(t). We consider, as above, the smooth projective curve C d given in an affine patch by z d = f(x). A simple Riemann-Hurwitz calculation gives us the following proposition. Proposition 1. If f is a degree m > 1 rational function with distinct zeros and poles, and C d is the smooth projective curve with affine model z d = f(x), then the genus of C d is (m 1)(d 1). Thus we find that the dimension of Jac(C d ) is (m 1)(d 1), for d Endomorphism algebras of Jacobians of the curves C d We are interested in understanding the endomorphism algebras of the Jacobians of the curves C d. The behavior of this algebra will be different in characteristic 0 and in positive characteristic. As discussed earlier, each ζ µ d gives an automorphism of C d defined over k(µ d ) via the morphism given affinely by (x, z) (x, ζz).

22 21 Hence we have a map µ d End k(µd )(Jac(C d )), which induces a homomorphism τ : Qµ d End 0 k(µ d )((Jac(C d )). (4.1) We emphasize that we use the notation Qµ d for the group ring, not a field. Hence Qµ d is a d-dimensional algebra over Q. Although it is a slight abuse of notation, we will frequently suppress the τ when denoting the images of elements of Qµ d in End 0 k(µ d )(Jac(C d )). Suppose now the field k is finite. In this case, the curve C d also possesses a Frobenius endomorphism given by (x, z) (x k, z k ) which we denote by π. We will denote the Frobenius endomorphism upon extending the scalars to k(µ d ) by π d. Note that π d = π r where r is the order of k modulo d. Note that π and π d induce endomorphisms of Jac(C d ) and that π d commutes with the action of µ d on Jac(C d ).

23 22 5. The Special Cases f = g and g = 1/f Throughout this section we will assume we are working over a ground field k which is algebraically closed. We will thus often suppress the k from the notation for homomorphisms and endomorphisms of abelian varieties. Resuming the study of the construction above, we restrict first to the case where we choose C = D and f = g. Thus C d is isomorphic to D d via the obvious isomorphism. Hence we may simplify the term Hom 0 k(jac(c d ), Jac(D d )) in formula 3.4 to End 0 k(jac(c d )), which is more easily analyzed. Further inspection reveals that Hom 0 (Jac(C d ), Jac(D d )) µ d is isomorphic to End 0 (Jac(C d )) µ d, i.e., the part of the endomorphism algebra of the Jacobian of C d which commutes with the action of µ d. As noted above, in the case that k is finite the Frobenius endomorphism π d is an element of End 0 (Jac(C d )) µ d. We also note that τ(µd ) End 0 (Jac(C d )) µ d, and hence we also have have τ(qµ d ) End 0 (Jac(C d )) µ d. If we instead choose to let C = D and g = 1/f then we still have C d = Dd, with an explicit isomorphism ι : C d D d given by ι : (x, z) (x, 1/z) (5.1) and hence Hom 0 (Jac(C d ), Jac(D d )) is isomorphic to End 0 (Jac(C d )) as above. Unlike

24 23 the case above, however, the µ d action does not commute with the isomorphism ι between C d and D d but instead anti-commutes with it in the sense that if ζ µ d then ι ζ = ζ 1 ι. Hence Hom 0 (Jac(C d ), Jac(D d )) µ d is isomorphic to the subalgebra of End 0 (Jac(C d )) which anti-commutes with the µ d action. We will denote this subalgebra by End 0 (Jac(C d )) anti µ d. Note that unlike in the f = g case above, there is no a priori guarantee that End 0 (Jac(C d )) anti µ d contains any elements other than 0.

25 24 6. An Observation about Endomorphism Algebras The purpose of this section is to prove a relatively simple result about the endomorphism algebras of abelian varieties over finite fields. This section relies heavily on Honda-Tate theory, an exposition on which can be found in [Tat68]. The proposition we wish to prove is the following. Proposition 2. Suppose that A is an abelian variety of dimension n over a finite field k. Then every maximal commutative subalgebra of End 0 (A) has dimension 2n over Q. Proof. We note first that replacing A with an isogenous abelian variety does not change the endomorphism algebra. We then recall that every abelian variety A is isogenous to a product of simple abelian varieties, say A e A em m with A i not isogenous to A j, and hence we may replace A by A e A em m. We note further that it suffices to show the proposition is true for isotypical abelian varieties, that is abelian varieties which are isogenous to a power of a single simple abelian variety. This is because there are no non-zero homomorphisms between abelian varieties that do not share an isogenous simple factor and thus End 0 (A e A em m ) is isomorphic to End 0 (A e 1 1 )... End 0 (A em m ). Hence we assume that A A e 1 1. Recall that by the main theorem of Honda Tate theory (Theorem 2),

26 25 E := End 0 (A 1 ) is a central simple algebra over F := Q(π) (the Q-span of the Frobenius endomorphism relative to k in E). By part 2 of the theorem, we have the relation 2 dim(a 1 ) = [E : F ] 1/2 [F : Q] on the dimensions. Hence, we find that End 0 (A e 1 1 ) = M e1 (E) is a central simple algebra over F of dimension (2 dim(a 1 )/[F : Q]) 2 e 2 1. By standard facts about central simple algebras, this means that every maximal commutative subalgebra of End 0 (A e 1 1 ) has dimension 2e 1 dim(a 1 )/[F : Q] over F, i.e., dimension 2e 1 dim(a 1 ) over Q. Upon noting that 2e 1 dim(a 1 ) is precisely 2 dim(a), we have proven the proposition.

27 26 7. The f = g Case In order to compute the functions c 1 and c 2 in formula 3.4 for the ranks of curves given by the construction of Berger it is necessary to understand the geometry of the fibers of the morphism of the surface S to P 1. For ease of computation, we will usually work over the algebraic closure k of our ground field k. We now make several assumptions about the inputs to the construction. First we will assume that C = D = P 1 k. We will also assume that f and g are rational functions of degree m with no repeated zeroes or poles. Proposition 3. Under the above hypotheses, the genus of the generic fiber of the morphism S P 1 is (m 1) 2, and hence the generic fibers, X d, of the morphisms S d P 1 also have genus (m 1) 2. In particular, if m = 2 the construction yields a tower of curves of genus one. Proof. This is a simple consequence of Theorem 4. We will now further assume that f = g. Let us take f to have the form f(x) = x(x 1) x a for some a k with a / {0, 1}. This is a generic degree two rational function on P 1 in the sense that any separable degree two rational function on P 1 can be transformed to one of this form by composition with a linear fractional transformation.

28 27 We find that S is given affinely by the equation x(x 1)(y a) y(y 1)(x a) = t. (7.1) We wish to construct S more precisely. Recall that S was attained by resolving the rational map C D P 1 to a morphism of regular surfaces via blow-ups. In this case, we have the rational map π : P 1 P 1 P 1 : (x, y) [f(x) : f(y)]. This morphism is undefined precisely at the eight points (0, 0), (0, 1), (1, 0), (1, 1), (a, a), (a, ), (, a), and (, ). We will denote by S O the complement of these eight points in P 1 P 1. It is easy to verify that a blow-up at each of these eight points resolves the rational map π : P 1 P 1 P 1 : (x, y) [f(x) : f(y)] to a morphism π : where we denote by P1 P 1 P 1, P 1 P 1 the surface which results from blowing up P 1 P 1 once at each of the points listed above. The result is clearly regular, and so we may take S to be P 1 P 1. We will denote by E the union of the exceptional divisors of the eight blow-ups, a closed subvariety of P1 P 1. Set theoretically, the closed points of S are the closed points of S O and the closed points of E. Each of the eight disjoint copies of P 1 which make up E maps isomorphically to P 1 under π. Let us now consider the fibers of the morphism π : S P 1. Note that for each t,

29 28 π 1 (t) = (π 1 (t) S O ) (π 1 (t) E). (7.2) For fixed t 0,, it is easy to see that π 1 (t) S O is isomorphic to the zero locus of the polynomial x(x 1)(y a) ty(y 1)(x a) in A 2. As π 1 (t) E consists only of eight isolated points, to determine the number of irreducible components in π 1 (t) it is sufficient to determine the number of irreducible components in the affine model x(x 1)(y a) ty(y 1)(x a) of the fiber π 1 (t). Hence, the reducible fibers correspond to the values of t which make the polynomial x(x 1)(y a) ty(y 1)(x a) (7.3) reducible. Proposition 4. Suppose the characteristic of k is not 2. If a / {0, 1, 2 1 } and t k, then the polynomial x(x 1)(y a) ty(y 1)(x a) is irreducible in k[x, y] except when t = 0 or t = 1. When t = 1 the polynomial factors as x(x 1)(y a) y(y 1)(x a) = (x y)(xy ax ay + a), (7.4) with the latter polynomial being irreducible. If a = 2 1 then when t = 1 the polynomial

30 29 factors as above, but additionally the polynomial factors when t = 1 as x(x 1)(y 2 1 ) + y(y 1)(x 2 1 ) = (x + y 1)(xy 2 1 x 2 1 y). (7.5) Proof. First note that if x(x 1)(y a) ty(y 1)(x a) factors then it must factor into a linear and a quadratic factor because it is a degree three polynomial. Hence we wish to know the solutions of the equation below with a, b, c, d, e, f, g, h, i, j, t k and a 0, 1 and t 0. x(x 1)(y a) ty(y 1)(x a) = (bx + cy + d)(ex 2 + fxy + gy 2 + hx + iy + j) (7.6) We break our proof into two cases. First we assume that b 0. We then see immediately that e = 0 and that f 0. We normalize so that b = f = 1. We then see h = a. Further, one sees that cg = 0 and dj = 0. If c = 0 then one sees that d 0 so that j = 0, and thus d = 1. But x 1 can be a factor of this polynomial only if a = 1, so we see that c 0. Thus g = 0, and hence c = t. Thus i = a from examination of the y 2 coefficient. If d = 0 we see j = a. Thus from the xy coefficient we see that a + at = t 1, which implies either a = 1 or t = 1. We have explicitly ruled out a = 1 and the t = 1 case gives the factorization stated in the proposition. If instead j = 0 then we see d = 1. Thus from the y coefficient we see t = 1. Then looking at the xy coefficient we see that a = 2 1, leading to the other case in

31 30 the proposition. If we assume instead that b = 0, it follows that c 0. We thus normalize so that c = e = 1. The x 2 coefficient forces d = a, but y a can be a factor of this polynomial only if a = 0 or a = 1, and these cases are already excluded. Remark: We note that one could also prove this result by computing the discriminant of this elliptic curve with a computer algebra system. We give the above proof because it generalizes to the case where the construction produces a higher dimensional abelian variety. Hence we find that for this particular set of inputs, after extending the ground field to k, we have c 1 (d) = d by formula 3.5, provided a / {0, 1, 2 1 }. It is also easy to compute from the earlier formula 3.6 that c 2 (d) = 2. By combining the above facts, we find the following theorem. Theorem 5. Let k = F q with q not a power of 2, a k \ {0, 1, 2 1 }, and f(x) = x(x 1). Then the elliptic curve X x a d defined over the field k(t) by the affine equation f(x) f(y) = td has rank at least d for every integer d > 2 relatively prime to q. Proof. By formula 3.4, the rank of X d is given by rank(x d ) = dim Q (Hom 0 k (Jac(C d), Jac(D d )) µ d ) c 1 (d) + c 2 (d), (7.7)

32 31 which in this case simplifies to rank(x d ) = dim Q (End 0 k (Jac(C d)) µ d ) d + 2. (7.8) We will temporarily restrict to k(µ d ), a finite extension of k, so that we may apply Proposition 2. Note that, by Proposition 1, Jac(C d ) is a (d 1)-dimensional abelian variety, and hence, by Proposition 2, every maximal commutative subalgebra of End 0 k(µ d )(Jac(C d )) has dimension 2d 2. It is certainly the case that End 0 k(µ d )(Jac(C d )) µ d contains a maximal commutative subalgebra of End 0 k(µ d )(Jac(C d )) (since the image of Qµ d in End 0 k(µ d )(Jac(C d )) must be contained in some maximal commutative subalgebra), and hence dim(end 0 k(µ d )(Jac(C d )) µ d ) 2d 2. It is easy to see that End 0 k(µ d )(Jac(C d )) µ d is a subalgebra of End 0 k (Jac(C d)) µ d, and hence dim(end 0 k (Jac(C d)) µ d ) 2d 2. Thus we find that the rank of the generic fiber of the map S d P 1 is at least 2d 2 d + 2 = d, as was to be shown. Remark: There is no reason to expect the above inequality to be an equality. If, for example, µ d lies in the center of the endomorphism algebra, then the rank may be much larger. The disposition of µ d in End 0 (Jac(C d )) is an interesting problem and warrants further investigation.

33 32 8. Comparison to Previous Results This construction provides an example of a tower of elliptic curves where the rank grows at least linearly for every d prime to the characteristic of the ground field. Hence, unlike previous constructions, the rank of the associated curves is large for every d independent of the relationship between q and d. This is in contrast to the constructions of Tate and Shafarevich [ST67] and Ulmer [Ulm10]. We recall the theorem of Ulmer, from the introduction: Theorem 6. Let K = F q (t) and let E/F q (t) denote a non-isotrivial elliptic curve. Let n denote the part of the conductor of E/K prime to 0 and. Suppose that the degree of n is odd and that p 5. Then for d = q n + 1, ord s=1 L(E/K(t 1/d ), s) d 2n c, where c is a constant independent of n. We wish to note that the hypotheses of this theorem do apply to the curves constructed in the previous section. Indeed, let E be the elliptic curve with equation f(x) f(y) x(x 1) = t with f(x) =. A Weierstrass form for E is given by: x a

34 33 y 2 + (t 1)xy + (t 2 t)a 2 y = x 3 + (ta 2 + 2ta)x 2 + (2t 2 a 3 + t 2 a 2 )x + t 3 a 4. From this, one can compute the discriminant of E. It is given by: = a 4 (a 1) 4 t 4 (t 1) 2 (t 2 (16a 2 16a + 2)t + 1). Recall that the degree of each point ν in the conductor is equal to v ν ( ) f ν + 1, where v ν denotes the valuation associated to the prime (t ν) in the ring k(t) and f ν denotes the number of components in the t = ν fiber of the morphism E P 1 t. The t = 1 fiber has two components, as shown above, and hence contributes 1 to the degree of the conductor. The polynomial t 2 (16a 2 16a + 2)t + 1 is separable provided a 2 1. Thus, this polynomial has two distinct roots, each distinct from 1. We have already shown that neither corresponding fiber is reducible, and hence each root contributes 1 to the degree of the conductor. Thus, the degree of the part of the conductor of E that is prime to 0 and is 3, and is odd as in the hypothesis of the theorem. Remark: Note that in the a = 2 1 case the t 2 (16a 2 16a + 2)t + 1 term in the discriminant contributes only 1 to the degree of the conductor and hence the degree of the part of the conductor prime to 0 and is even, so that neither the theorem of Ulmer nor Theorem 5 predict rank in this case.

35 34 9. The m > 2 Case 9.1. General Discussion In the above discussion, we have focused on the choice f(x) = x(x 1), with g = f. As x a discussed earlier, we may also wish to consider choosing g = 1/f. However, in this particular case this is not profitable because the elliptic curve E with the equation f(x) f(y) = td is isomorphic to the elliptic curve E with equation f(x)f(y) = t d for this choice of f. This is because f( a) = 1, and hence one may give an isomorphism x f(x) φ : E E via the map (x, y) (x, a ). We omit further consideration of this case. y We now relax our assumption on f, and instead allow f to be any degree m rational function on P 1 with distinct zeroes and distinct poles. In this case, one sees from Theorem 4 that the genus of the curves f(x) f(y) = td and f(x)f(y) = t d are each (m 1) 2. However, when m > 2 these curves are not, in general, isomorphic. Recall that in the case of the f = g construction, the key positive term in the rank formula is dim(end 0 (Jac(C d )) µ d ), where Cd is the smooth projective curve with model z d = f(x). On the other hand, in the g = 1/f case, the key positive term is dim(end 0 (Jac(C d )) anti- µ d ). Note that End 0 (Jac(C d )) µ d and End 0 (Jac(C d )) anti µ d are linearly disjoint subspaces of End 0 (Jac(C d )). Note that by Proposition 1 the genus of C d is (m 1)(d 1). Therefore, the

36 35 dimension of the Jacobian of C d is (m 1)(d 1). We will now restrict to the case that our ground field k is finite, say F q. Then by Proposition 2 and reasoning similar to that in the proof of Theorem 5 above, the dimension of End 0 k (Jac(C d)) µ d is at least as large as 2(m 1)(d 1). We then make the following observation: Proposition 5. If d q n + 1 for some n, then dim(end 0 (Jac(C d )) µ d ) = dim(end 0 (Jac(C d )) anti- µ d ). (9.1) In particular, if d (q n + 1), then dim(end 0 (Jac(C d )) anti- µ d ) 2(m 1)(d 1). Proof. Define π to be the Frobenius morphism on C d. In coordinates, π : (x, z) (x q, z q ). Supposing, as in the proposition, that d q n + 1, we see that if ζ µ d then ζ π n : (x, z) (x qn, ζz qn ). Similarly we see that π n ζ : (x, z) (x qn, ζ qn z qn ) = (x qn, ζ 1 z qn ) = ζ 1 π n (x, z). Note that this requires d to divide q n + 1 so that we have ζ qn = ζ 1. This shows that π n anti-commutes with µ d. Thus, we see that End 0 (Jac(C d )) anti- µ d = π n End 0 (Jac(C d )) µ d, and hence the two spaces have the same dimension, and thus the proposition is proved.

37 36 Remark: Note that the hypothesis on d in this proposition is the same as the hypothesis on d in the theorem of Ulmer, Theorem 1. This proposition is insufficient to prove that there are ever large ranks in the m > 2 and f = 1/g case. However, computational evidence suggests that if m > 2, then for almost all 1 choices of f, the negative term in the rank formula, c 1 (d), is identically zero in this case. For a specific family of examples of this phenomenon, see Section 9.2. When this occurs, the previous proposition implies that there is linear growth of rank for those d which divide q n + 1 for some n. Remark: Note that in the f = g case, c 1 (d) d because in this case the t = 1 fiber is always reducible. Thus if we fix an f where c 1 (d) = 0 for all d in the g = 1/f case, the rank of the d th abelian variety constructed by taking g = 1/f will be at least d greater than the rank of the d th abelian variety constructed by taking g = f if d q n + 1 for some n. In order to demonstrate the principles here more clearly, we will give a family of examples in the m = 3 case in the next section. 1 By almost all here, we mean the set of f with this property is Zariski dense in the space of possible choices for f. This is unproven, but is born out in the example provided in the next section.

38 A family of cases with m = 3 For this section, we take f to be the degree 3 rational function given by 2 f = x(x 1)(x + 1) (x a)(x b), with a, b 0, ±1 and g to be 1/f. We will denote the smooth proper curve over k(t) with affine model f(x)f(y) = t d by X d. Note that by Theorem 4 the Jacobian of X d is a four-dimensional abelian variety for all d. It is easy to see that, in this case, c 2 (d) = 8. By reasoning similar to that in Chapter 7, we can compute the term c 1 (d) by examining the reducibility of the polynomial F = (x 3 x)(y 3 y) t(x a)(x b)(y a)(y b). (9.2) In particular, for fixed a and b, c 1 (d) is identically equal to 0 if F is irreducible for all t 0. The following proposition can be verified by hand, using the same idea as in the proof of Proposition 4, but the calculations are somewhat more involved. Proposition 6. Suppose the characteristic of k is either 0 or greater than or equal to 5, that a, b, t k, that a b, that a, b 0, 1, 1 and that t 0. Then the two 2 This particular form was chosen as it makes the associated computations tractable. The results should be similar if one took an arbitrary degree three rational function with distinct zeroes and poles.

39 38 variable polynomial F = (x 3 x)(y 3 y) t(x a)(x b)(y a)(y b) is irreducible in k[x, y] except in the following cases: 1. If a + b = 0 and t = ± 1 a 2. If a 2b + 1 = 0 and t = 2 1 b 3. If b 2a + 1 = 0 and t = 2 1 a 4. If a 2b 1 = 0 and t = 2 b+1 5. If b 2a 1 = 0 and t = 2 a+1 The figure below shows the set of pairs (a, b) in R 2 for which the polynomial F factors for some non-zero value of t. The picture over other fields is similar. Figure 9.1. Pairs (a, b) for which the polynomial F factors for a non-zero value of t

40 39 Remark: In this case one sees that the set of pairs (a, b) which lead to a reducible fiber has codimension one. We expect this statement to generalize to larger m, but no proof of this is known. Combining the ideas above, we get the following theorem. Theorem 7. Let k = F q, and suppose d q n +1 for some n. Let a and b be elements of F q \{0, 1, 1} such that a b, a + b 0, a 2b + 1 0, b 2a + 1 0, a 2b 1 0, and b 2a 1 0. Let f(x) = x3 x, and let X (x a)(x b) d be the Jacobian of the smooth proper curve over k(t) of which f(x)f(y) = t d is an affine model. Then the rank of the Jacobian of X d is greater than or equal to 4d + 4. Remark: Note that for d q n + 1 if one takes g = f the rank of the constructed abelian variety is strictly smaller rank than the abelian variety constructed in the theorem above. Remark: A similar theorem should be true in the f = g case, but with the conclusion being that the rank of the Jacobian is at least 3d + 4. However, the computations involved in finding an analogue to Proposition 6 are less tractable than those in the g = 1/f case.

41 Future Directions The results presented here lead to many interesting unanswered questions. We outline a few here. Question 1: For a Q, what values do the ranks of the elliptic curves E d : x(x 1)(y a) t d y(y 1)(x a) attain over Q(t) as d varies across the positive integers? We know from Theorem 5 that the reduction of the curve E d modulo p has rank at least d for all but finitely many primes p. Question 2: It was shown above that if k = F q, f is a rational function on P 1 k, and C d is the smooth proper curve with equation z d = f(x), then when d is relatively prime to q we have dim(end 0 (Jac(C k d)) µ d ) 2g(Cd ). It would be fascinating to know when this bound is tight, and especially to know when it is exceeded. Question 3: The non-torsion part of the Mordell-Weil group of an elliptic curve over F q (t) attains the structure of a lattice. What is the structure of the lattices constructed in Theorem 5?

42 41 References [Ber08] L. Berger. Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields. Journal of Number Theory, 128(12): , [LE06] Q. Liu and R. Erné. Algebraic geometry and arithmetic curves. Oxford University Press, USA, [Mil] J.S. Milne. Abelian varieties. Arithmetic, Geometry, pages [Sil99] [ST67] [Tat68] A. Silverberg. Open questions in arithmetic algebraic geometry. Arithmetic algebraic geometry (Park City, UT, 1999), , IAS/Park City Math. Ser., 9, AMS, pages , I. Shafarevich and J. Tate. The rank of elliptic curves. In Soviet Mathematics Doklady, volume 8, pages , J. Tate. Classes d isogenie des varietes abeliennes sur un corps fini. Seminaire Bourbaki, 352:95 110, [Ulm07] D. Ulmer. L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields. Inventiones mathematicae, 167(2): , [Ulm10] D. Ulmer. On Mordell-Weil groups of Jacobians over function fields