Selmer Ranks of Quadratic Twists of Elliptic Curves

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1 UNIVERSITY OF CALIFORNIA, IRVINE Selmer Ranks of Quadratic Twists of Elliptic Curves DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics by Zev Klagsbrun Dissertation Committee: Professor Karl Rubin, Chair Professor Alice Silverberg Professor Daqing Wan 2011

2 c 2011 Zev Klagsbrun

3 Dedication To my father, for asking me what I would do if I knew I could not fail. ii

4 Table of Contents Acknowledgements Curriculum Vitae Abstract of the Dissertation iv v vi 1 Introduction and Background Introduction Controlling Selmer Rank Controlling Mordell-Weil Rank Layout Background Selmer Groups The Methods of Rubin and Mazur The Case Where E(K)[2] = The Results of Mazur and Rubin Raising the 2-Selmer Rank The Case Where E(K) Z/2Z How the case where E(K)[2] Z/2Z differs from that of E(K)[2] = The φ-selmer Group The Selmer Group arising from a 2-isogeny Duality Between Hφ 1(E/K) and H1 φ (E /K) Relationship Between Hφ 1(K v, C) and Hφ 1(K v, C F ) Proof of Theorem Characterization of curves with E(K) Z/2Z Twisting to Decrease Selmer Rank Twisting to Increase Selmer Rank Minimal Tamagawa Ratios Curves which acquire a cyclic 4-isogeny over K(E[2]) Local conditions of curves which obtain a cyclic 4-isogeny over K(E[2]) Upper Bounds on min{d 2 (E F /K) : F/K quadratic} A special family of curves Bibliography 45 iii

5 Acknowledgments I would like to express my deepest gratitude to all of those who made this thesis possible. First and foremost, I would like to thank Karl Rubin for serving as my thesis advisor and for acting as my mentor throughout my graduate career. His knowledge and valuable insights have been critical to my development as a mathematician. I am also grateful to him for thoroughly reading my thesis, pointing out errors, and suggesting various important improvements. Without his assistance this work would not have been possible. I greatly benefitted from the stimulating academic environment at UC Irvine. I would like to especially single out for thanks the members of the number theory group, Professors Alice Silverberg, Karl Rubin, and Daqing Wan, as well as the graduate students, particularly Nick Alexander and Sunil Chetty, for the time, advice, and knowledge they have shared with me. I am also indebted to a number of others in the mathematical community: Barry Mazur for his initial suggestion that I consider the existence of a cyclic 4-isogeny as well as for additional guidance he provided along the way. Ken Kramer for guiding me through his earlier work referenced in this thesis and for a series of useful discussions. Bjorn Poonen for showing me how to arrive at the explicit formula for the quadratic form appearing in Remark The creators and maintainers of the Magma computations algebra system, which was used for computations underlying most of the results in this paper. I would also like to thank the mathematics and computer science faculty at CUNY Queens College, especially Bojana Obrenic, Alex Ryba, Kent Boklan, and Ken Kramer for sparking my interest in number theory and for urging me to pursue higher mathematics. Additionally I would like express my appreciation to Karl Rubin and the National Science Foundation for providing me with financial support under NSF grants DMS and DMS Last but not least, I would like to extend my thanks to my friends and family who have provided me with the emotional support that has sustained me over the last four years. iv

6 Curriculum Vitae Zev Klagsbrun B.A. in Mathematics and Computer Science, CUNY Honors College, 2006 M.A. in Mathematics, CUNY Queens College, 2007 M.S. in Mathematics, University of California, Irvine, 2009 Ph.D. in Mathematics, University of California, Irvine, 2011 v

7 Abstract of the Dissertation Selmer Ranks of Quadratic Twists of Elliptic Curves By Zev Klagsbrun Doctor of Philosophy in Mathematics University of California, Irvine, 2011 Professor Karl Rubin, Chair This thesis investigates the 2-Selmer rank in quadratic-twist families of elliptic curves defined over number fields, presenting new results in this area for curves having E(K)[2] = 0 and E(K)[2] Z/2Z. In particular, we show that all elliptic curves with E(K)[2] = 0 have twists with 2-Selmer rank equal to r for every r 0 subject to the condition of constant 2-Selmer parity, and give a lower bound on the number of such twists as a function of the conductor. We do the same for all elliptic curves with E(K)[2] Z/2Z that do not have a cyclic 4-isogeny defined over K(E[2]). Lastly, we present an infinite family of elliptic curves with coefficients in Q such that if 2 splits completely in K, then the 2-Selmer rank of E F /K is bounded below by r 2 (K) for every quadratic F/K. vi

8 Chapter 1 Introduction and Background 1.1 Introduction Questions relating to Mordell-Weil ranks of elliptic curves over number fields are among the most important unsolved problems in number theory, among them the questions of whether the ranks are unbounded, and more generally, the overall distribution of ranks. These questions remain unanswered in part due to the difficulty of computing the rank of an elliptic curve, for which an efficient method is not yet known. This often leads to looking at the 2-Selmer rank of an elliptic curve, which is computable and serves and an upper bound for the rank of the curve, instead. We are able to ask similar questions about 2-Selmer ranks of elliptic curves as we are about the ranks of elliptic curves. For instance, it has been long known that the 2-Selmer ranks of elliptic curves over number fields are unbounded [2]. More recently, Bhargava and Shankar proved that taken over all elliptic curves, the average size of the 2-Selmer group is 3 [1]. We can ask the same questions about 2-Selmer ranks within families of quadratic twists Controlling Selmer Rank Let E be an elliptic curve defined over a number field K and let Sel 2 (E/K) be its 2-Selmer group (see section for its definition). We define the 2-Selmer rank of E/K, denoted d 2 (E/K) by d 2 (E/K) = dim F2 Sel 2 (E/K) dim F2 E(K)[2]. There are some restrictions on 2-Selmer ranks within a quadratic twist family. Certain curves exhibit a phenomenon called constant 2-Selmer parity where d 2 (E/K) d 2 (E F /K)(mod 2) for all quadratic F/K. Mazur and Rubin showed in [10] that if E/K has constant 2-Selmer parity, then K is totally imaginary and E has additive reduction at all bad primes. Work of Dokchitser and Dokchitser [4] combined with the BSD and Tate-Shafaravich conjectures implies that E/K has constant 2-Selmer parity if and only if K is totally imaginary and E acquires everywhere good reduction over an abelian extension of K. 1

9 Additionally, Section exhibits the existence of a previously unknown obstruction for certain curves having cyclic 4-isogenies defined over K(E[2]), displaying a non-trival lower bound for the 2-Selmer rank within the quadratic twist families of certain elliptic curves. However, outside of these two obstructions, the expectation is that the number of curves in a twist family with 2-Selmer rank equal to r grows linearly with the number of quadratic extensions F/K for every r 0. We make this precise with the following definition and conjecture. The conjecture appears with a slightly different formulation in [10]. Definition For X R +, define a set S(X) = {Quadratic F/K : N K/Q f(f/k) < X} where f(f/k) is the finite part of the conductor of F/K. For each r Z 0 define a quantity N r (E, X) by N r (E, X) = {F/K S(X) : d 2 (E F /K) = r}. Conjecture 1.1. Let E be an elliptic curve defined over a number field K that does not have a cyclic 4-isogeny defined over K(E[2]). (i) If r d 2 (E/K) (mod 2), then N r (E, X) S(X). (ii) If K has a real embedding or if E does not acquire everywhere good reduction over an abelian extension of K, then N r (E, X) S(X) for all r Z 0. Heath-Brown [6] showed that for K = Q and E given by y 2 = x 3 x, lim Nr(E,X) S(x) α r for an explicit α r for every r Z 0. Kane [7], building on work of Swinnerton-Dyer [17], recently extended that same result to all elliptic curves over Q with E(Q)[2] Z/2Z Z/2Z that do not have a cyclic 4-isogeny defined over Q. The picture for curves over general number fields and for curves that do have full two-torsion in the base field is considerably more murky. We present the following results in the direction of this conjecture for such cases. Theorem 1.2 (Mazur, Rubin 2008, Klagsbrun 2010). Let E be an elliptic curve defined over a number field K with E(K)[2] = 0. Then N r (E, X) X (log X) 2 3 have constant 2-Selmer parity, then N r (E, X) for all non-negative r d 2 (E/K) (mod 2). If E does not X (log X) 2 3 for all r Z 0. A version of this theorem for a special class of curves with no two-torsion was proved by Mazur and Rubin in [10]. Theorem 1.3. Let E be an elliptic curve defined over a number field K with E(K)[2] Z/2Z that does not possess a cyclic 4-isogeny defined over K(E[2]). Then N r (E, X) X log X (mod 2). If E does not have constant 2-Selmer parity, then N r (E, X) for all non-negative r d 2(E/K) X log X for all r Z 0. Additionally, if E does not possess a cyclic 4-isogeny defined over K we are able to get the following three results. 2

10 Theorem 1.4. Let E be an elliptic curve defined over a number field K with E(K)[2] Z/2Z that does not possess a cyclic 4-isogeny defined over K. If E has a twist E F such that d 2 (E F ) = r, then N r (E, X) X log X. Theorem 1.5. Let E be an elliptic curve defined over a number field K with E(K)[2] Z/2Z that does not possess a cyclic 4-isogeny defined over K but acquires one over K(E[2]). Then N r (E, X) X log X for all r r 2 with r d 2 (E/K) (mod 2), where r 2 is the number of conjugate pairs of complex embeddings of K. If E does not have constant 2-Selmer parity, then N r (E, X) X log X for all r r 2. Theorem 1.6. Let E be an elliptic curve defined over a number field K with E(K)[2] Z/2Z that does not possess a cyclic 4-isogeny defined over K but acquires one over K(E[2]). (i) Then either N r (E, X) X log X for all r Z 0 with r d 2 (E/K) (mod 2) or N r (E, X) X log X all r Z 0 with r d 2 (E/K) (mod 2), where E is the curve 2-isogenous to E. (ii) If E does not have constant 2-Selmer parity, then we additionally have that either N r (E, X) for all r Z 0 with r d 2 (E/K) (mod 2) or N r (E, X) (mod 2). X log X for X log X for all r Z 0 with r d 2 (E/K) (iii) If either K has a real place or E has a place of multiplicative reduction, then N r (E, X) r Z 0 or N r (E, X) X log X for all r Z 0. X log X for all We also present some negative results, showing that the 2-Selmer ranks within the quadratic twist families of certain elliptic curves have a non-trival lower bound. Theorem 1.7. For any number field K, there exist infinitely many elliptic curves E defined over Q and nonisomorphic over K that do not have constant 2-Selmer parity such that d 2 (E F /K) r 2 for every quadratic F/K. When K has a complex embedding, Theorem 1.7 provides a counterexample to Conjecture 1.3 of [10] Controlling Mordell-Weil Rank The 2-Selmer rank of E is an upper bound for the Mordell-Weil Rank of E. We can therefore prove the following corollary based on the results of the previous section. Corollary 1.8. Let E be an elliptic curve defined over a number field K with E(K)[2] Z/2Z that does not possess a cyclic 4-isogeny defined over K. If either d 2 (E/K) 0 (mod 2) or E does not have constant 2-Selmer parity, then the number of twists E F X log X. of E having Mordell-Weil rank 0 grows at least as fast as In order to say something about E having twists of rank one, we need to rely on the following well-known conjecture that is a consequence of the Tate-Shafarevich conjecture. 3

11 Conjecture 1.9 (Conjecture XT 2 (K)). For every elliptic curve E defined over K, dim F2 X(E/K)[2] is even. The 2-Selmer group sits inside the exact sequence 0 E(K)/2E(K) Sel 2 (E/K) X(E/K)[2] 0. As dim F2 E(K)/2E(K) = rank E/K + dim F2 E(K)[2], if Conjecture XT 2 (K) holds and d 2 (E/K) = 1, then the Mordell-Weil rank of E will be 1. We can therefore state the following: Corollary Let E be an elliptic curve defined over a number field K with E(K)[2] Z/2Z that does not possess a cyclic 4-isogeny defined over K. Assuming conjecture XT 2 (K) holds and either d 2 (E/K) 1 (mod 2) or E does not have constant 2-Selmer parity, then the number of twists E F grows at least as fast as X log X. of E having rank one Other results similar to Corollaries 1.8 and 1.10 are due to Ono and Skinner in [12] when K = Q and E(Q)[2] = 0 and Skorobogotav and Swinnerton-Dyer in [16] when E(K)[2] Z/2Z Z/2Z Layout An introduction to Selmer groups and their behavior under quadratic twist is given in section and section presents the methods of Mazur and Rubin which form the basis for much of this work. Chapter 2 concerns the case where E(K)[2] = 0. Section 2.1 presents the contribution of Mazur and Rubin in the direction of Theorem 1.2 and sketches the proofs of some of these results. Section 2.2 presents some new results of Poonen and Rains [13] and shows how these can be used to complete the proof of Theorem 1.2. Chapter 3 considers the case where E(K)[2] Z/2Z. It begins with a brief discussion of how this case differs from the case where E(K)[2] = 0 in section 3.1. Section 3.2 introduces the φ-selmer group, a Selmer group connected to the 2-isogeny on E, that will be crucial to controlling the 2-Selmer rank under twisting. Section 3.3 provides an example of how the φ-selmer group can be used to control the 2-Selmer rank of a twist of E by using it to prove Theorem 1.4 and provides a important characterization of curves with E(K)[2] Z/2Z in the process. In sections 3.5 and 3.6 we extend the techniques presented in the earlier sections to produce twists with smaller and larger 2-Selmer ranks than a given curve. Section 3.7 discusses applicability of these techniques to a given elliptic curve and presents the proof of Theorem 1.3. Section 3.8 considers a special class of curves, those which do not have a cylic 4-isogeny defined over K but acquire one over K(E[2]). We discuss some special properties of these curves and use them to prove Theorem 1.6 and Corollaries 1.8 and 1.10 in section These special properties are further exploited in section to prove Theorem 1.5. Section proves Theorem 1.7 by exhibiting a special family of elliptic curves with a remarkable property. 4

12 1.2 Background Selmer Groups We begin by defining Selmer structures and the Selmer groups associated to them. Let K be a number field and A a G K -module. For each finite place v of K, define the unramified local cohomology group H 1 u(k ur v, A) H 1 (K v, A) by H 1 u(k v, A) = ker where K ur v is the maximal unramified extension of K v. ( ) H 1 (K v, A) res H 1 (Kv ur, A), Definition A Selmer structure F for A is a collection of local cohomology subgroups H 1 F (K v, A) H 1 (K v, A) for each place v of K such that H 1 F (K v, A) is equal to the unramified local subgroup H 1 u(k v, A) for all but finitely many places v. We define the Selmer group associated to the Selmer structure F as ( Ker resv : H 1 (K, A) ) H 1 (K v, A)/HF(K 1 v, A) and denote this group by HF 1 (K, A). v of K One of the most important examples of a Selmer group is the n-selmer group of an elliptic curve. If E is an elliptic curve, then E(K)/nE(K) maps into H 1 (K, E[n]) via the Kummer map coming from the long exact sequence of cohomology groups 0 E(K)[n] E(K) E(K) H 1 (K, E[n]) H 1 (K, E).... We can study the rank of E(K) by studying the image of E(K)/nE(K) in H 1 (K, E[n]). One way to study this image is to define a finite subgroup of H 1 (K, E[n]) that contains it. For each place v of K, the diagram below commutes, where δ is the Kummer map. E(K)/nE(K) δ H 1 (K, E[n]) E(K v )/ne(k v ) δ Res v H 1 (K v, E[n]) We would like to examine the subgroup of H 1 (K, E[n]) consisting of elements whose restrictions come from local points on E(K v ) for each place v of K. For each place v of K we define a distinguished local subgroup Hf 1(K v, E[n]) H 1 (K v, E[n]) by Image ( δ : E(K v )/ne(k v ) H 1 (K v, E[n]) ). By Lemma 4.1 in [3], Hf 1(K v, E[n]) = Hu(K 1 v, E[n]) for all v n where E has good reduction and the collection of Hf 1(K v, E[n]) therefore forms a Selmer structure on E[n]. The Selmer group attached to this Selmer structure is referred to as the n-selmer group of E and denoted Sel n (E/K). The n-selmer group is a finitely generated Z/nZ module and it sits inside the exact sequence 0 E(K)/nE(K) Sel n (E/K) X(E/K)[n] 0 5

13 where X(E/K) is the Tate-Shafaravich group of E. The Tate-Shafaravich conjecture states that X(E/K) is always finite. The finiteness of X(E/K) would imply that the order of X(E/K)[n] is a square by the Cassels pairing. We will restrict our attention to the case where n = 2. Definition We define the 2-Selmer rank of E, denoted d 2 (E/K) by d 2 (E/K) = dim F2 Sel 2 (E/K) dim F2 E(K)[2]. The following lemma provides us with a way to study Hf 1(K v, E[2]). Lemma (i) If v 2, then dim F2 Hf 1(K v, E[2]) = dim F2 E(K v )[2] (ii) If v 2 and E has good reduction at v then Hf 1 (K v, E[2]) E[2]/(F rob v 1)E[2] with the isomorphism given by evaluation of cocycles in Hf 1(K v, E[2]) at the Frobenius automorphism F rob v. Proof. This is Lemma 2.2 in [10]. We would like to examine the behavior of Sel 2 (E/K) under the action of twisting by a quadratic extension. Definition Let E be given by E : y 2 = x 3 + Ax 2 + Bx + C and F/K be a quadratic extension given by F = K( d). The quadratic twist of E by F denoted E F is the elliptic curve given by the model y 2 = x 3 + dax 2 + d 2 Bx + d 3 C. There is an isomorphism E E F given by (x, y) (dx, d 3/2 y) defined over F. Restricted to E[2], this map gives a canonical G K isomorphism E[2] E F [2]. This allows us to view Hf 1(K v, E F [2]) as sitting inside H 1 (K v, E[2]). Remark It is rarely the case that E[n] and E F [n] are G K -isomorphic for n > 2. We will study Sel 2 (E F /K) by considering the relationship between Hf 1(K v, E[2]) and Hf 1(K v, E F [2]). This relationship is addressed by the following lemma: Given a place w of F above a place v of K, we get a norm map E(F w ) E(K v ), the image of which we denote by E N (K v ). Lemma Viewing Hf 1(K v, E F [2]) as sitting inside H 1 (K v, E[2]), we have Hf 1 (K v, E F [2]) Hf 1 (K v, E[2]) = E N (K v )/2E(K v ) Proof. This is Proposition 7 in [8] and Proposition 5.2 in [9]. The proof in [9] works even at places above 2. 6

14 This equality gives rise to the following lemma: Lemma Let E be an elliptic curve defined over K, v a place of K, and F/K be a quadratic extension. Then (a) Hf 1(K v, E[2]) = Hf 1(K v, E F [2]) if any of the following hold: (i) v 2 and E(K v )[2] = 0 (ii) E has multiplicative reduction at v with ord v ( E ) odd and v unramified in F/K. (iii) v is real with ( E ) v < 0 (iv) v splits in F/K (v) v is a prime where E has good reduction and v is unramified in F/K (b) Hf 1(K v, E[2]) Hf 1(K v, E F [2]) = 0 if v 2, E has good reduction at v, and v is ramified in F/K. Proof. Part (a) is Lemma 2.10 in [10] and part (b) is Lemma 2.11 in [10]. Lemma Suppose E has good reduction at a prime v away from 2 and F/K is a quadratic extension ramified at v. Then E F (K v ) contains no points of order 4. It follows that Hf 1(K v, E F [2]) is the image of E F (K v )[2] under the Kummer map. Proof. Let p be the rational prime sitting below v and k v be the residue field of K v. The sequence 0 E F 1 (K v ) E F 0 (K v ) ẼF ns(k v ) 0 is exact (Proposition VII.2.1 in [14]), where Ẽns(k v ) is the group of non-singular points on the reduction of E to the residue field k v of K v, E 0 (K v ) is the group of points on E(K v ) that reduce into Ẽns(k v ), and E 1 (K v ) is the group of points that reduce to the identity in Ẽns(k v ). Since E F has additive reduction at v, Ens(k F v ) k v + which has order prime to 2 since v 2. Further, by Proposition VII.3.1 in [14], E1 F (K v ) has no torsion of order prime to p, so we get that E0 F (K v )[2 ] = 0. Next, we consider the quotient E F (K v )/E0 F (K v ). Since E had good reduction at v, we get that 12 ord v ( E ). The extension F/K is given by K( d) for some d K and since F/K is ramified at v and v 2, we know that ord v (d) is odd. If we take a model y 2 = x 3 + Ax + B for E and let y 2 = x 3 + d 2 Ax + d 3 B be our model for E F, then E F = d 6 E, giving that ord v ( E F ) 6 (mod 12). This allows us to conclude that the order of v in the discriminant of the minimal model of E F is 6. Since E F has additive reduction at v, this tells us that E F (K v )/E0 F (K v ) Z/2Z Z/2Z, and in particular that E F (K v ) has no points of order 4 (See Table 4.1 in section IV.9 of [15]). Lastly, by Proposition 6.3 in [14], E F (K v ) can be written as the direct sum of E F (K v )[2 ] and something that is 2-divisible, thereby giving the result. 7

15 1.2.2 The Methods of Rubin and Mazur Let A be a finitely generated G K -module ramified at a finite set of primes. Define the Cartier dual of A of A as A = Hom(T, µ p ). For a place v of K we have a perfect Tate pairing H 1 (K v, A) H 1 (K v, A ) H 1 (K v, µ p ) Q p /Z p arising from the cup-product pairing. If F is a Selmer structure on A, then we we define a dual Selmer structure F on A by setting HF 1 (K v, A ) = HF 1 (K v, A) for each place v of K where is taken with respect to the Tate pairing. If F 1 and F 2 are Selmer structures, then we say F 1 F 2 if HF 1 1 (K v, A) HF 1 2 (K v, A) for each place v of K. Theorem (Theorem in [9]). Suppose F 1 and F 2 are Selmer structures with F 1 F 2, then 1. The sequences 0 H 1 F 1 (K, A) H 1 F 2 (K, A) P resv v H 1 F 2 (K v, A)/H 1 F 1 (K v, A) and 0 H 1 F 2 (K, A ) H 1 F 1 (K, A ) P resv v H 1 F 1 (K v, A )/H 1 F 2 (K v, A ) are exact where the sum is taken over all places v such that HF 1 1 (K v, A) HF 1 2 (K v, A). 2. The images of the right hand maps are orthogonal complements with respect to the sum of the local Tate pairings. Proof. The first part follows immediately from the definition of Selmer structures. The second is part of Poitou-Tate global duality; see Theorem I.4.10 in [11] for example. We will apply this theorem in a number of contexts throughout this paper, the first of which is to prove an important proposition due to Mazur and Rubin. Let E be an elliptic curve defined over a number field K and T a finite set of places of K. We define a localization map loc T : H 1 (K, E[2]) H 1 (K v, E[2]) v T as the sum of the restriction maps over the places v in T. We define the strict Selmer group S T H 1 (K, E[2]) as Ker ( loc T : Sel 2 (E/K) v T H 1 (K v, E[2]) ). Observe that S T is the Selmer group associated to the Selmer structure with trivial distinguished local subgroups for all places v in T and distinguished local subgroups Hf 1(K v, E[2]) for all places v T. Noting that E[2] = E[2] under the Weil pairing and that Hf 1(K v, E[2]) is self-dual under the local Tate-pairing, we get that the Selmer group associated to the dual Selmer structure for Sel 2 (E/K) is equal to Sel 2 (E/K). This gives the following Lemma. 8

16 Lemma (Lemma 3.2 in [10]). dim F2 S T dim F 2 S T = v T dim F 2 H 1 f (K v, E[2]), where S T Selmer group associated to the dual Selmer structure for S T. is the Proof. We apply part (ii) of Theorem to S T S T giving us that 2(dim F2 S T dim F2 S T ) = v T dim F2 H 1 (K v, E[2]). As Hf 1(K v, E[2]) is self-dual in H 1 (K v, E[2]), we therefore get that dim F2 H 1 (K v, E[2]) = 2 dim F2 Hf 1 (K v, E[2]) v T v T and the result follows. The following theorem of Kramer provides an important relationship between the parities of d 2 (E/K) and d 2 (E F /K). Theorem (Kramer). We have d 2 (E/K) d 2 (E F /K) + dim F2 E(K v )/E N (K v ). v of K Proof. This is Theorem 2.7 in [10]; also see Remark 2.8 there as well. Proposition (Proposition 3.3 in [10]). Suppose E is an elliptic curve defined over a number field K. Suppose F/K is a quadratic extension such that all of following places split in F/K. places where E has additive reduction places above 2 real places v with ( E ) v > 0 places v where E has multiplicative reduction and ord v ( E ) is even. where is the discriminant of some model of E. Further assume that F/K is unramified at all places v where E has multiplicative reduction at v and ord v ( E ) is odd. Let T be the set of (finite) primes p of K such that F/K is ramified at p and E(K p )[2] 0. Let V T = loc T (Sel 2 (E/K).Then d 2 (E F /K) = d 2 (E/K) dim F2 V T + d for some d satisfying and where V T = loc T (Sel 2 (E/K)). 0 d dim F2 Hf 1 (K p, E[2])/V T p T ( d dim F2 Hf 1 ) (K p, E[2])/V T (mod 2) p T 9

17 Proof. Let VT F = loc T (Sel 2 (E F /K)). Lemma gives us that Hf 1(K v, E F [2]) = Hf 1(K v, E[2]) for all v T and therefore S T Sel 2 (E F /K). This gives us that the sequences 0 S T Sel 2 (E/K) V T 0 and 0 S T Sel 2 (E F /K) V F T 0 are exact. We therefore get that We will let d = dim F2 V F T d 2 (E F /K) = d 2 (E/K) dim F2 V T + dim F2 V F T. and show that it satisfies the conditions above. Since H 1 f (K v, E[2]) H 1 (K v, E F [2]) = 0 for all v T by part (ii) of Lemma 1.2.7, we have dim F2 V T +dim F2 V F T = dim F2 Sel 2 (E/K)/S T +dim F2 Sel 2 (E F /K)/S T dim F2 (S T /S T ) = v T dim F2 H 1 f (K v, E[2]). The last equality follows from Lemma This gives us that dim F2 V F T v T dim F2 Hf 1 (K v, E[2]) dim F2 V T = dim F2 Hf 1 (K p, E[2])/V T, p T giving us the first condition above. To get the parity result, we will apply Kramer s Theorem. Observe that Lemma and part (i) of Lemma tell us that dim F2 E(K v )/E N (K v ) = 0 for all v T and that dim F2 E(K v )/E N (K v ) = dim F2 Hf 1(K v, E[2]) for all v T. We therefore have that dim F2 V T + dim F2 VT F dim F2 Hf 1 (K v, E[2]) (mod 2) v T so dim F2 V F T ( dim F2 Hf 1 ) (K p, E[2])/V T p T (mod 2) 10

18 Chapter 2 The Case Where E(K)[2] = 0 The goal of this chapter is to outline a proof of the Theorem 1.2, offering detailed proofs of the parts not present in [10]. We will rely on (but not prove) Theorem 1.4 in [10] which states that if E has at least one twist E F with d 2 (E F /K) = r then N r (E, X) 2-Selmer rank equal to each r. X (log X) 2 3. The task is then reduced to showing that E has one twist with 2.1 The Results of Mazur and Rubin Mazur and Rubin proved the following result in the in the direction of Theorem 1.2. Theorem (Theorems 1.4, 1.7, 1.5, and 1.6 in [10]). Let E be an elliptic curve defined over a number field K with E(K)[2] = 0. (i) If E has a twist E F with d 2 (E / K) = r, then N r (E, X) X N r (E, X) X (log X) 1 3 if Gal(K(E[2])/K) Z/3Z (log X) 2 3 if Gal(K(E[2])/K) S 3 and (ii) N r (E, X) X (log X) 2 3 for all r d 2 (E/K) with r d 2 (E/K) (mod 2). (iii) If E has a place of multiplicative reduction or K has a real place, then N r (E, X) r d 2 (E/K). X (log X) 2 3 for all (iv) If Gal(K(E[2])/K) S 3 and K has a place v 0 such that either E has multiplicative reduction at v 0 with ord v0 ( E ) odd or v 0 is real with ( E ) v0 < 0, then N r (E, X) X (log X) 1 3 for all r 0. Proof. In order, these are Theorems 1.4, 1.7, 1.5, and 1.6 in [10]. For the purposes of proving Theorem 1.2, it suffices to utilize parts (i) and (ii) of Theorem Proving part (ii) is an easy consequence of the following proposition, the proof of which we will sketch. 11

19 Proposition (Proposition 5.2 in [10]). Let E be an elliptic curve defined over a number field K with E(K)[2] = 0 and d 2 (E/K) 2. Then E has a twist E F such that d 2 (E F /K) = d 2 (E/K) 2. The proof of Proposition will rely on the following Lemma. Lemma (Lemma 3.6 in [10]). Suppose E(K)[2] = 0 and c 1, c 2 are cocycles representing distinct nonzero elements of H 1 (K, E[2]). Then there is some γ G K such that c 1 (γ) and c 2 (γ) are an F 2 basis for E[2] and γ MK ab = 1, where M = K(E[2]). We include the proof of this lemma taken from [10] in order to later show where it breaks down when E(K)[2] 0. Proof. Let Γ = Gal(M/K). We either have Γ S 3 or Γ Z/3Z and in either case E[2] is an irreducible Γ-module and H 1 (Γ, E[2]) = 0. Since H 1 (Γ, E[2]) = 0, the inflation restriction exact sequence gives us that the restriction map H 1 (K, E[2]) Hom(G M, E[2]) Γ is injective. Let c 1 and c 2 be the distinct non-zero elements of Hom(G M, E[2]) Γ obtained by restricting c 1 and c 2 to G M. For i = 1, 2, let N i be the fixed field of ker( c i ). As c i : Gal(N i /M) E[2] is non-zero and Γ-equivariant, it must be an isomorphism. We can therefore find τ Gal(N 1 N 2 /M) such that c 1 (τ) and c 2 (τ) are distinct and non-zero and therefore generate E[2]. Since Γ acts trivially on Gal(MK ab N 1 N 2 /M) and Gal(N 1 N 2 /M) has no non-zero quotients on which Γ acts trivially, we get that MK ab N 1 N 2 = M. We can therefore find γ G M such that γ MK ab = 1 and γ N1N 2 = τ. We present the proof in [10] in order to give some context to the other proofs that appear in this paper. Proof of Proposition Let M = K(E[2]) and E the discriminant of some model of E. Since d 2 (E/K) 2, we can find cocycles c 1 and c 2 representing F 2 independent elements of Sel 2 (E/K) and by Lemma we can find γ G K such that c 1 (γ) and c 2 (γ) form an F 2 basis for E[2] and γ MK(8 E ) = 1 where K(8 E ) is the ray-class field modulo 8 E. Let N be a finite Galois extension of MK(8 E ) such that the restriction of Sel 2 (E/K) to N is trivial. Choose a prime p of K away from 2 E such that the F rob p in Gal(N/K) is the class of γ. Then p has a totally positive generator π 1 (mod 8 E ). Letting F/K be defined by F = K( π), we get that all places above 2 E split in F/K and that p is the only prime that ramifies in F/K. We apply Proposition to E with T = {p}. Since E has good reduction at p, part (ii) of Lemma gives us that Hf 1 (K p, E[2]) = E[2]/(F rob p 1)E[2] = E[2]/(γ 1) = E[2] 12

20 and that the localization map loc T is given by evaluation of cocycles representing elements of Sel 2 (E/K) at F rob p = γ. By our choice of γ, we get that loc T is surjective and therefore dim F2 V T = dim F2 H 1 f (K p, E[2]) = 2. Proposition then gives us that d 2 (E F /K) = d 2 (E/K) 2. Part (ii) follows from iteratively applying Proposition d2(e/k) r 2 times to E. 2.2 Raising the 2-Selmer Rank It is more difficult to use Proposition to find twists of E with larger 2-Selmer rank than it is to find twists with smaller 2-Selmer rank. Following the structure of the proof of Proposition 2.1.2, the most straightforward way to do so would be to find some F/K ramified at a single place p in which all of the places listed in Proposition split in F/K, dim F2 E(K p )[2] = 1, and loc {p} (Sel 2 (E/K)) = 0. Proposition would then give that d 2 (E F /K) = d 2 (E/K) + 1. Mazur and Rubin were able to show that such an F/K can always be found if Gal(K(E[2])/K) S 3 and K has a place v 0 such that either E has multiplicative reduction at v 0 with ord v ( E ) odd or v 0 is real with ( E ) v < 0, allowing them to prove part (iv) of Theorem However, finding such an F/K is not possible in general. Alternatively, if F/K is a quadratic extension ramified at a single place p in which all of the places listed in Proposition split in F/K, dim F2 E(K p )[2] = 2, and loc {p} (Sel 2 (E/K)) = 0, then Proposition would then give that d 2 (E F /K) = d 2 (E/K) or d 2 (E F /K) = d 2 (E/K) + 2. However, there does not seem to be a good method in general to distinguish between the two cases. Instead, we develop the following approach. Rather than choosing a single extension F/K ramified at p, we choose two extensions F 1 /K and F 2 /K such that F 1 /K is ramified at a single place p with dim F2 E(K p )[2] = 2 and F 2 /K is ramified at two places, p as well as a carefully chosen auxiliary prime q with E(K q )[2] = 0. By a careful choice of p and q, we are able to show that exactly one of Sel 2 (E F1 /K) and Sel 2 (E F2 /K) has 2-Selmer rank equal to d 2 (E/K) + 2 and that the other has 2-Selmer rank equal to d 2 (E/K). If loc {p} (Sel 2 (E/K)) = 0 for some prime p 2 where E has good reduction, then Sel 2 (E/K) = S {p}, where S {p} is the strict Selmer group defined in section If we further have dim F2 E(K p )[2] = 2, then Lemma tells us that loc {p} (S {p} ) is disjoint from H1 u(k p, E[2]) and has F 2 dimension 2 in H 1 (K p, E[2]). If F/K were a quadratic extension ramified only at p in which all places above 2 E split and H 1 f (K p, E F [2]) was equal to loc {p} (S{p} ), then recalling from the proof of Proposition that the correction term d is given by dim F2 loc {p} Sel 2 (E F /K), we would see that d 2 (E F /K) = d 2 (E/K) + 2. It turns out that if p 2 is a prime where E has good reduction, dim F2 E(K p )[2] = 2, and loc {p} (Sel 2 (E/K)) = 0, then we can use work of Poonen and Rains [13] to show that H 1 f (K p, E F [2]) will be one of three mutually disjoint subspaces of H 1 (K p, E[2]). 13

21 Definition If V and T are abelian groups, then a function Q : V T is called a quadratic map if Q(av) = a 2 v for all a Z, v V and the form, : V T given by x, y = Q(x + y) Q(x) Q(y) is bilinear. A subgroup W V is called a maximal isotropic subspace of V with respect to Q if it is its own orthogonal complement with respect to, and Q W = 0. Theorem Let E be an elliptic curve over a number field K and v and place of K. (i) The Tate pairing on H 1 (K v, E[2]) arises from a quadratic map q v : H 1 (K v, E[2]) F 2 and Hf 1(K v, E[2]) is maximal isotropic with respect to q v. (ii) The image of H 1 (K, E[2]) is isotropic in v K H1 (K v, E[2]) with respect to the sum of the q v. Proof. Part (i) is Proposition 6.5 in [13] and Part (ii) is part (a) of Proposition 6.8 in [13]. Remark When E(K v ) Z/2Z Z/2Z, then we can write H 1 (K v, E[2]) as { (a, b, c) K v /(K v ) 2 K v /(K v ) 2 K v /(K v ) 2 abc (K v ) 2}. In this case, q v is given by q v (a, b, ab) = (a, b) v + (ab, 1) v where (, ) v is the Hilbert symbol. If we choose a basis for H 1 (K v, E[2]) to set H 1 (K v, E[2]) K v /(K v ) 2 K v /(K v ) 2, then we get q v (a, b) = (a, b) + (a, 1) + (b, 1). This leads to the following corollaries. Corollary Let p 2 be a prime where E has good reduction such that dim F2 E(K p )[2] = 2. If F 1 /K v and F 2 /K v are the two different quadratic extensions ramified at v, then H 1 u(k v, E[2]), H 1 f (K v, E F1 [2]), and H 1 f (K v, E F2 [2]) are all maximal isotropic subspaces of H 1 (K v, E[2]) with respect to q v that have trivial mutual intersection. Further, any maximal isotropic subspace of H 1 (K v, E[2]) with respect to q v must intersect at least one of these non-trivially. Proof. As seen from Remark 2.2.3, the definition of q v on H 1 (K v, E F [2]) is dependent only on H 1 (K v, E[2]) and not on F/K. Theorem therefore gives us that H 1 f (K v, E F [2]) is maximal isotropic with respect to q v for any quadratic F/K. Part (b) of Lemma gives that H 1 u(k v, E[2]) intersects trivially with both H 1 f (K v, E F1 [2]) and H 1 f (K v, E F2 [2]). To see that H 1 f (K v, E F1 [2]) and H 1 f (K v, E F2 [2]) have trivial intersection, we observe that E F2 is a twist of E F1 by the unique unramified quadratic extension F/K v. Lemma then tells us that H 1 f (K v, E F1 [2]) H 1 f (K v, E F2 [2]) = E F1 N F/Kv (K v )/2E F1 (K v ). To show that H 1 f (K v, E F1 [2]) and H 1 f (K v, E F2 [2]) have trivial intersection, it therefore suffices to show that every element of E F1 N F/Kv (K v ) is two-divisible. By Proposition 6.3 in [14], E F1 (K v ) can be written as the direct sum of E F1 (K v )[2 ] and something that is 2-divisible. It therefore suffices to consider points in E F1 N F/Kv (K v ) that are norms of points in E F1 (F )[2 ]. Applying Lemma to (E/F ) F1, we get that E F1 (F ) 14

22 has no points of order 4. As E F1 (F )[2] Z/2Z Z/2Z, the norm on E F1 (F ) is given by multiplication by 2, and therefore all points in E F1 N F/Kv (K v ) that are norms of points in E F1 (F )[2 ] are 2-divisible. This gives that all of H 1 u(k v, E[2]), H 1 f (K v, E F1 [2]), and H 1 f (K v, E F2 [2]) have mutually trivial intersection. Lastly, since v 2 and dim F2 E(K p )[2] = 2, H 1 (K v, E[2]) K v /(K v ) 2 K v /(K v ) 2 has 16 elements. Since H 1 u(k v, E[2]), H 1 f (K v, E F1 [2]), and H 1 f (K v, E F2 [2]) have mutually trivial intersection, their union consists of 10 elements of H 1 (K v, E[2]), all of which map to 0 under q v. By Remark 2.2.3, we observe that every a K v (K v ) 2 has some b K v such that q v (a, b) 0. Therefore H 1 (K v, E[2]) must have at most 10 elements that map to 0 under q v. If A H 1 (K v, E[2]) is maximally isotropic with respect to q v, then it must contain some non-trivial element x H 1 (K v, E[2]) such that q v (x) = 0. However, since all of the elements of H 1 (K v, E[2]) for which q v is 0 sit in the union of H 1 u(k v, E[2]), H 1 f (K v, E F1 [2]), and H 1 f (K v, E F2 [2]), A must intersect one of these non-trivially. Corollary In the setup of Propostion , if T consists of a single prime p, then loc T (S T ) is maximal isotropic in H 1 (K p, E[2]). Proof. Take c S T. By part (ii) of Theorem 2.2.2, v q v(c) = 0. As c H 1 f (K v, E[2]) for all v p, part (i) of Theorem tells us that q v (c) = 0 for all v p. We therefore get that q p (c) = 0 as well. By Lemma , dim F2 loc T (S T ) = dim F 2 H 1 (K p,e[2]) 2, so it is in fact maximal. We are finally ready to prove the following proposition. Proposition If E is an elliptic curve defined over a number field K with E(K)[2] = 0, then E has a twist E F with d 2 (E F ) = d 2 (E) + 2. Proof. Let M = K(E[2]). Take N to be any finite extension of M such that the restriction of Sel 2 (E/K) to N is trivial. Choose σ Gal(MK[8 ]/K), where K[8 ] is the 2-power part of the ray class field K(8 ), such that the order of σ M is either 3 or 6 and σ K[8 ] = 1. As [M K[8 ] : K] is a power of 2, this is always possible. Now choose a prime q away from 2 such that F rob q in Gal(MK[8 ]/K) is the conjugacy class of σ. Since [K(8 ) : K[8 ]] is odd and σ K[8 ] = 1, there is some odd positive integer h such that F rob h q K(8 ) = 1 and therefore q h has a totally positive generator π with π 1 (mod 8 ). Define L = K( π ). Now take γ Gal(NLK(8 )/K) such that γ NK(8 ) = 1 and γ L 1. Since q is ramified in L/K and not in NK(8 )/K, we have L NK(8 ) = K and we can therefore always find such a γ. Let p be any prime away from 2 q such that the image of F rob p in Gal(NLK(8 )/K) is γ. As F rob p in Gal(K(8 )/K) is trivial, p has a totally positive generator π with π 1 (mod 8 ). Define two quadratic extensions of K, F 1 and F 2 by F 1 = K( π) and F 2 = K( ππ ). Observe that all places above 2 split in both F 1 /K and F 2 /K. The only prime ramified in F 1 /K is p and the only primes ramified in F 2 /K are p and q. Further realize that since the image of F rob p2 in Gal(M/K) has order either 3 or 6, E(K q )[2] = 0. 15

23 We then apply Proposition to both E F1 and E F2 with T = {p}. Since the image of F rob p in Gal(N/K) is trivial, we get that G Kp G N and therefore loc T (Sel 2 (E/K)) is trivial. Therefore d 2 (E F1 /K) = d 2 (E/K) + d 1 and d 2 (E F2 /K) = d 2 (E/K) + d 2 with d 1 d 2 0 (mod 2) and 0 d 1, d 2 2. As loc T (Sel 2 (E/K)) is trivial, we get that loc T (S T ) H1 u(k p, E[2]) = 0. Next, observe that since F rob p L 1, we get that p doesn t split in L/K and therefore x 2 π (mod p) is irreducible and π (K p ) 2. This means that the two extensions F 1 and F 2 give different ramified extensions of K p. By Corollaries and 2.2.5, loc T (S T ) must intersect either H1 f (K v, E F1 [2]) or H 1 f (K v, E F2 [2]) non-trivially. Let F be one of F 1, F 2 such that H 1 f (K v, E F [2]) loc T (S T ) contains some element c 0. and let c S T be a lift of c. Observe that c Sel 2 (E F /K), Sel 2 (E/K) Sel 2 (E F /K), and that c Sel 2 (E/K). We therefore get that d 2 (E F /K) > d 2 (E/K) and that therefore d 2 (E F /K) = d 2 (E/K) + 2. Remark In the above proof, if F is taken to be either F 1 of F 2 such that F F, then d 2 (E F /K) = d 2 (E/K). Since Sel 2 (E F /K)) = S T, we get that loc T (S T ) = H1 f (K p, E F [2]), and therefore Sel 2 (E F /K) = S T = Sel 2 (E/K). We conclude by proving Theorem 1.2. Proof of Theorem 1.2. Let E be an elliptic curve defined over a number field K with E(K)[2] = 0 and r 0 with r d 2 (E/K) (mod 2). If r d 2 (E/K), then part (ii) of Theorem gives the result. If r > d 2 (E/K), we iteratively apply Proposition r d2(e/k) 2 times to obtain a curve E F with d 2 (E F ) = r. We then apply part (i) of Theorem to get the result. If r d 2 (E/K) (mod 2) and E does not have constant 2-Selmer parity, then we can replace E by a twist E F such that d 2 (E F /K) d 2 (E/K) (mod 2) and then the result will follow by applying the above to E F. 16

24 Chapter 3 The Case Where E(K) Z/2Z 3.1 How the case where E(K)[2] Z/2Z differs from that of E(K)[2] = 0 The case where E(K)[2] Z/2Z is significantly more complex than the cases where E(K)[2] Z/2Z Z/2Z and E(K)[2] = 0. Like the case where E(K)[2] = 0, we do not have the same two-descent formalism used by Swinnerton-Dyer and Kane in [17] and [7]. However, unlike the case where E(K)[2] = 0, the action of Gal(K(E[2])/K) on E[2] is not irreducible, causing many of the techniques used by Mazur and Rubin in [10] to break down. This is demonstrated by the following example. Example Let E be the elliptic curve defined by y 2 = x 3 +8x 2 +81x over Q. The group of coboundaries for E[2] is generated by a homomorphism g : G Q E[2](Q) Z/2Z where g is the map defining the quadratic field Q(E[2]) = Q( 65). The three homomorphisms f 1, f 2, f 3 : G Q E[2](Q) Z/2Z defining the quadratic fields Q( 1), Q( 5), and Q( 2) respectively are cocycle representatives for a basis of Sel 2 (E/K). The image of the point (0, 0) on E in Sel 2 (E/K) can be identified with a map G Q E[2](Q) Z/2Z defining the quadratic extension Q( 26). This shows that not only can Lemma fail when E(K)[2] 0, but it can do so in spectacular fashion. There are some elliptic curves E with Sel 2 (E/K) Image(E(K)[2]) for which the map c : G K E[2] is not surjective for every cocycle c Sel 2 (E/K)! This leads to the question of when a cocycle representing an element in H 1 (K, E[2]) gives a surjective map G K E[2]. 3.2 The φ-selmer Group If E is an elliptic curve defined over K with E(K)[2] Z/2Z, then E can be given by an integral model over y 2 = x 3 + Ax 2 + Bx defined over K. The subgroup C = E(K)[2] is then generated by the point P = (0, 0). 17

25 Given this model, we are able to define an isogeny φ : E E ( with Kernel C, where E is given by a ( ) ) 2 model y 2 = x 3 2Ax 2 + (A 2 x 4B)x and φ is given by φ(x, y) = y, y(b x 2 ) x for (x, y) C. If we 2 define C = φ(e[2]), then we get a short exact sequence of G K modules 0 C E[2] φ C 0. This short exact sequence gives rise to a long exact sequence of cohomology groups (3.1) 0 C E(K)[2] φ C δ H 1 (K, C) H 1 (K, E[2]) φ H 1 (K, C )... The map δ is given by δ(q)(σ) = σ(r) R where R is any point on E with φ(r) = Q. We can observe from this exact sequence that the elements of H 1 (K, E[2]) represented by cocycles given by maps G K C E[2] are exactly those elements of H 1 (K, E[2]) that come from H 1 (K, C). This indicates that we would like to identify the part of Sel 2 (E/K) that comes from elements in H 1 (K, C). This leads to defining the φ-selmer group of E The Selmer Group arising from a 2-isogeny Given an elliptic curve E with a point P of order two defined over K, there is a Selmer group that arises naturally from the 2-isogeny φ with kernel P. The isogeny φ gives a short exact sequence of G K modules (3.2) 0 C E(K) φ E (K) 0. This sequence gives rise to a long exact sequence of cohomology groups 0 C E(K) φ E (K) δ H 1 (K, C) H 1 (K, E) H 1 (K, E )... The map δ is the same as the map δ in Sequence (3.1). This sequence remains exact when we replace K by its completion K v at any place v, which gives rise to the following commutative diagram. E (K)/φ(E(K)) δ H 1 (K, C) Res v E (K v )/φ(e(k v )) δ H 1 (K v, C) This diagram allows us to define a natural Selmer structure which we will call φ by abuse of notation. The distinguished local subgroups Hφ 1(K v, C) H 1 (K v, C) are defined to be the image of E (K v )/φ(e(k v )) under δ for each place v of K. For places away from 2 E, we have Hφ 1(K v, C) = Hu(K 1 v, C) by Lemma 4.1 in [3]. 18

26 Definition We define the φ-selmer group of E, denoted Sel φ (E/K), as a subgroup of H 1 (K, C) given by the exactness of the sequence 0 Sel φ (E/K) H 1 (K, C) vres v of K H 1 (K v, C)/H 1 φ(k v, C). As can be seen, Sel φ (E/K) is the Selmer group associated to the Selmer structure φ. As in the case of the 2-Selmer group, the φ-selmer group is finite dimensional F 2 vector space. For reasons that will be addressed later, we define the quantity d φ (E/K) = dim F2 Sel φ (E/K) + dim F2 E(K)[2] 2. For curve with E(K)[2] Z/2Z, we have d φ (E/K) = dim F2 Sel φ (E/K) 1. For the remainder of this chapter we will pick an explicit model y 2 = x 3 + Ax 2 + Bx for E and let φ be the isogeny with kernel (0, 0) defined above. The group H 1 (K v, C) can be explicitly identified with H 1 (K v, ±1) K v /(K v ) 2 and the transition map δ : E (K v ) H 1 (K v, C) is given by (x, y) x(k v ) 2 for x 0 and (0, 0) (A 2 4B)(K v ) 2 (Proposition X.4.9 in [14]). The isogeny φ on E gives gives rise to the dual isogeny ˆφ on E with kernel C = φ(e[2]). Exchanging the roles of (E, C, φ) and (E, C, ˆφ) in the above defines the ˆφ-Selmer group as a subgroup of H 1 (K, C ). Sequence (3.1) will allow us to relate the φ-selmer group, the ˆφ-Selmer group, and the 2-Selmer group. Proposition The sequence (3.3) 0 C /φ (E(K v )[2]) H 1 φ(k v, C) H 1 f (K v, E[2]) φ H 1ˆφ(K v, C ) 0 sitting inside the exact sequence (3.4) 0 C /φ (E(K v )[2]) H 1 (K v, C) H 1 (K v, E[2]) φ H 1 (K v, C ) H 2 (K v, C) is exact. Proof. This is Remark X.4.7 in [14]. Corollary The sequence (3.5) 0 C /φ(e(k)[2]) Sel φ (E/K) Sel 2 (E/K) φ Sel ˆφ(E/K) sitting inside the exact sequence (3.6) 0 C /φ(e(k)[2]) H 1 (K, C) H 1 (K, E[2]) φ H 1 (K, C ) is exact. Proof. We begin by showing that we can restrict sequence (3.6) to sequence (3.5). The commutativity of the diagram below for every place v of K combined with sequence (3.3) gives that Sel φ (E/K) Sel 2 (E/K) and that Sel 2 (E/K) φ Sel ˆφ(E /K). Since C Sel φ (E/K) we get that (3.5) is in fact a well-defined sequence. 19

27 H 1 (K, C) H 1 (K, E[2]) φ H 1 (K, C ) Res v H 1 (K v, C) Res v H 1 (K v, E[2]) φ Res v H 1 (K v, C ) Exactness at C /φ(e(k)[2]) and at Sel φ (E/K) follow immediately from the exactness of (3.6) and from C mapping into Sel φ (E/K). The fact that follow from the exactness of (3.6). ( ) Image (Sel φ (E/K) Sel 2 (E/K)) Ker Sel 2 (E/K) φ Sel ˆφ(E/K) To conclude that sequence (3.5) is exact at Sel 2 (E/K) we will rely on Proposition If c v H 1 f (K v, E[2]) such that c v maps to 0 in H 1ˆφ(K v, C ), then there is some c v H 1 φ (K v, C) such that c v c v. Moreover, if d v c v for some d v H 1 (K v, C), we have c v d v Image(C ) so d v Hφ 1(K v, C) as well. ( ) Now take c Ker Sel 2 (E/K) φ Sel ˆφ(E /K). We get that c comes from c H 1 (K, C) by exactness of (3.6). By the argument in the previous paragraph, we get that c Sel φ (E/K) thereby giving exactness at Sel 2 (E/K). Corollary (i) d 2 (E/K) d φ (E/K) + dim F2 E(K)[2] with equality if and only if Sel φ (E/K) surjects onto Sel 2 (E/K). (ii) d 2 (E/K) d φ (E/K) + d ˆφ(E/K) + 2 dim F2 E(K)[2] dim F2 E (K)[2] Proof. (i) This follows from Corollary and from d φ (E/K) being defined in such a way that it is equal to the F 2 -dimension of the image of Sel φ (E/K) in Sel 2 (E/K). (ii) Adding dimensions from the sequence (3.5), we get that dim F2 Sel 2 (E/K) + (2 dim F2 E(K)[2]) dim F2 Sel φ (E /K) + dim F2 Sel ˆφ(E/K). It follows that d 2 (E/K) + 2 d φ (E/K) + 2 dim F2 E(K)[2] + d ˆφ(E /K) + 2 dim F2 E (K)[2] giving that d 2 (E/K) d φ (E/K) + d ˆφ(E/K) + 2 dim F2 E(K)[2] dim F2 E (K)[2]. Remark In Example 3.1.1, we in fact had Sel φ (E/K) surjecting onto Sel 2 (E/K). 20

28 3.2.2 Duality Between Hφ 1(E/K) and H1 φ (E /K) Lemma (Local Duality). For each place v of K there is a local Tate pairing H 1 (K v, C) H 1 (K v, C ) {±1} induced by a pairing [, ] : C C {±1} given by [Q, R] = Q, R where Q, R is the Weil pairing and R is any pre-image of R under φ. The subgroups defining local conditions Hφ 1(K v, C) and H 1ˆφ(K v, C ) are orthogonal complements under this pairing. Proof. Orthogonality is equation (7.15) and the immediately preceding comment in [3]. Combining this orthogonality with exact sequence (3.3) gives that Hφ 1(K v, C) and H 1ˆφ(K v, C ) are orthogonal complements. This local duality gives rise to a Tamagawa ratio T (E/E ) that gives a second relationship between d φ (E/K) and d ˆφ(E/K). Definition The ratio is called the Tamagawa ratio of E. T (E/E ) = Sel φ(e/k) Sel ˆφ(E /K) The Tamagawa ratio can be computed using a local product formula. Definition Let ω be a differential on E, v a place of K, d + v x the additive Haar measure normalized so that the measure of O Kv is 1 for v finite, the Lebesgue measure for v real, and twice the Lebesgue measure for v complex. We define a local Tamagawa factor µ v (ω, E(K v )) by µ v (ω, E(K v )) = f(a) v d + v x(a) where ω = fdx. a E(K v) Lemma (Cassels). Let ω be any differential on E and ω be any differential on E. Then T (E/E ) is given by a local product T (E/E ) = v µ v (ω, E (K v )) µ v (ω, E(K v )). Proof. This is equation (1.22) in [3]. Cassels actually uses this as the definition of the Tamagawa ratio, and proves that the definition given above is equivalent to this in Theorem 1.1 of [3]. If the differentials on E and E are chosen in concert, then we get the following: Lemma (Cassels). If ω is a differential on E and ω = φ 1 ω, then µ v (ω, E (K v )) H 1 µ v (ω, E(K v )) = φ (K v, C). 2 and T (E/E ) = v Hφ 1(K v, C) Proof. The first statement is equation (3.4) in [3] and the product formula then follows from Lemma