On the Joint Distribution Of Sel φ (E/Q) and Sel φ (E /Q) in Quadratic Twist Families

Transcription

1 On the Joint Distribution Of Sel φ E/Q and Sel φ E /Q in Quadratic Twist Families Daniel Kane and Zev Klagsbrun Introduction Recently, there has a lot of interest in the arithmetic statistics related to the quadratic twist family of a given ellitic curve E/Q Much rogress has been made towards understanding how 2-Selmer ranks are distributed in these families when either EQ[2] Z/2Z Z/2Z or E[2] has an S 3 Galois action In both of these cases, there are exlicit constants α r summing to one such that the roortion of twists with 2-Selmer rank r is given by α r [Kan0], [KMR] Strikingly, this is not true when E has a single rational oint of order two In this case E has a degree two isogeny φ : E E and an associated Selmer grou Sel φ E/Q Work of Xiong shows that if E does not have a cyclic 4-isogeny dened over QE[2], then the distribution of the ranks of Sel φ E d /Q as d varies among the squarefree integers less than X tends to the distribution Max{0, N 0, 2 log log X} as X, where N µ, σ2 is the normal distribution with mean µ and variance σ 2 [Xio3] In this case, Sel φ E/Q mas 2 to into Sel 2 E/Q, showing that for any xed r, at least half of the quadratic twists of E have 2-Selmer rank greater than r This same result can be deduced by studying how dim F2 Sel φ E/Q dim F2 Sel ˆφE /Q varies under quadratic twist, where Sel ˆφE /Q is the Selmer grou associated to the dual isogeny ˆφ of φ In [Kla2], the second author shows that as d varies among the squarefree integers less than X, the distribution of dim F2 Sel φ E d /Q dim F2 Sel ˆφE d /Q tends to N 0, log log X as X 2 This article studies the joint distribution of Sel φ E d /K and Sel ˆφE d /K conditional on a xed value of dim F2 Sel φ E d /K dim F2 Sel ˆφE d /K In articular, we rove the following: Theorem Suose E/Q is an ellitic curve with EQ[2] Z/2Z that does not have a cyclic 4-isogeny dened over QE[2] and u Z Dene SX, u {d squarefree, d X, dim F2 Sel φ E d /Q dim F2 Sel ˆφE d /Q u} Then for any r Max{, u + }, {d SX, u : dim F2 Sel φ E d /Q, dim F2 Sel ˆφE d /Q r, r u} lim X SX, u α r,u,

2 2 DANIEL KANE AND ZEV KLAGSBRUN where α r,u 2 r r u s 2 s r s 2 s r u s 2 s Remark 2 The constants α r,u aear in Cohen and Lenstra's original aer about the distribution of class grous of quadratic elds and α r, u is equal to what is described there as the u-robability that a nite abelian 2-grou has rank r See Theorem 63 in [CL84] 2 φ-descent We begin by dening the Selmer grous Sel φ E/Q and Sel ˆφE /Q and then giving an exlicit descrition of the Selmer grous Sel φ E d /Q and Sel ˆφE d /Q associated to the quadratic twist of an ellitic curve by a squarefree integer d Let E be an ellitic curve with a single oint of order two dened by y 2 x 3 + Ax 2 + Bx and set C EQ[2] 0, 0 There is an isogenous curve E given by a model y 2 x 3 2Ax 2 + A 2 4Bx and an isogeny φ : E E with kernel C There is a Kummer ma via κ : E Q/φEQ Q /Q 2 κx, y { if x, y 0, 0 x if x, y 0, 0 where is the discriminant of E We have similarly dened local mas κ : E Q /φeq Q /Q 2 for every comletion Q of Q which give a commutative diagram for every comletion Q of Q, where the restriction ma Res is the natural ma Q /Q 2 Q /Q 2 E Q/φEQ κ Q /Q 2 Res E Q /φeq κ Q /Q 2 The φ-selmer grou Sel φ E/Q is dened as Sel φ E/Q { c Q /Q 2 : Res c κ E Q /φeq for all laces of Q } Exchanging the roles of E and φ for those of E and the dual isogeny ˆφ : E E yields a ˆφ-Selmer grou Sel ˆφE /Q via the same construction Standard descent technology tells us that the images of κ and κ in Q /Q 2 are dual to each other via the Hilbert symbol airing Further, when is a rime away from 2 where E has good reduction, the images of κ and κ are both equal to the unramied subgrou of Q /Q 2 generated by the image of Z This last fact allows us to describe each of Sel φ E/Q and Sel ˆφE /Q as the intersection of two nite dimensional F 2 subsaces

3 ON THE JOINT DISTRIBUTION OF Sel φ E/Q AND Sel φ E /Q IN QUADRATIC TWIST FAMILIES 3 Let T be the set of laces of Q dividing 2 and dene V Q /Q 2 v T Dene a subsace U V as the image of the T -units Z T in V Next, for each lace T, dene W as W κ E Q /φeq and W κ EQ /φe Q, and set W W v T and W v T W It then follows that Sel φ E/Q U W and Sel ˆφE /Q U W Because the images of κ and κ were dual to each other under the Hilbert symbol airing, it follows that the subsaces W and W and therefore the Selmer grous Sel φ E/Q and Sel ˆφE /Q are orthogonal under the sum of the Hilbert symbol airings over the laces in T 3 Twisting Now suose that d is a squarefree integer relatively rime to 2 Let T d T { d} and observe that T d contains all of the laces of Q above 2 and the laces at which E d has bad reduction We dene V v T d Q /Q 2 and dene U d V d as the image of Z T d and and set in V For each lace in T d, we dene W d κ E d Q /φe d Q W d κ E d Q /φe d Q W v T d W d and W v T d W d We then get that Sel φ E d /Q U d W d and Sel ˆφE d /Q U d W d Moreover, if d, we can exlicitly describe the subsets W d and W d of Q /Q 2 By Lemma 37 in [Kla], the images of κ and κ are given by κ E d Q [2] and κ E d Q [2] resectively Exlicitly, we get if W d, da + 2 B if

4 4 DANIEL KANE AND ZEV KLAGSBRUN and W d if, 2d A + A 2 4B if where is the discriminant of E We note that the sum of the dimensions of W d and W d will always be equal to two We also note that u to squares, we have A 2 4BQ 2 and BQ 2 4 Tamagawa Numbers and Primes [Note: Zev, I'm utting this section here because I assume you are going to need some of this discussion] As seen above the role that an odd rime d has in the comutation of the φ Selmer grou of E d deends a lot on the residue symbols and here behave very dierently, we will consider them each searately Definition If is a rime relative rime to 2 We say that is tye if We say that is tye 2 if We say that is tye 3 if We say that is tye 4 if We note that W d is deendent on the tye of In articular, da + 2 B if is of tye W d if is of tye 2 Q /Q 2 if is of tye 3 Z /Z 2 if is of tye 4, Since the four ossibilities We note that there is a relationshi between dim F2 Sel φ E d dim F2 Sel φ E d and the number of rimes of various tyes Lemma 4 For each L Z/2 Z there exists an integer c L so that for any d n where i are distinct rimes so that the i contain n j rimes of tye j for j 4, and so that d L mod 2, then dim F2 Sel φ E d dim F2 Sel φ E d c L + n 3 n 2 Furthermore, for every u Z there exist d relatively rime to 2 so that dim F2 Sel φ E d dim F2 Sel φ E d u TODO: Zev, can you handle this assuming I got it right? It should also be noted that if d n for i are distinct rimes relatively rime to 2 that an element l of U d can only be in W d if i l for i of tye 2 or 4 Thus if we let U d be the san of then Sel φ E d U d W d {, 2} {q : q } { i : i is of tye or 3}

5 ON THE JOINT DISTRIBUTION OF Sel φ E/Q AND Sel φ E /Q IN QUADRATIC TWIST FAMILIES 5 5 A Markov Chain Aroach Based on the above descrition, it is easy to see that if d n for i distinct rimes relatively rime to 2 that the rank of Sel φ E d /Q deends only on the values of the i modulo 8, whether or not A + 2 B is a square mod i for rimes i for which i i i, and on j There is a natural robability distribution over ossible combinations of such values Namely, the i take random, indeendent congruence classes in Z/8 Z, for i with A+2 B i i j i i, the symbols are randomly + or, and the are random and indeendent u to the constraints imosed by quadratic recirocity In terms of this robability distribution, there are combinatorial means by which the distribution of Selmer ranks can be analyzed In articular, we get the following theorem Theorem 2 In terms of the robability distribution above let α r,u n : P dim F2 Sel φ E d r d n, dim F2 Sel φ E d dim F2 Sel φ E d u, there are at least n/0 i of tye i for all i, then lim α r,un α r,u n as it is clear that the last condition holds with robability aroaching as n TODO: Zev, I need to have this in order to get my result to work 6 Natural Density While Theorem 2 roves a limiting result along the lines of [SD08], it would be convenient to have a result in terms of natural density such as Theorem above We roceed in a manner analogous to that in [Kan0] with a few added comlications due to our slightly dierent context In articular, we attemt to get at the densities of ranks via moments of the actual sizes of the Selmer grous in question on a large subset In articular, letting ωn denote the number of distinct rimes dividing n, we dene: Definition 3 Let S X, u be the set of d satisfying the following roerties: 0 < d X 2 d is squarefree 3 d is relatively rime to 2 4 dim F2 Sel φ E d dim F2 Sel φ E d u 5 ωd log logx < log logx 5/8 6 d has more than ωd/0 rime factors of each tye ie tye through tye 4 Let S X, u be the set of d satisfying only the rst four of these roerties We note that SX, u is a roer subset of S X, u, but that the density of one within the other aroaches as X Lemma 6 For any E and u we have that S X, u lim X S X, u

6 6 DANIEL KANE AND ZEV KLAGSBRUN In order to rove this, we will need the following slight strengthening of [Kan0] Proosition 0: Proosition 62 Let n, N, D be integers with log log N >, and log log N/2 < n < 2 log log N Let G Z/DZ n Let f : G C be a function Let f G g G fg Let f 2 G g G fg 2 Then n N i distinct rimes i,d f,, n f n N i distinct rimes i,d + O f 2 N log log log N D log log N Proof The result follows from the roof of [Kan0] Proosition 0 There is a articularly, nice version of this result when f is symmetric Corollary 4 Let n, N, D be integers with log log N >, and log log N/2 < n < 2 log log N Let G Z/DZ n Let f : G C be a function symmetric in its inuts For d relatively rime to D with ωd n, let fd f,, n, where i are the rime factors of d Let f G d N d squarefree d,d g G fg Let f 2 f,, n f d N d squarefree d,d G g G fg 2 Then + O f 2 N log log log N D log log N Proof This follows immediately from Proosition 62 uon noting that each such d can be written as a roduct,, n in exactly ways We can now rove Lemma 6 Proof We begin by showing that S X, u is reasonably big, in articular, that S X, u ΩX log logx /2 Pick an L modulo D 4 so that it is ossible to have dim F2 Sel φ E d dim F2 Sel φ E d u for some d L mod D By Lemma 4, this will haen whenever d L and n 3 n 2 is equal to some articular constant, U In articular, this imlies that L U Consider the number d X with d squarefree and ωd n for some n log log X < log logx 5/8 with d L mod D and n 3 n 2 U Note that whether or not this holds for such a d deends only on the congruence classes of the rimes dividing d modulo D Thus, if we dene f,, n to be if it holds and 0 otherwise, we may aly Corollary 4 We note that if the i are icked randomly modulo D with robability Θ u log logx /2 that n 3 n 2 U and that at least one rime is not of tye 2 or 3 We note furthermore that uon xing the values of all of the i modulo D excet for one of tye or 4, there is a unique setting of the last rime modulo D so that d L mod D This setting is of tye or 4, since if d is the roduct of the other rimes dividing d, then d n 2+n 3 U L, and thus Therefore f Θ L/d D,u log logx /2, and thus f 2 Θ D,u log logx /4 Hence, alying Corollary 4 for each n with n

7 ON THE JOINT DISTRIBUTION OF Sel φ E/Q AND Sel φ E /Q IN QUADRATIC TWIST FAMILIES 7 log logx < log logx 5/8 we nd that letting SX be the set of d satisfying Proerties,2 and 3 above that X log log log X S X, u Θ D,u log logx /2 SX + O D,u log log X 5/8 We note that by a slight modication of [Kan0] Corollary 8, we can show that the number of d X with d log logx > log logx 5/8 ex Ωlog logx /4 Thus, SX Ω D X, and thus S X, u Ω D,u X log logx /2 We have yet to show that S X, u S X, u is small In articular, by the above, ox log logx /2 of integers less than X fail to satisfy roerty 5 Of the numbers satisfying the Proerties,2,3 and 5, and ωd n, we can aly Corollary 4 to count the number that fail to satisfy Proerty 6 since it is clear that this roerty deends only on the rime factors modulo D It is also clear that f e Ωn and that f 2 f Therefore, we have that the number of d failing Proerty 6 is O D Xe Ωn Summing over all n with n log logx < log logx 5/8 tells us that the number of d satisfying the rst ve roerties but not the sixth is O D X logx ΩD, which is also much smaller than S X, u This comletes the roof Having restricted ourselves, to S X, u, we may now consider the average moments of twists of E by elements of this set In articular, the bulk of our work will be to rove the following roosition: Proosition 63 Let k be a non-negative integer, and u be an integer Then d S lim X,u Sel φe d k 2 kr α X S r,u X, u Note that the limit above is exactly what you would exect if an α r,u -fraction of the d S X, u had dim F2 Sel φ E d r Before roceeding with the roof, we show how Proosition 63 can be used to rove Theorem Proof of Theorem assuming Proosition 63 From Proosition 63 it is not hard to show along the same lines as [Kan0], Section 5 that for any u, r r0 #{d S X, u : dim F2 Sel φ E d r} lim X S X, u [TODO: Should I go into more detail?] By Lemma 6, it immediately follows that #{d S X, u : dim F2 Sel φ E d r} lim X S X, u α r,u α r,u Writing S F, X, u to denote the version of S associated to a erhas dierent ellitic curve, F, we note that SX, u ms E m, X/m, u m 2 Therefore, the set of twists of the form {E d : d SX, u}

8 8 DANIEL KANE AND ZEV KLAGSBRUN can be written as a union m 2 {E m d : d SE m, X, u} Since an α r,u -fraction of the twists in each of these sets have Selmer grous of rank r, the same holds for the union This comletes the roof The rest of this section will be devoted to roving Proosition 63 We begin by further artitioning S X, u further In articular let S X, u, n be the subset of d S X, u so that ωd n We note that as long as n log logx < log logx 5/8 that each d S X, u can be written in exactly ways as d n where i are distinct rimes, relatively rime to 2, so that if n i of them are of tye i, then n i > n/0 and n 3 n 2 U d : u c L, where c L is as given in Lemma 4 for d L mod 2 Thus we have, letting D 8, that d S X,u,n Sel φ E d k,, n distinct rimes d n X, i,d n i of tye i,n 3 n 2 U d n i >n/0 Sel φ E d k We subdivide this sum further by conditioning on the values of each of the i modulo D In articular, we let Cu, n be the set of elements c,, c n Z/DZ n so that if there are n i c's of tye i, then n i > n/0 for all i and n 3 n 2 U d It is easy to verify that so long as n > 0U that Cu, n ΘφD n n /2 In any case, we can now rewrite the above equation as d S X,u,n Sel φ E d k c,,c n Cu,n,, n distinct rimes d n X i c i mod D Sel φ E d k We need to better understand Sel φ E n when i c i mod D If d n, this is the number of x U d so that x W d For each i, let ti,,, ti, n i be the distinct indices so c ti,j of tye i We note that any such x can be written uniquely as n x y i u i t,i n 3 i u n +i t3,i, where y is squarefree and divides D and u u,, u n +n 3 F n +n 3 2 We abbreviate the above as x y u In order for x to be in W d it must be the case that x Wq d for q D and for q i for each i We note that whether or not x Wq d for q D deends only on the congruence classes of i modulo D Thus, if c c,, c n, we let Uc denote the set of airs y, u as above so that y u is in Wq d for all q D It should also be noted that such x are automatically in W d i for i or tye 3 or 4 For i of tye, x W d i if and only if the Hilbert symbol x, da + 2 B i equals For i of tye 2, x W d i if and only if the Hilbert symbol x, i i equals Therefore, we have that if x y u for y, u Uc then if i c i

9 ON THE JOINT DISTRIBUTION OF Sel φ E/Q AND Sel φ E /Q IN QUADRATIC TWIST FAMILIES 9 mod D then w F n +n 2 2 n i x, da + 2 B w i t,i n 2 i x, t2,i w n +i t2,i Therefore we have that if d n with i c i mod D, Sel φ E d 2 n +n 2 Taking a k th ower yields 2 Sel φ E d k 2 kn +n 2 y,u Uc w F n +n 2 2 k y l,u l Uc l w l F n +n 2 2 l k n i n y u, da + 2 B w i t,i { 2 n +n 2 if x W d 0 else n 2 i y u, t2,i w n +i t2,i y l u l, da + 2 n 2 B w i,l t,i y l u l, t2,i w n +i,l i i t2,i Substituting this in to Equation, and interchanging the order of summation, we get that 3 d S X,u,n Sel φ E d k d n X i distinct rimes i c i mod D c,,c n Cu,n k l n 2 kn +n 2 y l,u l Uc w l F n +n 2 2 l k y l u l, da + 2 n 2 B w i,l t,i y l u l, t2,i w n +i,l i i t2,i Dene λ to be the function on rimes relatively rime to D so that λ { A+2 B if 0 otherwise We note that by quadratic recirocity and our knowledge of the i modulo D, we can rewrite the inner summand as zy l, u l, w l i<j n i j ei,j y l,u l,w l i T y l,u l,w l λ i For some zy l, u l, w l, e i,j y l, u l, w l e j,i y l, u l, w l F 2, and T y l, u l, w l {,, n} so that i T y l, u l, w l only if c i is of tye and e i,j or e j,i is for all j of tye 2

10 0 DANIEL KANE AND ZEV KLAGSBRUN We can now remove the conditioning on the congruence classes of the i with an aroriate character sum Namely, we have that: 4 d S X,u,n d n X i distinct rimes Sel φ E d k φd n zy l, u l, w l c,,c n Cu,n n χ i i /c i i i<j n 2 kn +n 2 i j y l,u l Uc w l F n +n 2 2 χ i mod D ei,j y l,u l,w l i T y l,u l,w l λ i The inner summand is now a constant of norm times a roduct of χ i i where the χ i are characters of modulus dividing D, times a roduct of Legendre symbols i j for i < j, times a roduct of terms of the form λ i i j where j t2, Note that this sum is very similar to the sum consider in [Kan0] Proosition 9, and can be bounded by similar means In articular, we have Lemma 64 Let χ i, z zy l, u l, w l, e i,j e i,j y l, u l, w l and T T y l, u l, w l be as above Let m be the number of indices i n so that at least one of the following holds: Then χ i e i,j for some j i d n X i distinct rimes z n χ i i /c i i i<j n i j ei,j λ i O c,d Xc m i T Proof We may assume without loss of generality that c n is of tye 2 Thus, we may merge terms to relace the λ i terms with terms of the form λ i i n The remainder of the roof is now comletely analogous to the roof of Proosition 9 in [Kan0] after noting that [Kan0] Lemma 5 also imlies that A, 2 2 X a b 2 λ 2 OX logxa /8 For given values of χ i,y l, u l, w l, let m be as given in Lemma 64, and let m be the number of indices i so that e i,j for some j i We would like to show the contribution to the sum in Equation 4 with m > 0 is negligible We begin by showing that the sum over terms with m > 0 is negligible In articular, we show that

11 ON THE JOINT DISTRIBUTION OF Sel φ E/Q AND Sel φ E /Q IN QUADRATIC TWIST FAMILIES Lemma 65 φd n c,,c n Cu,n d n X i distinct rimes 2 kn +n 2 zy l, u l, w l y l,u l Uc χ i mod D w l F n +n 2 2 m >0 n χ i i /c i i O D,k,U X logx 2 k i<j n i j ei,j y l,u l,w l i T y l,u l,w l λ i Furthermore, for xed c, the number of collections of u l, v l so that m 0 is O k,u 2 kn +n 2 Proof To understand the size of this sum, we must better understand the number of u l, w l with a given value of m In order to do this, we must better understand the terms e i,j We begin with the following denitions: For i n, let v,i F 2k 2 be given by w i,,, w i,l, u i,,, u il For i n 2, let v 2,i F k 2 be given by w n +i,,, w n +i,l For i n 3, let v 3,i F k 2 be given by u n +i,,, u n +i,l It is now easy to verify that: e t2,i,t3,j v 2,i, v 3,j, and e t,i,t,j φt, i + t, j where φ is the non-degenerate quadratic form φx,, x k, y,, y k k i x iy i Call an index, i between and n active if e i,j for any j i Let S F 2k 2 be the set of elements of the form v,i for i so that t, i is not active Let m i be the number of active indices of tye i Dene S 2, S 3 F k 2 similarly We make the following claim: Claim S 2 k, S 2 S 3 2 k Furthermore, the rst inequality is strict if m > 0 and the second inequality is strict if m 2 or m 3 is bigger than 0 Finally m 4 > 0 only if m > 0 Proof The rst inequality follows from noting that for any v, v 2 S that φv +v 2 0, and thus that S is contained in a translation of a Lagrangian subsace of φ If m > 0 then there is some t, i which is active, and thus v,i S On the other hand, by the above reasoning S {v,i } is contained in a translate of a Lagrangian subsace for φ, imlying that the inequality is strict The second inequality follows from the observation that S 2 is contained in the orthogonal comliment of the san of S 3 If e t2,i,t3,j for some i, j, then S 2 is also orthogonal to v 3,j S 3, from which we infer that either S 3 is strictly contained in it's san, or that S 2 is strictly contained in the orthogonal comlement of S 3, either of which imly that S 2 S 3 < 2 k Finally, note that e t4,i,t4,j is always 0, and thus if m 4 > 0 then some other m i must also be ositive

12 2 DANIEL KANE AND ZEV KLAGSBRUN Note that this claim immediately imlies the second art of the Lemma We are now ready to rove our Proosition We write the sum over u l, w l in a articular way First we roduce an outer sum over the values of m, m 2, m 3, m Next we sum over ossible choices of the sets S, S 2, S 3 consistent with the above claim We note that there are only O k many ossibilities Then we count the number of choices of u l, w l, y l consistent with these choices We note that for each choice of u l, w l there are O k,d ossible valid choices for y l We note that making choices of u l and w l is equivalent to icking values for the v i,j To do this we rst decide which of the indices contribute to m, which can be done in at most n m many ways Next, we ick the values of the vi,j consistently with our choices of S i, which can be done in at most S n S 2 n 2 S 3 n 3 2 km many ways Finally, we note that By Lemma 64, the inner sum is then O k,c,d Xc m Finally, we choose the values of χ i, noting that χ i unless i contributes to m Thus, the χ s can be icked in at most φd m ways Thus the sum in question is at most n φd n 2 kn +n 2 O k,d,c X2 k φdc m S n S 2 n 2 S 3 n 3 m c Cn,u 0<m +m 2 +m 3 m n n2 n O k,d,c,u X2 k φdc m S S2 S 3 m 2 k 2 k 0<m +m 2 +m 3 m 2 k minn,n 2 n O k,d,c,u Xc m m 3 m m 2 k n/0 O c,k,d,u X + c 3n 2 k n/0 O k,d,u X + 2 k 7 n O k,d,u X logx 2 k 5 This comletes our roof Now that we have shown that the contribution from terms with m > 0, we can deal with the sum in question Lemma 66 For n log logx < log logx 5/8, d S X,u,n Furthermore, Sel φ E d k S X, u, n r 2 kr α r,u n + O D,k,U X log log logx log logx 5/4 2 kr α r,u n O D,k,U r Proof By Lemma 65, we know that we can already safely ignore the terms with m > 0 Also by Lemma 65, the number of such terms in the sum over y l, u l, v l is O k,d 2 kn +n 2 Thus, u to negligible error the sum in question without the m > 0 restriction is 5 d n X i distinct rimes f,, n,

13 ON THE JOINT DISTRIBUTION OF Sel φ E/Q AND Sel φ E /Q IN QUADRATIC TWIST FAMILIES 3 where f : Z/DZ n C is some function with f O k,d and f suorted on Cu, n By Proosition 62, this is fg φd n +O X log log logx f suf D log logx φd n g Z/DZ n d n X i distinct rimes The error term here is clearly seen to be O D,k,U X log log logx log logx 5/4 First we note that r 2kr α r,u n is the exectation over i as described in Theorem 2 of the k th moment of the size of the Selmer grou times the indicator function of the event that there are more than n/0 rimes of each tye, given that n 3 n 2 U This is the exectation of the k th ower of Selmer times the indicator function that all n i are more than n/0 and that n 3 n 2 U, divided by the robability that n 3 n 2 U The former exectation can be comuted via a formula similar to Equation 3 in which the inner sum and the is relaced by an exectation In this case, the sum over terms with m > 0 is exactly 0, and thus is equal to an exression analogous to that in Equation 5 Thus, it is easy to see that this sum is exactly fg φd n g Z/DZ n Thus, we have that 2 kr α r,u n fg C n, u r g Z/DZ n Where C n, u is the set of congruence classes with n 3 n 2 U This is clearly O D,k,U By Proosition 62 with f the indicator function of Cu, n, we nd that that S X, u, n is Cu, n #{d X squarefree, d, D, ωd n} + O φd n D,k,U Combining these last two lines with Equation 5 yields the desired result X log log logx log logx 5/4 We are almost ready to rove Proosition 63 First we need one more Lemma Lemma 67 For any k, u lim n 2 kr α r,u n r r 2 kr α r,u Proof Alying the second art of Lemma 66, with one higher k, we nd that 2 kr+r α r,u n O k,d,u and thus r 2 kr α r,u n O D,k,U 2 r

14 4 DANIEL KANE AND ZEV KLAGSBRUN Theorem 2 tells us that 2 kr α r,u n converge to 2 kr α r,u ointwise Our result now follows from the Dominated Convergence Theorem We are now ready to rove Proosition 63 Proof By Lemma 67, for any ɛ > 0 there is an N so that whenever n > N, r 2kr α r,u n r 2kr α r,u < ɛ Take X so that log logx > 2N Then Sel φ E d k d S X,u n log logx <log logx 5/8 d S X,u,n n log logx <log logx 5/8 n log logx <log logx 5/8 Sel φ E d k S X, u, n r S X, u, n r X log log logx 2 kr α r,u n + O D,k,U log logx 5/4 X log log logx 2 kr α r,u + Oɛ + O D,k,U log logx 5/8 S X, u 2 kr α r,u + Oɛ S X, u + O D,k,U S X, u log logx /0 r Thus for X suciently large, d S X,u Sel φe d k Oɛ S X, u This comletes the roof References [CL84] H Cohen and H Lenstra Heuristics on class grous of number elds Number Theory, Noordwijkerhout 983, ages 3362, 984 [Kan0] DM Kane On the ranks of the 2-Selmer grous of twists of a given ellitic curve Prerint available at htt://arxiv org/abs/009365, 200 [Kla] Z Klagsbrun Selmer ranks of quadratic twists of ellitic curves with artial rational two-torsion Prerint available at htt://arxivorg/abs/205408, 20 [Kla2] Z Klagsbrun On the distribution of 2-Selmer ranks within quadratic twist families of ellitic curves with artial rational two-torsion Prerint available at htt://arxivorg/abs/203030, 202 [KMR] Z Klagsbrun, B Mazur, and K Rubin A markov model for selmer ranks in families of twists Prerint available at htt://arxivorg/df/ vdf, 20 [SD08] P Swinnerton-Dyer The eect of twisting on the 2-Selmer grou In Mathematical Proceedings of the Cambridge Philosohical Society, volume 45, ages Cambridge Univ Press, 2008 [Xio3] Maosheng Xiong On Selmer grous of quadratic twists of ellitic curve a two-torsion over q Mathematika, 203