THE DISTRIBUTION OF 2-SELMER RANKS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO-TORSION

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1 THE DISTRIBUTION OF -SELMER RANKS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO-TORSION ZEV KLAGSBRUN AND ROBERT J LEMKE OLIVER Abstract This paper presents a new result concerning the distribution of -Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number eld K with a single point of order two that does not have a cyclic 4-isogeny dened over its twodivision eld We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large -Selmer rank, showing that the distribution of -Selmer ranks in the quadratic twist family of such an elliptic curve diers from the distribution of -Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion or full rational two-torsion Introduction Distributions of Selmer Ranks Let E be an elliptic curve dened over a number eld K and let Sel E/K) be its -Selmer group see Section for its denition) We dene the -Selmer rank of E/K, denoted d E/K), by d E/K) = dim F Sel E/K) dim F EK)[] In 994, Heath-Brown [4] proved that the -Selmer ranks of all of the quadratic twists of the congruent number curve E/Q given by y = x 3 x had a particularly nice distribution In particular, he showed that there are explicits constants α 0, α, α, summing to one such that {d squarefree d < X : d E d /Q) = r} lim = α r X {d squarefree d < X} for every r Z 0, where E d is the quadratic twist of E by d This result was extended by Swinnerton-Dyer [0] and Kane [5] to all elliptic curves E over Q with EQ)[] Z/Z Z/Z that do not have a cyclic 4-isogeny dened over K A similar result was obtained by Klagsbrun, Mazur, and Rubin [7] for elliptic curves E over a general number eld K with GalKE[])/K) S 3, where squarefree d are replaced by quadratic characters of K and a suitable ordering of all such characters is taken In this work we show that this type of result does not hold when EK)[] Z/Z In particular, we prove the following: Theorem For d O K, let χ d be the quadratic character of K that cuts out the extension K d) and dene CK, X) := {χ d : N K/Q d < X} Let E be an elliptic curve dened over K with EK)[] Z/Z that does not have a cyclic 4-isogeny whose kernel contains EK)[] dened over KE[]) Then for any xed r, lim inf X {χ CK, X) : d E χ /K) r} where E χ is the quadratic twist of E by any d O K with χ d = χ

2 ZEV KLAGSBRUN AND ROBERT J LEMKE OLIVER In particular, this shows that there is not a distribution function on -Selmer ranks within the quadratic twist family of E Theorem is an easy consequence of the following result Theorem Let E be an elliptic curve dened over K with EK)[] Z/Z that does not have a cyclic 4-isogeny whose kernel contains EK)[] dened over KE[]) Then the normalized distribution P r T E/E ), X) log log X converges weakly to the Gaussian distribution where Gz) = π z e w dw, P r T E/E ), X) = {χ CK, X) : ord T E χ /E χ ) r} for X R +, r Z 0, and T E χ /E χ ) as dened in Section In turn, Theorem follows from a variant of the Erd s-kac theorem for quadratic characters of number elds Let CK) denote the set of all quadratic characters of K For any χ CK), we can associate a unique squarefree ideal D χ to χ by taking D χ to be the squarefree part of the ideal d, where d is any element of O K such that χ d = χ We say that a function f on the ideals of O K is additive if faa ) = fa) + fa ) whenever the ideals a and a are relatively prime; we dene fχ) for χ CK) to be fd χ ) To an additive function f, we attach quantities µ f X) and σ f X), dened to be µ f X) := ) /, and we prove the following fp) Np, σ fx) := fp) Np Theorem 3 Suppose that f is an additive function such that 0 fp) for every prime p If σ f X) as X, then {χ CK, X) : fχ) µ f X) z σ f X)} lim X = Gz), In the special case where K = Q, Theorem follows from results recently obtained by Xiong about the distribution of dim F Sel φ E/Q) in quadratic twist families [] We are able to obtain results over general number elds by applying dierent methods to the coarser question about the distribution of ord T E/E ) in twist families see Theorem ) Remark 4 The methods in the paper can easily be adapted to studying cubic twists of the j-invariant 0 curve y = x 3 +k for any number eld K with K k, 3)/K biquadratic In that case, Theorem holds with -Selmer rank replaced by 3-Selmer rank and quadratic characters replaced by cubic characters Remark 5 This work is the rst of a series of papers which develop techniques for using additive functions to study questions relating to the distribution of -Selmer ranks of elliptic curves The next paper in this series will use these techniques to prove variants of Theorems

3 SELMER RANKS OF TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO-TORSION 3 and for the set of all elliptic curves with a rational point of order two [8] The third and nal paper in this series will address a number of questions relating to the joint distribution of -Selmer ranks within quadratic twist families [9] Layout We begin in Section by recalling the denitions of the -Selmer group and the Selmer groups associated with a -isogeny φ and presenting some of the connections between them In Section 3, we examine the behavior of the local conditions for the φ- Selmer group under quadratic twist and show how that quantity T E χ /E χ ) can be related to the value fχ) of an additive function f on CK) Theorem 3 is proved in Section 4 and we conclude with the proofs of Theorems and in Section 5 Acknowledgement The rst author would like to express his thanks to Karl Rubin for his helpful comments and suggestions, to Ken Kramer for a series of valuable discussions, and to Michael Rael and Josiah Sugarman for helpful conversations regarding the Erd s- Kac theorem The rst author was supported by NSF grants DMS , DMS , and DMS The second author was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship, DMS Selmer Groups We begin by recalling the denition of the -Selmer group If E is an elliptic curve dened over a eld K, then EK)/EK) maps into H K, E[]) via the Kummer map The following diagram commutes for every place v of K, where δ is the Kummer map EK)/EK) δ H K, E[]) EK v )/EK v ) δ Res v H K v, E[]) We dene a distinguished local subgroup Hf K v, E[]) H K v, E[]) as the image δ EK v )/EK v )) H K v, E[]) for each place v of K and we dene the -Selmer group of E/K, denoted Sel E/K), by Sel E/K) = ker H K, E[]) P resv v of K H K v, E[])/H f K v, E[]) The -Selmer group is a nite dimensional F -vector space that sits inside the exact sequence of F -vector spaces 0 EK)/EK) Sel E/K) XE/K)[] 0 where XE/K) is the Tate-Shafarevich group of E If EK) has a single point of order two, then there is a two-isogeny φ : E E between E and E with kernel C = EK)[] This isogeny gives rise to two Selmer groups We have a short exact sequence of G K modules ) 0 C EK) φ E K) 0 which gives rise to a long exact sequence of cohomology groups 0 C EK) φ E K) δ H K, C) H K, E) H K, E ) The map δ is given by δq)σ) = σr) R where R is any point on EK) with φr) = Q )

4 4 ZEV KLAGSBRUN AND ROBERT J LEMKE OLIVER 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)/φEK)) δ H K, C) Res v E K v )/φek v )) δ H K v, C) In a manner similar to how we dened the -Selmer group, we dene distinguished local subgroups Hφ K v, C) H K v, C) as the image of E K v )/φek v )) under δ for each place v of K We dene the φ-selmer group of E, denoted Sel φ E/K) as Sel φ E/K) = ker H K, C) P resv v of K H K v, C)/H φk v, C) The isogeny φ on E gives gives rise to a dual isogeny ˆφ on E with kernel C = φe[]) Exchanging the roles of E, C, φ) and E, C, ˆφ) in the above denes the ˆφ-Selmer group, Sel ˆφE /K), as a subgroup of H K, C ) The groups Sel φ E/K) and Sel ˆφE /K) are nite dimensional F -vector spaces and we can compare the sizes of the φ-selmer group, the ˆφ- Selmer group, and the -Selmer group using the following two theorems Theorem The φ-selmer group, the ˆφ-Selmer group, and the -Selmer group sit inside the exact sequence ) 0 E K)[]/φEK)[]) Sel φ E/K) Sel E/K) φ Sel ˆφE /K) Proof This is a well known diagram chase See Lemma in [] for example The Tamagawa ratio T E/E ) dened as T E/E Selφ E/K) ) = Sel ˆφE /K) gives a second relationship between the F -dimensions of Sel φ E/K) and Sel ˆφE /K) Theorem Cassels) The Tamagawa ratio T E/E ) is given by T E/E ) = H φ K v, C) v of K Proof This is a combination of Theorem and equations ) and 34) in [] Stepping back, we observe that if T E/E ) r+, then d φ E/K) r +, and therefore by Theorem, d E/K) r If E does not have a cyclic 4-isogeny dened over K then we can in fact show that T E/E ) r implies that d E/K) r, but this is entirely unnecessary for our purposes) 3 Local Conditions at Twisted Places For the remainder of this paper, we will let E be an elliptic curve with EK)[] Z/Z and let φ : E E be the isogeny with kernel C = EK)[] If p is a prime where E has good reduction, then H φ K p, C) is a -dimensional F - subspace of H K p, C) equal to the unramied local subgroup H uk p, C) If such a p is ramied in the extension F/K cut out by a character χ, then the twisted curve E χ will have bad reduction at p The following lemma addresses the size of H φ K p, C χ ) )

5 SELMER RANKS OF TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO-TORSION 5 Lemma 3 Suppose p is a prime where E has good reduction and p is ramied in the extension F/K cut out by χ i) If EK p )[] Z/Z E K p )[], then dim F H φ K p, C χ ) = ii) If EK p )[] Z/Z Z/Z E K p )[], then dim F H φ K p, C χ ) = iii) If EK p )[] Z/Z and E K p )[] Z/Z Z/Z, then dim F H φ K p, C χ ) = iv) If EK p )[] Z/Z Z/Z and E K p )[] Z/Z, then dim F H φ K p, C χ ) = 0 Proof From Lemma 37 in [6], we have E χ K p )[ ]/φe χ K p )[ ]) = E χ K p )[]/φe χ K p )[]) All four results then follow immediately Remark 3 Suppose that and are discriminants of any integral models of E and E respectively The condition that EK p )[] Z/Z resp E K p )[] Z/Z) is equivalent to the condition that resp ) is not a square in K p Which case of Lemma 3 we are is therefore determined by the Legendre symbols p ) and We use Theorem and Lemma 3 to relate ord T E χ /E χ ) to the value gχ) of an additive function g on CK) dened as follows: For χ CK) cutting out F/K, let ) ) p p 3) gχ) = p ramied in F/K p That is, gχ) roughly counts the dierence between the number of primes ramied in F/K where condition iii) of Proposition 3 is satised and the number of primes ramied in F/K where condition iv) is satised We then have the following: Proposition 33 The order of in the Tamagawa ratio T E χ /E χ ) is given by ord T E χ /E χ ) = gχ) + dimf HφK v, C χ ) ) v Proof By Thereom, ord T E χ /E χ ) is given by ord T E χ /E χ ) = dimf HφK v, C d ) ), v F/K where F/K is the relative discriminant of the extension F/K cut out by χ By Remark 3, Lemma 3 gives us that ) ) dim F HφK p, C χ p p ) = for places p F/K with p and the result follows Because the sum 4 The Erd s-kac Theorem For Quadratic Characters v dimf H φk v, C χ ) ) p )

6 6 ZEV KLAGSBRUN AND ROBERT J LEMKE OLIVER can be bounded uniformly for a xed elliptic curve, Proposition 33 suggests that we should study the distribution of the additive function gχ) in order to understand the distribution of T E χ /E χ ) as χ varies When f : N C is an additive function on the integers, then under mild hypotheses, the classical Erd s-kac Theorem tells us that the distribution of f on natural numbers less than X approaches a normal distribution with mean µx) and variance σ X) as X, where µx) and σx) are the rational analogues of µ f X) and σ f X) dened in the introduction Theorem 3 is the statement that the same type of result holds for additive functions on quadratic characters of a number eld We begin by observing that if χ d = χ d, then, if d ) = ab and d ) = ab, b and b lie in the same ideal class of O K Conversely, if b and b lie in the same class, then χ d = χ εd for some ε O K Thus, we see that elements of CK, X) correspond to triples b, a, ε) with b a representative of its ideal class of minimal norm, a a squarefree ideal of norm Na < X/Nb such that ab is principal, and ε an element of O K /O K ) From this discussion, it is apparent, for a prime ideal p and a xed choice b 0 of b, that {χ CK, X) b 0, a, ε) : p a} lim X = Prp a : a is squarefree, ab 0 is principal) = Np +, whence, ranging over all χ, the probablity that p a is Np+ Thus, treating the events p a and p a as independent, we might predict that where µ f X) := fχ) µ f X), fp) Np + = µ f X) + O) In fact, this prediction is true, which we establish using the method of moments, following the blueprint of Granville and Soundararajan [3] This requires the following technical result about the function g p χ), dened to be g p χ) := ) fp) Np+ ) fp) Np+ if p a, and if p a Theorem 4 With notation as above, uniformly for k σ f z) /3, we have that ) k )) k g p χ) = c k σ f z) k 3 + O + O X λ 3 k π σ f z) K z) k) if k is even, and g p χ) ) k c k σ f z) k k 3/ + X λ 3 k π K z) k

7 SELMER RANKS OF TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO-TORSION 7 if k is odd Here, c k = Γk + )/ k/ Γ k + ) and λ < depends only on the degree of K The proof of Theorem 4 relies upon a result about the distribution of squarefree ideals To this end, given ideals c and q and squarefree d q, dene N sf X; c, q, d) to be the number of squarefree ideals a of norm up to X in the same class as c and such that a, q) = d We then have: Lemma 4 With notation as above, we have that deg K deg K+ N sf X; c, q, d) = ClK) res s= ζ K s) φq, d)x + OX λ 3 ωq) ), ζ K ) where λ = if deg K 3 and λ = / otherwise, ωq) denotes the number of distinct primes dividing q, and φq, d) = Np Np + Np + p d p q,p d Proof This follows from elementary considerations and the classical estimate = ClK) res s=ζ K s) X + OX deg K deg K+ ) Na<X ac prin Proof of Theorem 4 For any ideal q, dene g q χ) := p α q g pχ) α We then have that ) k g p χ) = ) k O K /O K ) g p a) = O K /O K ) b b Na< X Nb ab prin Np,,Np k <z Na< X Nb ab principal g p p k a), where, as expected, the summation over b is taken to be over representatives of minimal norm for each ideal class and the prime on the summation over a indicates it is to be taken over squarefree ideals Given an ideal c, we now consider for any q the more general summation Na<Y ac principal g q a) = d q g q d)n sf Y ; c, q, d) = =: Y res s= ζ K s) ClK) ζ K ) Y ClK) g q d)φq, d) + O d q res s= ζ K s) Gq) + O Y λ 3 ωq)), ζ K ) Y λ 3 ωq) d q say, where q = p q p We note that Gq) is multiplicative, and is given by Gq) = ) fp) α α + Np ) α ) Np + Np + Np + p α q g q d),

8 8 ZEV KLAGSBRUN AND ROBERT J LEMKE OLIVER Thus, Gq) = 0 unless each α is at least two, ie q is square-full Returning to the original problem, we nd that k g p χ)) = ck) X Gp p k ) + O X λ 3 k π K z) k), where π K z) := #{p : Np < z} and ck) := O K /O K ) Np,,Np k <z res s= ζ K s) ClK) ζ K ) Nb b Noting that the above discussion also proves that = ck) X + OX λ ), the goal is to estimate the summation over p,, p k Since Gq) = 0 unless q is square-full, we have that Gp p k ) = Np,,Np k <z s k/ α + +α s=k, each α i k! α! α s! p < <p s Np s<z Gp α p αs s ), where p < p is determined by a norm-compatible linear ordering on the prime ideals of O K We note that, since Gp α ) fp), the inner summation contributes no more than Np Oσ f z) s ), which will be an error term unless s = k ; the dependence of this error on k can be sussed out exactly as in Granville and Soundararajan's work In fact, the handling of the main term, arising when k is even and s = k, is also essentially the same In particular, the inner summation is equal to ) k)! Gp p k ) = k/ k)! Gp ) + Olog log k) Np,,Np k <z distinct This yields Theorem 4 We are now ready to prove Theorem 3 = k )! σf z) + Olog log k) ) k/ Proof of Theorem 3 Recall that we wish to show that the quantity fχ) µ f X), χ CK, X), σ f X) is normally distributed as X As remarked above, we will do so using the method of moments In particular, we have that fχ) µ f X)) k = fp) k fp) Np + + O) p a ) ) k log X = g p a) + O log z

9 SELMER RANKS OF TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO-TORSION 9 Considering the error term in Theorem 4, we take z = X λ k With this choice, the inner summation becomes k k g p a)) k j + O k j g p a), j=0 hence Theorem 4, the Cauchy-Schwarz inequality, and the fact that σ f z) = σ f X) + Olog k) yield that )) k fχ) µ f X)) k = c k σ f X) k 3/ + O σ f X) if k is even, and if k is odd This proves the theorem fχ) µ f X)) k c k σ f X) k k 3/ 5 Proof of Main Theorems In order to apply Theorem 3 to the additive function gχ) dened in 3), we need to evaluate the quantities µ g X) and σ g X) We begin with the following Proposition 5 Let c K be non-square Then ) c + p = log log X + O) Np Np X Proof This is a consequence of the prime ideal theorem and the fact that the Hecke L- function attached to the non-trivial character of GalK c)/k) is analytic and non-vanishing on the line Rs) = We now decompose µ g X) as µ g X) = p + Np ) p p + and it immediately follows from Proposition 5 that µ g X) = O) We also rewrite σ g X) as / ) / σ g X) = = p Np Np Np X,p Np X p ) p p ) In order to apply Proposition 5 to σ g X), we therefore need to be non-square in K This will be the case when E does not have a cyclic 4-isogeny dened over KE[]) Proposition 5 If E is an elliptic curve with EK)[] Z/Z that does not have a cyclic 4-isogeny whose kernel contains EK)[] dened over KE[]), then K ) Proof This follows from Lemma 4 in [6] Np ) p,

10 0 ZEV KLAGSBRUN AND ROBERT J LEMKE OLIVER By Proposition 5, we therefore get that σ g X) = not have a cyclic 4-isogeny dened over KE[]) log log X + O) whenever E does Proof of Theorem Applying Theorem 3 to gχ), we get that { )} χ CK, X) : gχ) O) z log log X + O) 4) lim = Gz) X for every z R By Proposition 33, there is some contant C, independent of χ, such that gχ) ord T E χ /E χ ) < C, so in fact 4) holds with gχ) replaced by ord T E χ /E χ ) and the result follows Proof of Theorem By Theorem, {χ CK, X) : ord T E χ /E χ ) r} lim = X for any xed r 0 As d E χ /K) ord T E χ /E χ ), this shows that for any ɛ > 0, for suciently large X {χ CK, X) : d E χ /K) r} References ɛ [] JWS Cassels Arithmetic on curves of genus VIII: On the conjectures of Birch and Swinnerton-Dyer Journal für die reine und angewandte Mathematik Crelles Journal), 9657):8099, 965 [] EV Flynn and C Grattoni Descent via isogeny on elliptic curves with large rational torsion subgroups Journal of Symbolic Computation, 434):93303, 008 [3] A Granville and K Soundararajan Sieving and the Erd skac theorem Equidistribution in number theory, an introduction, pages 57, 007 [4] DR Heath-Brown The size of Selmer groups for the congruent number problem, II Inventiones mathematicae, 8):33370, 994 [5] Daniel Kane On the ranks of the -Selmer groups of twists of a given elliptic curve Algebra Number Theory, 75):5379, 03 [6] Z Klagsbrun Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion Preprint available at 0 [7] Z Klagsbrun, B Mazur, and K Rubin A Markov model for Selmer ranks in families of twists Compositio Math, to appear [8] Z Klagsbrun and R L Oliver The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point In preparation [9] Z Klagsbrun and R L Oliver Elliptic curves and the joint distribution of additive functions In preparation [0] P Swinnerton-Dyer The eect of twisting on the -Selmer group In Mathematical Proceedings of the Cambridge Philosophical Society, volume 45, pages 5356 Cambridge Univ Press, 008 [] Maosheng Xiong On Selmer groups of quadratic twists of elliptic curves with a two-torsion over Q Mathematika, 59):30339, 03 Center for Communications Research, 430 Westerra Court, San Diego, CA 9 address: zdklags@ccrwestorg Department of Mathematics, Stanford University, Building 380, Stanford, CA address: rjlo@stanfordedu