Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions


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1 Numbers Volume 2015, Article ID , 4 pages Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International Management, Ritsumeikan Asia Pacific University, Oita , Japan Correspondence should be addressed to Rintaro Kozuma; Received 21 July 2015; Accepted 6 September 2015 Academic Editor: Andrej Dujella Copyright 2015 Rintaro Kozuma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give a method for explicitly constructing an elementary cubic extension L over which an elliptic curve E D :y 2 +Dy=x 3 (D Q ) has MordellWeil rank of at least a given positive integer by finding a close connection between a 3isogeny of E D and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [L : Q] often becomes small. 1. Introduction Let E be an elliptic curve defined over a number field F.Itis wellknown that the MordellWeil group E(F) of Frational points on E forms a finitely generated abelian group, and its rank rank E(F) = dim Q E(F) Z Q is of great interest in arithmetic geometry. The present paper is motivated by the following general problem: To understand the behavior of ranks of elliptic curves in towers of finite extensions over F. This problem dates back at least to [1], which first introduced an Iwasawa theory for elliptic curves. In the context of Iwasawa theory, Kurcanov [2] showed that if an elliptic curve E over Q without complex multiplication has good reduction at a prime number p > 3 satisfying a mild condition then E has infinite rank in a certain Z p extension, and Harris [3] showed that if E is a modular elliptic curve over F having good ordinary reduction at p and a specific Frational point arisingfromthemodularcurvethenthereisapadic Lie extension of F in which the rank of E grows infinitely. On the other hand, Kida [4] constructed a tower of elementary 2extensions in which the rank of y 2 = x 3 n 2 x becomes arbitrarily large by using a result in theory of congruent numbers. Also, Dokchitser [5] proved that for any elliptic curve E over F there are infinitely many cubic extensions K/F so that the rank of E increases in K/F. In the present paper, we consider the specific curve E D :y 2 +Dy=x 3, D Q, (1) and give an explicit construction of an elementary cubic extension over which the elliptic curve E D is of rank of at least a given positive integer l by finding a close connection between a 3isogeny of E D and a generic polynomial for cyclic cubic extensions. We shall call an extension K/F an elementary cubic extension if K is a finite compositum of cubic extensions over F. Compared with related results as above, in our method, the extension degree [L : Q] often becomes small. The question of finding a small elementary cubic extension with rank l seems to be of independent interest. 2. Main Results Let S D ={b Z D=a 2 +3ab+9b 2 (a, b Z, b>0), where a is relatively prime to b}. We denote by F a fixed algebraic closure of F. LetF( 3 S D ) denote the field obtained by adjoining a real number 3 b for each b S D to F. SincethesetS D is finite, the extension F( 3 S D )/F is also finite. The main result of the paper is the following theorem. Theorem 1. Let D be any squarefree product of distinct primes p 1,...,p k Z satisfying p i 1(mod.Then,foranypositive (2)
2 2 Numbers Table 1: rank k. D L rank Q Q Q Q Q( 17) Q( Q( 61) Q( Q( 307) 3 3 integer l( k),wecanconstructasubfieldl of Q( 3 S D ) such that rank l, [L :Q] 3 l. ( As we will see in Section 4, the field L in Theorem 1 is effectively computable from the decomposition of the primes p 1,...,p k in Z[ζ 3 ],whereζ 3 denotes a primitive cube root of unity. See Remark 4 for this computation. It will also turn out in Section 4 that the field L is constructed using certain products of prime elements in Z[ζ 3 ] above each prime p i, and in particular L depends only on the primes p 1,...,p k ( 1(mod ) and a positive integer l( k). One question arises here: how small can [L : Q] be when p 1,...,p k vary through the set of all primes p 1(mod for a fixed integer l( k)? We give some examples in the case l=k(table 1). Corollary 2. Let D be any squarefree product of distinct primes p 1,...,p k Z satisfying p i 1(mod.Then,forany positive integer l,wecanconstructasubfieldl of Q( 3 S dd, 3 d) with some integer d such that rank E D (L ) l, [L : Q] 3 l+1. (4) Note that our construction in Theorem1 (resp., Corollary 2) differs from Dokchitser s [5], and the extension degree [L : Q] (resp., [L : Q])isoftensmallerthanorequal to 3 l 1 (resp., 3 l ). 3. A Connection between the Elliptic Curve E D and a Generic Polynomial for Cyclic Cubic Extensions In this section, let F be a perfect field with characteristic not equal to 3,andletD be any element in F.Theellipticcurve E D has a rational 3torsion point (0, 0) and admits a 3isogeny φ:e D E D =E D / (0, 0) over F.The3isogeny φ can be explicitly written as φ:e D E D (x, y) ( x3 +D 2 Dx 2, ( y/d)3 + 3 ( y/d) 1 ( y/d) ( y/d 1) ). (5) The isogenous elliptic curve is given by the form E D :t 2 +3t+9=Ds 3. (6) Taking Galois cohomology of the short exact sequence 0 φ E D [φ] E D E D 0yields a Kummer map E D (F) φ(e D (F)) H1 (F, E D [φ]) (7) P {σ φ 1 (P) σ 1 }. Combining this map with the map H 1 (F, E D [φ]) = Hom (Gal ( F, Z 3Z ) Gal ( C 3 ξ F Ker ξ, where Gal(C 3 /F) denotes the set of all cyclic cubic extensions over F together with F, Gal ( C 3 := { K F : a Galois extension in F Gal (K C 3 Z 3Z }, we have a map δ F : E D (F) φ(e D (F)) Gal (C 3 P F (φ 1 (P)). (8) (9) (10) It is wellknown that the set Gal(C 3 /F) is parametrized by a generic polynomial for C 3 extensions [6] g (x; t) =x 3 +tx 2 (t+ x+1 with t F, (11) which has discriminant (t 2 +3t+9) 2.WeshalldenotebyK t the minimal splitting field of g(x; t) over F 0 (t), wheref 0 stands for the prime field in F. Bythepropertyofg(x; t), forany K/F Gal(C 3 /F) there is some t Fso as to be K=K t F in F. The following proposition is the key observation in the present paper. Proposition 3. The connecting homomorphism δ F is given by δ F (s, t) = K t F for any (s, t) E D (F). Furthermore,inthe case where F is a number field, for any prime ideal p of F not dividing 3,thepcomponent of the conductor of K t F/F is p if ] p (Ds 3 )>0, ] p (D) 0(mod (12) 1 otherwise. Here ] p :F p Z denotes the normalized valuation of the padic completion F p.
3 Numbers 3 Proof. For any (s, t) E D (F), takeapoint(x, y) on E D satisfying φ(x, y) = (s, t). Bytheformoftheycoordinate of (5), the splitting field K t coincides with F 0 (t, y). Here F(x, y) = F(x) = F(y). Thelatterpartoftheproposition follows from the equation t 2 +3t+9 = Ds 3 by using Proposition 8.1 in [7]. 4. A Method for Constructing an Elementary Cubic Extension From now on, let D be a squarefree product of distinct primes p 1,...,p k Z satisfying p i 1(mod, and let l be a positive integer k. RecallaresultofKomatsu[8]thatthesetofallcycliccubic fields (i.e., cyclic cubic extensions over Q) of conductor D= p 1 p k has a onetoone correspondence to the subset of the algebraic torus T(Q) =Q { } consisting of 2 k 1 elements of the form c p1 ± T c p2 ± T ± T c pk 1 ± T c pk, (1 where + T is the composition of T(Q) given by a/b+ T c/d = (a/b c/d 1)/(a/b + c/d + 1), T denotes the inverse of + T given by T a/b = a/b 1, andc pi =e pi /f pi T(Q) is a rational number so that the pair (e pi,f pi ) of integers is unique in the sense of Lemma 2.1 in [8] (especially f pi 0(mod, whichwillbeusedinthefollowing)andsatisfies e 2 p i +e pi f pi +f 2 p i =(e pi ζ 3 )(e pi ζ 2 3 )=p i. (14) There is a group isomorphism φ : T G m mapping t to (t ζ 3 )/(t ζ 1 3 ),whereg m denotes the multiplicative group. Then the composition + T is written as a/b+ T c/d = φ 1 (φ(a/b)φ(c/d)). Now, let π i := e pi ζ 3, which is a prime element in Z[ζ 3 ] dividing p i,andletα denote the complex conjugation of α.foreachi(1 i l),let t i := { 3(c p1 + T + T c pk ) if i=1 { 3(c { p1 + T + T c pi 1 T c pi + T c pi+1 + T + T c pk ) if i =1. (15) Since f pi 0(mod for each i, there exist unique integers a i, b i so that a i 3b i ζ 3 = { π 1 π k if i=1 { π { 1 π i 1 π i π i+1 π k if i =1, (16) where a i is relatively prime to 3b i.thenφ(t i / = (a i 3b i ζ 3 )/(a i 3b i ζ 3 ),andthust i =3 φ 1 (φ(t i /) = a i /b i. Let L:=Q({ 3 b i 1 i l}). Remark 4. In order to calculate t i =a i /b i,wehaveonlyto know the values c pi by the definition of t i. For example, one can use Table 5.1 in [8], in which the values c p for p < 1000 arelisted.fromtheconstructionofl,thefieldl depends only on D=p 1 p k and a positive integer l( k). Lemma 5. Every prime number p i (1 i l)is unramified in L. Proof. Since p i does not divide 3b 1 b l and is unramified in Q(ζ 3 ),itturnsoutthatp i is unramified in Q(ζ 3,{ 3 b i 1 i l}) by Kummer theory. Hence, p i is also unramified in L=Q({ 3 b i 1 i l}). Proposition 6. K ti L Im δ L {L} for any i(1 i l). Proof. Since (a i 3b i ζ 3 )(a i 3b i ζ 3 )=D,wehave t 2 i +3t i +9=D( 3 b i 2 ) 3. (17) Thus ( b 3 2 i,ti ). ItfollowsfromProposition3that δ L ( b 3 2 i,ti ) = K ti L, and hence K ti L Im δ L.Combining Proposition 3 with Lemma 5 yields K ti L =L. Proposition 7. dim /φ() l. Proof. One can easily verify that (0, 3ζ 3 ) is a 3torsion point in E D (L(ζ 3 )), and the rational function f:=(t 3ζ 3 )/D 2 on E D has a zero of order 3 at (0, 3ζ 3 ) and a pole of order 3 at infinity and satisfies f φ=( y Dζ 3 3 Dx ). (18) Then there exists an injective homomorphism (see [9] for details): E D (L (ζ 3 )) φ(e D (L (ζ 3 ))) L(ζ L(ζ 3 ) 3 (s, t) D (t 3ζ 3 )L(ζ 3 ) 3 (0, 3ζ 3 ) D 2 L(ζ 3 ) 3. (19) Combining this with the natural map φ() E D (L (ζ 3 )) φ(e D (L (ζ 3 ))), (20) we have the homomorphism φ() L(ζ L(ζ 3 ) 3 (21) (s, t) D (t 3ζ 3 )L(ζ 3 ) 3. By Proposition 6, the point ( b 3 2 i,ti ) corresponds to D(t i 3ζ 3 )L(ζ 3 ) 3 = D(a i 3b i ζ 3 )L(ζ 3 ) 3 via the above homomorphism. It thus suffices to show that {D(a i 3b i ζ 3 ) 1 i l}is linearly independent in L(ζ 3 ) /L(ζ 3 ) 3.Thenthe points {( b 3 2 i,ti ) 1 i l}must be linearly independent in /φ(). Assume that γ:= {D (a i 3b i ζ 3 )} r i L(ζ 3 ) 3 (22) 1 i l
4 4 Numbers for some integers {r i } 1 i l.foreachi, letp i be a prime ideal of L(ζ 3 ) above the prime element π i Q(ζ 3 ).Forany prime ideal p of L(ζ 3 ), we denote by ] p : L(ζ 3 ) p Z the normalized (additive) valuation of the padic completion L(ζ 3 ) p.bylemma5,wehave] pi (π i )=1.Then l r { i 0(mod i=1 ] pj (γ) = l if j=1 { r i +r j 0(mod if j =1, { i=1 (2 and hence r i 0(mod for each i.thisprovesthat{d(a i 3b i ζ 3 ) 1 i l}is linearly independent in L(ζ 3 ) /L(ζ 3 ) 3. Let φ : E D E D be the dual isogeny to φ. Sincethe composition φ φ(resp., φ φ) is the multiplicationby3 map on E D (resp., E D ), we have an exact sequence of F 3 vector spaces 0 and thus [ φ] φ( [3]) 0, φ() rank = dim dim φ() φ [3] φ() + dim F 3 φ() [ φ] φ( [3]) dim [3]. (24) (25) Since ζ 3 L,thegroup[ φ] is trivial and the group [3] is generated by the point (0, 0).Weseethat rank = dim φ() + dim F 3 φ() 1. (26) ProofofCorollary2. By Theorem 1, we have only to consider the case l>k.(inthecasewherel k,taked=1.) Take distinct prime numbers {q 1,...,q l k } not dividing D with q i 1(mod. Letd:=q 1 q l k. Then, applying Theorem 1 to the elliptic curve E dd,thereexistssomesubfieldl Q( 3 S dd ) so as to be rank E dd (L) l with [L : Q] 3 l.letl := L( 3 d). Then, it is easily seen that E dd is isomorphic over L to E D. Therefore, rank E D (L )=rank E dd (L ) rank E dd (L) l. Here [L : Q] 3 [L:Q] 3 l+1. Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper. Acknowledgment The author is supported by APU Academic Research Subsidy. References [1] B. Mazur, Rational points of abelian varieties with values in towers of number fields, Inventiones Mathematicae, vol. 18, pp , [2] P. F. Kurcanov, Elliptic curves of infinite rank over Γ extensions, Mathematics of the USSRSbornik, vol. 19, no. 2, pp , [3] M. Harris, Systematic growth of MordellWeil groups of abelian varieties in towers of number fields, Inventiones Mathematicae,vol.51,no.2,pp ,1979. [4] M. Kida, On the rank of an elliptic curve in elementary 2extensions, Japan Academy. Proceedings A. Mathematical Sciences,vol.69,no.10,pp ,1993. [5] T. Dokchitser, Ranks of elliptic curves in cubic extensions, Acta Arithmetica,vol.126,no.4,pp ,2007. [6] J.P. Serre, Topics in Galois Theory, Second Edition,vol.1of Research Notes in Mathematics,CRCPress,2007. [7] R. Kozuma, Elliptic curves related to cyclic cubic extensions, International Number Theory, vol.5,no.4,pp , [8] T. Komatsu, Cyclic cubic field with explicit Artin symbols, Tokyo Mathematics, vol. 30, no. 1, pp , [9]J.H.Silverman,The Arithmetic of Elliptic Curves, vol.106of Graduate Texts in Mathematics, Springer, ProofofTheorem1. By Proposition 7, it suffices to show that (0, 0) φ().then dim φ() 1, and hence rank l. (27) Assume that there exists some point Q such that φ(q) = (0, 0). ThenQ [3]. Itfollowsfromthe3 division polynomial 3x(x 3 27D 2 ) for E D that there is no nontrivial 3torsion point in. Therefore, Q must be the trivial element in [3]. This contradicts the assumption, and hence (0, 0) φ().
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