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UC Irvine UC Irvine Electronic Theses and Dissertations UC Irvine UC Irvine Electronic Theses and Dissertations Title Selmer Parity of Quadratic Twists of Elliptic Curves Permalink https://escholarship.org/uc/item/35z7784k Author Su, Heng Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, IRVINE Selmer Parity of Quadratic Twists of Elliptic Curves DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics by Heng Su Dissertation Committee: Professor Karl Rubin, Chair Professor Alice Silverberg Professor Daqing Wan 2015 c 2015 Heng Su Dedication To my parents, for their care in my pursuit of my dreams. ii Table of Contents Acknowledgments iv Curriculum Vitae v Abstract of the Dissertation vi 1 Introduction and Background 1 1.1 Introduction . 1 1.2 Background . 3 2 Selmer Parity Densities of Quadratic Twists 7 2.1 Counting by elements . 7 2.2 Counting by fair counting functions . 17 3 Cokernel of Local Norm Map 22 3.1 Introduction . 22 3.2 Calculating jE(L)=NE(F )j ............................ 23 3.3 Proof of Theorem 3.1.1 . 30 3.4 Examples and remarks . 36 A Convex Bodies 39 iii Acknowledgments First and foremost, I would like to thank Professor Karl Rubin for his advice and support over the years. However busy he was, he met with me every week and patiently explained mathematical concepts to me. He guided me to develop the theories in my thesis step by step. Every time I submitted a draft of paper, he read word by word. He can always give me valuable advice and detailed explanation, including grammar mistakes. Without his tremendous effort and patience, the completion of my thesis is not possible. I would also thank Professor Daqing Wan and Professor Alice Silverberg for giving me advice on my mathematical career in UC Irvine. I am also grateful to Professors Vladimir Baranovsky, Christopher Davis, Dennis Eichhorn and Liang Xiao for introducing me to the various aspects of algebra and number theory and giving me comprehensive advice on every aspect of mathematics. I would not find algebra and number theory so interesting without their guidance. I would like to express my special thanks to my academic brother Myungjun Yu. To me he is both brother and mentor. He is always trying to be helpful and we spent many memorable days and nights working together and encouraging each other. I would thank Rufei Ren, Hayan Nam other friends and fellows in the UCI math circle for their encouragement and support. I appreciate the accommodation and valuable lectures provided by the Arizona Winter School. My works in Section 2.2 were inspired by the talks given by Melanie Wood during the 2014 Arizona Winter School. Lastly, I would express my deepest appreciation to my family and friends for their spiritual support. Without their care, support and most importantly love, I would not be able to complete my Ph.D. study on the other side of the planet. iv Curriculum Vitae Heng Su B.S. in Mathematics and Applied Mathematics, Beihang University, 2011 Ph.D. in Mathematics, University of California, Irvine, 2015 v Abstract of the Dissertation Selmer Parity of Quadratic Twists of Elliptic Curves By Heng Su Doctor of Philosophy in Mathematics University of California, Irvine, 2015 Professor Karl Rubin, Chair Inspired by the paper of Klagsbrun, Mazur and Rubin [5], this thesis investigates the disparity of 2-Selmer ranks of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. In the first part, we calculate the density of quadratic twists of E with even 2-Selmer ranks under two different counting methods. First we count twists by elements inside a large convex body of the Euclidean space that contains the integer lattice of K. The second counting method is counting quadratic twists EL by the norms of the finite part of conductors of quadratic extensions L=K. Under both counting methods we give an explicit formula for the densities, which are finite products of local factors. In the second part of the paper we give a method that uses Tate's algorithm to calculate the size of the cokernel of the local norm maps of E at places over 2, assuming that E has good reduction. With this method we can extend Kramer's early work on the cokernel of the local norm maps, and compute the local factors mentioned above in some additional cases. vi Chapter 1 Introduction and Background 1.1 Introduction We investigate the distribution of the parities of 2-Selmer ranks in the quadratic twist family of an arbitrary elliptic curve E over an arbitrary number field K. Let Eχ (resp. Ed) be the quadratic twist (Definition 1.2.3) of E by the quadratic character χ (resp. an element d 2 OK , d 6= 0) of K, denote r(χ) (resp. r(d)) the 2-Selmer rank (Definition 1.2.4) of Eχ (resp. Ed). In Chapter 2, we will give an explict formula for the density of quadratic d twists E with even 2-Selmer ranks, as d varies over a large subset of OK , and a formula for the density of quadratic twists Eχ with even 2-Selmer ranks counting by the finite part of conductor of χ. To be precise, in Section 2.1, we identify the ring of integers OK of K with an N- dimensional lattice in RN under the map: d 7! (σ1d; :::; σr1 d; Re(τ1d); Im(τ1d); :::; Re(τr2 d); Im(τr2 d)) ; where N is the extension degree [K : Q], σ1; ··· ; σr1 are the real embeddings of K and N τ1; ··· ; τr2 are the complex embeddings of K. Let C be a convex body in R . For real numbers λ > 0, let λC := fλv : v 2 Cg ⊂ RN . Define jfd 2 O \ λC : r(d) is evengj ρ(E=K; C) := lim K ; λ!1 jOK \ λCj if the limit exists. We get the following theorem that evaluates ρ(E=K; C). 1 Theorem A. Suppose that E is an elliptic curve defined over K. Then for any convex body C in RN 1 ξ(C) ρ(E=K; C) = + : 2 2 r(1K ) Q Here ξ(C) = (−1) ι(C) v2Σ ξv is a finite product of local data (see Definition 2.1.5). In Section 2.2, we count quadratic twists Eχ of E by \fair" counting functions c of χ. In particular, c could be the norm of finite part of the conductor or norm of product of ramified primes of χ (see definition of fairness in Definition 2.2.2). Define jfquadratic characters χ over K : r(χ) is even, c(χ) < Xgj Pc(E=K) := lim ; X!1 jfquadratic characters χ over K : c(χ) < Xgj if the limit exists. We will show Theorem B. Suppose that E is an elliptic curve defined over a number field K. Then 1 β P (E=K) = + c : c 2 2 r(1K ) Q Here βc = (−1) v2Σ βc;v is a finite product of local probabilities (see Definition 2.2.7). Theorem A is based on the joint work of Klagsbrun, Mazur and Rubin [5], who proved an analogue of Theorem A with a different ordering of characters and different values of ξ (see Theorem 1.2.9 below). They showed that Y (1.1.1) r(χ) ≡ r(1K ) (mod 2) () !v(χv) = 1; v2Σ where Σ is a finite set of places depending only on E and K, χv's are localizations of χ and !v's are local factors (see Definition 1.2.6). Theorem B is a combination of [5] and work of Wood [13]. Chapter 3 deals with the problem of calculating the size of the cokernel of local norm map N : E(F ) ! E(L) of an arbitrary elliptic curve E over an arbitrary finite extension L of Q2 and F=L quadratic, inspired by the work of Kramer [6][Proposition 4]. If we add condition that the minimal model of E has no xy term, then we have the following theorem Theorem C. Let L be a finite extension of Q2 with residue field k and valuation v. Let p F = L( d) be a quadratic extension of L. Suppose E=L has good reduction and has the 2 following minimal Weierstrass model 2 3 2 y + a3y = x + a2x + a4x + a6: Then dimF2 E(L)=NE(F ) ≡ [k : F2] · ordvd (mod 2) The above theorem is Theorem 3.1.1. This will allow us to calculate !v in (1.1.1) and deal with the case when E has good reduction and vj2, which is not given in the table of [5][Propostion 7.9]. More generally, Theorem 3.2.1 shows that using Tate's algorithm we can effectively cal- culate the size of the cokernel E(L)=NE(F ) assuming that E has good reduction over L and F=L ramified. 1.2 Background The notations here are all from [5]. Let E be an elliptic curve defined over a number field K. Fix a finite set Σ of places of K, containing all places above 2, places of bad reduction and all archimedean places. Let Kv denote the completion of K at place v of K. Fix an algebraic closure K¯ of K. Definition 1.2.1. For every place v of K, let 1 1 Hf (Kv;E[2]) = image(E(Kv)=2E(Kv) ,! H (Kv;E[2])) be the image of Kummer map obtained from GKv -cohomology of the exact sequence 2 0 / E(Kv)[2] / E(Kv) / E(Kv) / 0: Define the 2-Selmer group of E=K as 1 1 Sel2(E=K) := fc 2 H (K; E[2]) : resv(c) 2 Hf (Kv;E[2]) for all v of Kg; 1 1 where resv : H (K; E[2]) ! H (Kv;E[2]) is the restriction map at v.
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