2^\infty-Selmer groups, 2^\infty-class groups, and Goldfeld's conjecture
aa r X i v : . [ m a t h . N T ] J un ∞ -SELMER GROUPS, ∞ -CLASS GROUPS, AND GOLDFELD’SCONJECTURE ALEXANDER SMITHA
BSTRACT . We prove that the ∞ -class groups of the imaginary qua-dratic fields have the distribution predicted by the Cohen-Lenstra heuris-tic. Given an elliptic curve E/ Q with full rational -torsion and norational cyclic subgroup of order four, we analogously prove that the ∞ -Selmer groups of the quadratic twists of E have distribution as pre-dicted by Delaunay’s heuristic. In particular, among the twists E ( d ) with | d | < N , the number of curves with rank at least two is o ( N ) . C ONTENTS
1. Introduction 22. Algebraic tools 82.1. Sets of governing expansions 82.2. Sets of raw expansions 162.3. Raw expansions for class groups 192.4. Raw expansions for Selmer Groups 233. Additive-Restrictive systems 283.1. Additive-Restrictive systems for class and Selmer groups 304. Ramsey-Theoretic results 365. Prime divisors as a Poisson point process 425.1. Proof of Lemma 5.1 445.2. Proof of Proposition 5.2 465.3. Proof of Theorem 5.4 516. Equidistribution of Legendre symbols 566.1. Equidistribution via Chebotarev and the Large Sieve 606.2. Combinatorial approaches to small primes 656.3. Boxes of integers 677. Proofs of the main theorems 73References 83 D EPARTMENT OF M ATHEMATICS , H
ARVARD U NIVERSITY
E-mail address : [email protected] .I would like to thank Noam Elkies, Jordan Ellenberg, and Melanie Matchett Wood for theirsupport over the course of this project. I would also like to thank Brian Conrad, DorianGoldfeld, Ye Tian, and David Yang for their comments on prior versions of this paper. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS
1. I
NTRODUCTION
Recall that a positive integer is called a congruent number if it is the areaof some right triangle with rational side lengths. This paper was born as aneventually-successful attempt to prove the following theorem.
Theorem.
The set of congruent numbers equal to , , or mod has zeronatural density in N . Previously, the best upper bound on this density was due to Heath-Brown,who found the limit as N approaches ∞ of the distribution of -Selmergroups in the quadratic twist family E ( d ) : dy = x − x with | d | < N. He found that, among d equal to , , or mod , the minimal -Selmerrank of two was attained in the limit by about . of curves [12]. It iswell known that d is congruent if and only if E ( d ) has positive rank, and wealways have an inequalityrank ( E ( d ) ) ≤ − Sel E ( d ) , with dim denoting the dimension of the -Selmer group as an F -vectorspace. Then, from his Selmer group computations, Heath-Brown couldshow that at most . of d equal to , , or mod were congruent.The work of Heath-Brown was extended by Kane to families of the form E ( d ) : dy = x ( x + a )( x + b ) with | d | < N, where a and b are distinct nonzero rational numbers; that is to say, Kane as-sumed that E/ Q had full rational -torsion. With the additional hypothesisthat E had no rational cyclic subgroup of order four, Kane proved that thelimit of the distribution of the -Selmer groups in this family approachedthe distribution found by Heath-Brown [15]. With these results, Kane wasable to find upper bounds on the density of twists in this family with rank ≥ .Now, the -Selmer rank provides a coarse upper bound for the rank ofan elliptic curve. This bound can be improved by instead considering theranks of the k -Selmer groups, with larger k giving finer estimates for therank of the elliptic curve. In fact, if the Shafarevich-Tate conjecture is true,we expect that the Z -Selmer corankcorank Sel ∞ E = lim k →∞ dim 2 k − Sel k E should equal the rank of E for any elliptic curve E/ Q .With this in mind, the first goal of the paper is to find the distribution ofthe k -Selmer groups in the quadratic twist family of a curve E/ Q . To writedown the result, we will need some notation: ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 3 Definition.
For n ≥ j ≥ , take P Alt ( j | n ) to be the probability that aa uniformly selected alternating n × n matrix with coefficients in F haskernel of rank exactly j .In addition, given an elliptic curve E/ Q and integers n ≥ and k ≥ ,take R E, k ( n ) to be the set of squarefree d for which dim 2 k − Sel k ( E ( d ) ) = ( n + 2 if k = 1 n otherwise. Theorem 1.1.
Take E/ Q to be an elliptic curve with full rational -torsion.Assume that E has no rational cyclic subgroup of order four. Choose m ≥ , and choose any sequence of nonnegative integers n ≥ n ≥ · · · ≥ n m +1 for which the n k are either all even or all odd. Then lim N →∞ (cid:12)(cid:12) { , . . . , N } ∩ R E, ( n ) ∩ · · · ∩ R E, m ( n m ) ∩ R E, m +1 ( n m +1 ) (cid:12)(cid:12)(cid:12)(cid:12) { , . . . , N } ∩ R E, ( n ) ∩ · · · ∩ R E, m ( n m ) (cid:12)(cid:12) = P Alt ( n m +1 | n m ) . Together with Kane’s results, this Markov-chain behavior establishes thatthe ∞ -Selmer groups of the twists of such an elliptic curve E/ Q have thedistribution predicted by Delaunay [7] and Bhargava et al. [2]. This theoremalso gives us very fine control on the rank of elliptic curves in this family. Corollary 1.2.
Take E/ Q to be an elliptic curve with full rational -torsion.Assume that E has no rational cyclic subgroup of order four. Then, for any N > , we have (cid:12)(cid:12)(cid:8) ≤ d ≤ N : corank Sel ∞ E ( d ) ≥ (cid:9)(cid:12)(cid:12) = o ( N ) . By applying this corollary to E the curve y = x − x , we derive the zerodensity result that opened this paper. More generally, recall that Goldfeld’sconjecture states that, given an elliptic curve E/ Q , of the quadratictwists of E have analytic rank , have analytic rank , and havehigher analytic rank [10]. From global root number calculations, we knowthat of the twists will have even Z -Selmer corank, and have odd Z -Selmer corank. In light of this, we have the following. Corollary 1.3.
Take E/ Q to be an elliptic curve with full rational -torsion.Assume that E has no rational cyclic subgroup of order four. Then, if theBirch and Swinnerton-Dyer conjecture holds for the set of twists of E , Gold-feld’s conjecture holds for E . We will prove an explicit form of Theorem 1.1 as Theorem 7.1 and anexplicit form of Corollary 1.2 as Corollary 7.2. Neither of these results islikely to be sharp, with Corollary 7.2 particularly egregious in this manner. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS As detailed in [20], most heuristics for ranks of elliptic curves suggest that,for any ǫ > and any elliptic curve E , there is some N so, for N > N ,we have (cid:12)(cid:12)(cid:8) ≤ d ≤ N : rank ( E ( d ) ) ≥ (cid:9)(cid:12)(cid:12) < N / ǫ . We instead prove that, for any elliptic curve E/ Q as in Corollary 1.2 andany c < log 2log 6 , there is some N so, for N > N , we have (cid:12)(cid:12)(cid:8) ≤ d ≤ N : rank ( E ( d ) ) ≥ (cid:9)(cid:12)(cid:12) < N (log log log log log N ) c . If we assume the grand Riemann hypothesis, we can remove about threeof these logarithms. The remaining two logarithms come from the use ofRamsey theory in our arguments, and are likely unremovable without a newproof strategy.Our final main result concerns the class groups of quadratic fields. Fora positive integer k , the k -Selmer groups of quadratic twists of an ellipticcurve and the k +1 -class groups of imaginary quadratic fields are analogousfamilies of objects. The strength of this analogy can be seen in the work ofFouvry and Kl¨uners in [8]. By modifying the strategy used by Heath-Brownto find the distribution of -Selmer groups, this pair found that the distribu-tion of the -class groups in the family of imaginary quadratic fields wasconsistent with Gerth’s extension of the Cohen-Lenstra heuristic to p = 2 [9, 3]. Similarly, by modifying our approach to k -Selmer groups in The-orem 1.1, we can find the distribution of k +1 -class groups in the familyof imaginary quadratic fields. We start by introducing the notation we willuse. Definition.
For n ≥ j ≥ , take P Mat ( j | n ) to be the probability that auniformly selected n × n matrix with coefficients in F has kernel of rankexactly j .In addition, given k ≥ and n ≥ , take R Im , k ( n ) to be the set ofsquarefree d for which dim 2 k − Cl Q (cid:0) √− d (cid:1) [2 k ] = n. Theorem 1.4.
Take m ≥ , and choose any sequence of nonnegative inte-gers n ≥ n ≥ · · · ≥ n m +1 . Then lim N →∞ (cid:12)(cid:12) { , . . . , N } ∩ R Im , ( n ) ∩ · · · ∩ R Im , m ( n m ) ∩ R Im , m +1 ( n m +1 ) (cid:12)(cid:12)(cid:12)(cid:12) { , . . . , N } ∩ R Im , ( n ) ∩ · · · ∩ R Im , m ( n m ) (cid:12)(cid:12) = P Mat ( n m +1 | n m ) . This Markov-chain behavior is consistent with the Cohen-Lenstra heuris-tic and represents the third major result towards this heuristic for quadratic ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 5 fields, following the result of Fouvry-Kl¨uners for -torsion and the substan-tially earlier results of Davenport-Heilbronn for -torsion [6].Theorem 1.1 and 1.4 are generalizations of prior conditional results from[24], a paper by the author on -class groups and -Selmer groups. That pa-per was based on a result of R´edei that, for any negative squarefree d , thereis some number field M so, for any odd prime p not dividing d , the -classrank of Q ( √ dp ) is determined by the splitting behavior of p in M/ Q [21].In [4], it was conjectured that this result can be extended to higher classgroups. More specifically, it was conjectured that, for any k > and anysquarefree negative d , the structure of Cl Q ( √ dp )[2 k ] could be determinedfrom the splitting behavior of p in some governing field M/ Q determinedfrom d and k .For k > , this conjecture is likely to be false for all d , with compellingevidence found by Milovic in [17]. However, the concept of a governingfield remains useful for k > , as we can use splitting behavior to determinesome relative information about class groups. In particular, for d negativesquarefree, and for { p , p } , . . . , { p m , p m } some sequence of pairs ofdistinct primes, we can sometimes derive the m -class structure of Q d / Y i ≤ m p / i ! from the m -class structures of the m − fields Q d / Y i ≤ m p / if ( i ) ! with f ∈ F { ,...,m } − { } together with the splitting behavior of p and p in a governing field de-termined from the primes p , p , . . . , p m , p m . Thinking of the quadraticfields as lying at the vertices of some m -dimensional cube, we can rephrasethis result as finding the m -class structure at one vertex of the cube fromthe m -class structures at all the other vertices. We have a similar result forpredicting m -Selmer structure at one vertex of an m + 1 dimensional cubefrom the m -Selmer structures at all the other vertices in the cube.Making this relative governing field idea concrete takes up most of Sec-tions 2 and 3. The governing fields we need are constructed as the fields ofdefinition of certain Galois cochains that we call governing expansions . InSection 2.1, we prove the existence and basic properties of these cochains.Next, in Section 2.2, we study sets of Galois cocycles on cubes of qua-dratic twists of a given Galois module. We find that the na¨ıve way ofsumming this set of cocycles gives a cocycle under one set of hypothe-ses (Proposition 2.5) and gives a governing expansion under another set ofhypotheses (Proposition 2.6). These two simple propositions are the most ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS fundamental results in this paper. On the class side, the results are usedin Section 2.3 to prove Theorem 2.8, which allows us to control the Artinpairing on the k -class groups. On the Selmer side, the results are used inSection 2.4 to prove Theorem 2.9, which allows us to control the Cassels-Tate pairing on the k -Selmer groups.The conditions under which we can use either of these theorems are ex-tremely limited. In Section 3, we axiomatize some of the conditions with astructure that we denote an additive-restrictive system . Over the course ofthis technical section, we find additive-restrictive systems that handle setsof governing expansions and systems that handle sets of cocycles comingfrom either class structure or Selmer structure. Using this new terminology,we reduce Theorems 2.8 and 2.9 to Proposition 3.6.We have almost no control on the shape of these additive-restrictive sys-tems. That said, as we will show in Proposition 3.2, we do have somecontrol on their sizes. We can then prove the equidistribution results wewant on these arbitrarily-shaped additive-restrictive systems using Ramseytheory. This is the main goal of Section 4, a section that cumulates in theproof of Proposition 4.4. As a first step towards this proposition, we provethe following lovely result: Proposition.
Take d ≥ to be an integer, take − d − > δ > , and take X , . . . , X d to be finite sets with cardinality at least n > . Suppose that Y is a subset of X = X × · · · × X d of cardinality at least δ · | X | . Then, forany positive r satisfying r ≤ (cid:18) log n δ − (cid:19) / ( d − , there exists a choice of sets Z , . . . , Z d , each of cardinality r , such that Z × · · · × Z d ⊆ Y. This bound on r can be shown to be sharp up to a change of constantusing the probabilistic method.Through Sections 3 and 4, we are working with a grid of quadratic twists.We cannot explicitly find the k -Selmer structure or k +1 -class structure atany point in this grid. At the same time, under the condition that the corre-sponding grid of governing Artin symbols behaves generically, the resultsof these two sections let us say that the k -Selmer groups and k +1 -classgroups have the distribution we expect anyways.The next goal is to find a situation where this grid of Artin symbols usu-ally behaves generically. If we had assumed the grand Riemann hypothesis,this step would be straightforward. As we are not using this hypothesis, ittakes a three-logarithm detour to deal with this grid of Artin symbols. Tounderstand the issue, choose some large N , and choose n uniformly among ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 7 the positive squarefree integers less than n . Write p < · · · < p r for thesequence of prime factors of n . Choose k < r , and take M to be a num-ber field of discriminant near p · · · · · p k . Suppose we wish to control thesplitting of p k +1 , . . . , p r in M as we vary these primes in small intervals X k +1 , . . . , X r . Using the strongest form of unconditional Chebotarev avail-able to us (Proposition 6.5), we find that we can only do this if the gap log log p k +1 − log log p k is unusually large. We call tuples with such a sufficiently large gap extrava-gantly spaced . By carefully analyzing the Poisson point process that modelsprime divisors, we are able to show that most tuples are (just barely!) ex-travagantly spaced. This is the main focus of Section 5, with our main resultbeing Theorem 5.4.To avoid thinking about more complicated objects, we usually understand k -Selmer structure via the natural inclusion k − Sel k E ⊆ Sel E (cid:14) E [2] for k > , and we think about k +1 -class structure similarly. For this to work, we needgood ways to control -Selmer and -class structures. The distributional re-sults of Heath-Brown, Fouvry and Kl¨uners, and Kane are based on momentcalculations for these groups and are difficult to use for more specializedsets of integers. As an alternative, we take the following tack. Choosing p < · · · < p r as in the previous paragraph, we can define an r × r ma-trix M whose off-diagonal coefficient M ij is given by the Legendre symbol (cid:16) p i p j (cid:17) . With some extra quadratic-residue information, this matrix can beused to determine the -Selmer structure of E ( d ) ; our main aim of Section 6is to prove that the matrix M is almost equidistributed among all possibili-ties satisfying quadratic reciprocity after some of the p i are permuted. Themajor analytic ingredients for this work are Chebotarev’s density theoremand the large sieve. With these tools and a subtle induction argument, weprove Proposition 6.3, a weak equidistribution result for Legendre symbolmatrices. By accounting for the effect of permuting primes via some basiccombinatorics, we can strengthen this result to the form given in Theorem6.4. In Section 6.3, we make the transition from the set of all integers to cer-tain product spaces of integers that we call boxes . By applying Theorem 6.4to these boxes, we then rederive Kane’s results directly as Corollary 6.11.Finally, in Section 7, we use the results of Sections 5 and 6 to shave the setof integers { , . . . , N } to grids on which the additive-restrictive systems ofSection 3 can be defined, and on which the governing grid of Artin symbolsbehaves generically. Using this, we prove Theorem 7.1 and Corollary 7.2,and these give our main results on the Selmer side. We omit the analogous ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS arguments used for the results on the class side, as no new ideas are neededfor the translation. 2. A LGEBRAIC TOOLS
We will use the following notation: • X will denote a product X = X × X × · · · × X d , where each X i is a finite set. In all our applications, the X i will bedisjoint collections of odd primes. • For a a positive integer, [ a ] will denote the set { , . . . , a } . • For S ⊆ [ d ] , we define X S = Y i ∈ S ( X i × X i ) × Y i ∈ [ d ] − S X i . We use π i to denote projection to the i th factor. • We denote the projections of X i × X i to X i by π and π . • For
S, S ⊆ [ d ] , we take π S : X S → Y i ∈ S ∩ S ( X i × X i ) × Y i ∈ ([ d ] − S ) ∩ S X i to be the natural projection. • Given an element ¯ x ∈ X S and a subset T of S , and writing U = S − T , we define a subset b x ( T ) of X T by (cid:8) ¯ y ∈ X T : π i (¯ y ) ∈ π i (¯ x ) for i ∈ U and π [ d ] − U (¯ y ) = π [ d ] − U (¯ x ) (cid:9) . Fix some algebraic closure Q of Q , and take G Q = Gal ( Q / Q ) . All thenumber fields used in this paper will be Galois extensions of Q inside ofthis algebraic closure.Our main results will fall as a consequence of the Chebotarev densitytheorem. We begin by constructing the sets of governing fields we need todo this.2.1. Sets of governing expansions.
Take X , . . . , X d to be disjoint collec-tions of odd primes, and take X to be their product. Given a subset S of [ d ] and ¯ x ∈ π S ( X S ) , define K (¯ x ) = Y i ∈ S Q (cid:16)p π ( π i (¯ x )) · π ( π i (¯ x i )) (cid:17) where we use the Q symbol to denote a composition of number fields. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 9 For T ⊆ S ⊆ [ d ] and ¯ x ∈ X S , take χ T, ¯ x : G Q → F to be defined by χ T, ¯ x ( σ ) = ( if σ ( √ p i p i ) = −√ p i p i for i ∈ T otherwise,where we have taken ( p i , p i ) to be the coordinate π i (¯ x ) for i ∈ S .From the equation χ T ( στ ) = Y i ∈ T (cid:0) χ { i } ( σ ) + χ { i } ( τ ) (cid:1) = X U ⊆ T χ U ( σ ) χ T − U ( τ ) , we calculate the coboundary of χ T to be(2.1) dχ T ( σ, τ ) = X ∅6 = U ( T χ U ( σ ) · χ T − U ( τ ) . This equation is the backbone for the following definition.
Definition.
Choose S ⊆ [ d ] , choose ¯ x in X S , and choose some homo-morphism φ ∅ : G Q → F . Suppose that we have a set of maps φ S indexed by the subsets of S suchthat we have the coboundary relation(2.2) dφ S ( σ, τ ) = X ∅6 = T ⊆ S χ T, ¯ x ( σ ) · φ S − T ( τ ) for each subset S of S . Then, if φ S is defined, we call it a ( S, ¯ x ) -expansionof φ ∅ .Using (2.1), we can verify that the right hand side of (2.2) has zerocoboundary, so this definition is reasonable.There are two main ways to construct expansions. In this section, we willuse class field theory to construct ( S, ¯ x ) expansions from a set of smallerexpansions. The ramification of these governing expansions can be pre-cisely controlled, so their fields of definitions can be used as governingfields. In Section 2.2, we instead find expansions by summing cocycles rep-resenting Selmer or class elements over the points of b x ( ∅ ) . Such expansionsare less nicely behaved. However, if we calculate enough of these expan-sions within a small space, we can force some of the expansions to equal agoverning expansion. This gives us enough control over the Selmer groupsand class groups to prove our main theorems.We start with the class field theory we will need. We assume that thereader is familiar with the material in [23]. Proposition 2.1.
Take X , . . . , X d to be disjoint collections of odd primes,and write X for their product. Choose a subset S ⊆ [ d ] and a member ¯ x of ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS X S . Take φ ∅ ∈ H ( G Q , F ) . Suppose we have ( S − { i } , ¯ x ) expansions of φ ∅ for all i in S , and take M i to be the field of definition for φ S −{ i } . Write M = K (¯ x ) Y i ∈ S M i . Write ( p i , p i ) = π i (¯ x ) . Suppose that, for all i in S , • p i and p i split completely in the extension M i / Q , and • p i p i is a square at and at all primes where M i / Q is ramified.Then φ ∅ has an ( S, ¯ x ) expansion φ S whose field of definition is unramifiedabove M at all finite places.Proof. We need to check that the cocycle given on the right hand side of(2.2) is zero in H ( G Q , F ) . Call this cocycle ψ . Identifying F with ± and using the exact sequence −−→ ± −−→ Q × −−→ Q × −−→ , we find an exact sequence H (cid:0) G Q , Q × (cid:1) → H (cid:0) G Q , F (cid:1) → H (cid:0) G Q , Q × (cid:1) , with the left equality by Hilbert 90. But we know that the map H (cid:0) G Q , Q × (cid:1) → Y v H (cid:0) Gal ( Q v / Q v ) , Q × v (cid:1) is injective, where the product is over all places of Q . Furthermore, theconditions of the proposition imply that inv v ( ψ ) is zero at all places. Then ψ is the image of some -cochain. This cochain corresponds to a F centralextension of M .Write this extension as M ( √ α ) /M . This extension is Galois over Q , soif M ( √ α ) / Q is ramified at some place p other than or ∞ where M/ Q is unramified, we can lose the ramification by multiplying α by p . Now,suppose M/ Q is ramified at p . We see that the local conditions force ψ tobe trivial on Gal ( Q p / Q p ) , so M p ( √ α ) / Q p has Galois group ( Z / Z ) × Gal ( M p / Q p ) if M p ( √ α ) does not equal M p . But the inertia group cannot contain ( Z / Z ) for p = 2 , so M p ( √ α ) /M p is unramified. At p = 2 , we can avoid ramifica-tion by multiplying α by ± or ± . (cid:3) With this out of the way, we can define systems of governing expansions.
Definition 2.2.
Take X , . . . , X d to be disjoint collections of odd primes,and write X for their product. Fix i a ≤ d and Y ∅ ⊆ X . Suppose we choosethe following objects: ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 11 • For each subset S ⊆ [ d ] containing i a , we choose a subset Y S ⊆ X S . • For each S ⊆ [ d ] containing i a and each ¯ x ∈ Y S , we choose acontinuous function φ ¯ x : G Q → F , taking M (¯ x ) to be the minimal field of definition of this φ ¯ x .We call the collection of φ ¯ x a set of governing expansions if the followingcriteria are satisfied.(1) If ¯ x is in Y { i a } , then φ ¯ x = χ i a , ¯ x . (2) If S contains i a and ¯ x is in Y S , then b x ( T ) ⊂ Y T for i a ∈ T ⊆ S or for T = ∅ . Choosing ¯ x S − T arbitrarily in b x ( S − T ) , we have dφ ¯ x ( σ, τ ) = X i a T ⊂ S χ T, ¯ x ( σ ) · φ ¯ x S − T ( τ ) . (3) Suppose ¯ x , ¯ x are in X S , and suppose that (cid:26) π (cid:0) π i (¯ x ) (cid:1) , π (cid:0) π i (¯ x ) (cid:1)(cid:27) = (cid:26) π (cid:0) π i (¯ x ) (cid:1) , π (cid:0) π i (¯ x ) (cid:1)(cid:27) for all i ∈ S . Then, if b x ( ∅ ) ∪ b x ( ∅ ) ⊆ Y ∅ , we have an equivalence ¯ x ∈ Y S ⇐⇒ ¯ x ∈ Y S . If both lie in Y S , then they satisfy φ ¯ x = φ ¯ x . (4) (Additivity) Taking i ∈ S ⊆ [ d ] , suppose ¯ x , ¯ x , ¯ x ∈ Y S satisfy π S −{ i } (¯ x ) = π S −{ i } (¯ x ) = π S −{ i } (¯ x ) and π i (¯ x ) = ( p , p ) , π i (¯ x ) = ( p , p ) , π i (¯ x ) = ( p , p ) . Then φ ¯ x + φ ¯ x = φ ¯ x . (5) If ¯ x ∈ Y S , then M (¯ x ) K (¯ x ) /K (¯ x ) is unramified at all finite places. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS (6) Take ¯ x ∈ X S . Suppose that b x ( ∅ ) ⊆ Y ∅ and that, for all i ∈ S − { i a } , we have b x ( S − { i } ) ⊆ Y S −{ i } . Choosing ¯ x i ∈ b x ( S − { i } ) , suppose further that, for each i ∈ S , π ( π i (¯ x )) and π ( π i (¯ x )) split completely in M (¯ x i ) and π ( π i (¯ x )) π ( π i (¯ x )) is a quadratic residue at and at all primes ramifying in K (¯ x i ) / Q .Then ¯ x ∈ Y S . We will use the letter G to denote a set of governing expansions, writing Y S ( G ) , i a ( G ) , etc. to denote the data associated with G .Additivity reflects a natural tensor product structure present in a set ofgoverning expansions. We can explicitly uncover this linear structure viaiterated commutators. Definition.
Given a set of governing expansions, choose any S ∋ i a andany ¯ x ∈ Y S . Write k = | S | , and define(2.3) β k φ ¯ x ( σ , . . . , σ k ) = φ ¯ x (cid:0) [ σ , [ σ , [ . . . , [ σ k − , σ k ] . . . ]]] (cid:1) Note that φ ¯ x (cid:0) [ σ, τ ] (cid:1) = φ ¯ x ( στ ) + φ ¯ x ( τ σ ) + dφ ¯ x (cid:0) [ σ, τ ] , τ σ (cid:1) . From (2.2), we see that the coboundary above is zero since each χ T hasabelian field of definition. But we have φ ¯ x ( στ ) + φ ¯ x ( τ σ ) = dφ ¯ x ( σ, τ ) + dφ ¯ x ( τ, σ ) . Take Bij ∗ ([ k ] , S ) to be the set of bijective maps g from [ k ] to S such thateither g ( k − or g ( k ) equals i a . Then, in light of the above equation and(2.2), we can calculate β k φ ¯ x ( σ , . . . , σ k ) = X g ∈ Bij ∗ ([ k ] , S ) Y i ≤ k χ g ( i ) , ¯ x ( σ i ) . Write K ( X ) = Y ¯ x ∈ X [ d ] K (¯ x ) , and write V for the F vector space Gal ( K ( X ) / Q ) . Then β k can be consid-ered as a linear operator from the space generated by the φ ¯ x to O i ∈ S Hom ( V, F ) . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 13 If ¯ x , ¯ x , and ¯ x are as in part (4) of the definition above, we see that(2.4) β k φ ¯ x + β k φ ¯ x = β k φ ¯ x . This turns out to be a good way to force additivity on our set of governingexpansions.
Proposition 2.3.
For any choice of a product X of disjoint sets X , . . . , X d of odd primes, for any choice of i a ∈ [ d ] , and for any choice of Y ∅ , thereis a set of governing expansions G defined on X with i a ( G ) = i a and Y ∅ ( G ) = Y ∅ .Proof. We actually will prove something slightly stronger. Take W S to bethe space generated by the φ ¯ x for ¯ x ∈ Y S . In light of (2.4), we can proveadditivity by showing that we can choose the φ ¯ x so that β | S | is injective on W S .This is clear for S = { i a } . Now, suppose we had found φ ¯ y satisfying thisproperty for all ¯ y ∈ Y T and proper subsets T of S that contain i a , and wewish to prove the result for S . In light of Proposition 2.1, we certainly canfind expansions φ ¯ x for each ¯ x ∈ Y S . The only question is whether we canmake the map from W S injective.Take M to be the narrow Hilbert class field of K ( X ) . For each prime p that ramifies in K ( X ) / Q , choose P to be a prime of M over p , and take σ p to be the nontrivial inertia element corresponding to P . By adjusting the φ ¯ x , ¯ x ∈ Y S by the quadratic character χ ± p as needed, we can force φ ¯ x ( σ p ) = 0 ;we choose the sign for ± p to keep φ ¯ x unramified at .So suppose the set of φ ¯ x are zero at each σ p . We claim that this is suffi-cient for β | S | to be injective on W S .Take k = | S | . Suppose the map were not injective, with β k φ = 0 for φ = X j c j φ ¯ x j for some set of constants c j . We have β k φ ( σ , . . . , σ k ) = dφ ( σ , τ ) + dφ ( τ, σ ) where τ is the iterated commutator of σ , . . . , σ k .This splits into two cases depending on k . If k > , we always have that dφ ( τ, σ ) is zero, so β k φ = dφ ( σ , τ ) = X j X i ∈ S −{ i a } c j χ i, ¯ x j ( σ ) · φ ¯ x j , S −{ i } ( τ ) . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Using the independence of the sets of characters corresponding to each i ,we get that X j c j χ i, ¯ x j ( σ ) · φ ¯ x j , S −{ i } (cid:0) [ σ , [ . . . , [ σ k − , σ k ] . . . ]] (cid:1) = 0 for any choice of i ∈ S − { i a } . This can be reexpressed as X j c j χ i, ¯ x j ( σ ) · β k − φ ¯ x j , S −{ i } ( σ , . . . , σ k ) = 0 . By the induction hypothesis, we thus have X j c j χ i, ¯ x j ( σ ) · φ ¯ x j , S −{ i } ( τ ) = 0 for any choice of σ and τ . Taking coboundaries then gives X j c j X i ∈ T ⊆ S −{ i a } χ T, ¯ x j ( σ ) · φ ¯ x j , S − T ( τ ) = 0 . Again using the independence of these characters, we find X j c j χ T, ¯ x j ( σ ) · φ ¯ x j , S − T ( τ ) = 0 for any T ⊆ S − { i a } . Adding these together then gives that dφ = 0 .On the other hand, if k = 2 , we have τ = σ , and we still find dφ = 0 .Then X j c j φ ¯ x j is a Galois cocycle and hence corresponds to a quadratic extension of Q .But, from the φ ( σ p ) = 0 conditions, we find that it is unramified at all finiteprimes, so φ = 0 . Then β | S | is injective on W S , and this set of governingextensions is additive at level S . This gives the proposition by induction. (cid:3) There is one final result we need for sets of governing expansions. Toprove our main theorems, we apply Chebotarev’s density theorem to thecomposition of fields M (¯ x ) over a special set of ¯ x ∈ X S . For this reason,it is essential to have a sense of when a given field M (¯ x ) is not containedin the composition of all the other M (¯ x ) . The next proposition gives us theindependence result we need. Proposition 2.4.
Take X = X × · · · × X d to be a product of disjoint setsof odd primes, and take i a ∈ S ⊆ [ d ] . For i ∈ S , take Z i ⊆ X i × X i ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 15 to be the set of edges of some ordered tree in X i . Suppose we have a set ofgoverning expansions on X such that π S (cid:0) Y S (cid:1) ⊇ Z = Y i ∈ S Z i . For z in the latter product, choose ¯ x ( z ) so π S (¯ x ( z )) = z . Then, for anychoice of z ∈ Q i ∈ S Z i , and writing ¯ z = ¯ x ( z ) , we have that M S ( X ) Y z = z ∈ Z M (¯ x ( z )) does not contain the field M (¯ z ) , where M S ( X ) is as in the proof of theProposition 2.3.Proof. We need to check that dφ ¯ z is not in the span of the other dφ ¯ x ( z ) inside of H (cid:0) Gal ( M S ( X ) / Q ) , F (cid:1) . Since Gal ( M S ( X ) / Q ) has nilpotence degree | S | − , we see that the map β ( ψ )( σ , . . . , σ k )= ψ (cid:0) σ , [ σ , [ . . . , [ σ k − , σ k ] . . . ]] (cid:1) + ψ (cid:0) [ σ , [ . . . , [ σ k − , σ k ] . . . ]] , σ (cid:1) is trivial on any -coboundary, where we have taken k = | S | . That is, β isdefined on this cohomology group.Then we just need to check that β ( dφ ¯ z ) = β k φ ¯ z is not in the span of the other β k φ ¯ z . Taking K i ( X ) = Y ¯ x ∈ X [ d ] −{ i } K (¯ x ) , we define V i to be the associated F vector space Gal ( K ( X ) /K i ( X )) . Withthis notation, we can consider β | S | φ ¯ z restricted to V i × · · · × V i k − × V i a , where S = { i , . . . , i k − , i a } . Restricted to this domain, we find β | S | φ ¯ z = χ i , ¯ z ⊗ · · · ⊗ χ i k − , ¯ z ⊗ χ i a , ¯ z for any z in Z . The tree assumption implies that, for any i ∈ S , the set (cid:8) χ i, ¯ z : ¯ z ∈ Z (cid:9) is a linearly independent set; that is, once all the duplicate entries are re-moved, the remaining characters are linearly independent. Since each ¯ z corresponds to a distinct tuple of characters, the structure of tensor prod-ucts implies that φ ¯ z must be independent from the other φ ¯ z . This provesthe proposition. (cid:3) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Sets of raw expansions.
Take N to be a G Q module that is isomorphicto some power of Q / Z if the G Q structure is forgotten. Take X , . . . , X d to be disjoint sets of odd primes where N is not ramified, and take X to betheir product. For x ∈ X , we use N ( x ) to denote quadratic twist of N bythe quadratic character of Q (cid:16)p π ( x ) · · · · · π d ( x ) (cid:17) / Q . Note that, for any x ∈ X , N ( x )[2] = N [2] . We write β ( x , x ) for the isomorphism N ( x ) → N ( x ) that preservesGalois structure above K ( x , x ) = Q (cid:0)p π ( x ) π ( x ) . . . π d ( x ) π d ( x ) (cid:1) . Call the associated multiplicative quadratic character χ ( x , x ) For our next definition, we will need that N contains a copy of Z / Z . Definition.
Given N and X as above, takerk : X → Z + ∪ {∞} to be any function. For x in X and k ≤ rk ( x ) an integer, take ψ k ( x ) ∈ C (cid:0) G Q , N ( x )[2 k ] (cid:1) , where C denotes the set of -cocycles of G Q in N ( x )[2 k ] . This data willbe called a set of raw cocycles on X if, for x ∈ X and k < rk ( x ) , we have ψ k +1 ( x ) = ψ k ( x ) . Given S ⊆ [ d ] , we will call R consistent over S if ψ ( x ) = ψ ( x ′ ) whenever x, x ′ ∈ X satisfy π [ d ] − S ( x ) = π [ d ] − S ( x ′ ) Given i a ∈ S , we call our set of raw cocycles i a -consistent over S if thereis some injection of Galois modules ι : F → N [2] such that ψ ( x ) − ψ ( x ′ ) = ι ◦ χ π ia ( x ) π ia ( x ′ ) . We will use the letter R to refer to a set of raw cocycles, writing rk ( R ) and ψ k ( R , x ) for the data associated to R . We will also use the notation i a ( R ) and ι ( R ) for the corresponding data of i a -consistent R .The goal of this subsection is to compare sets of raw cocycles with setsof governing expansions. In our final results for sets of raw cocycles, wewill be interested in situations where i a ( R ) = i a ( G ) . First, we look at the simpler situation where i a plays no role. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 17 Definition.
Take R to be a set of raw cocycles on X , and take S to be anonempty subset of [ d ] , and take ¯ x ∈ X S . Take ¯ x ∈ X S , and suppose thatrk ( R )( x ) ≥ | S | for x ∈ b x ( ∅ ) . Choosing x ∈ b x ( ∅ ) , we then define ψ ( R , ¯ x ) = X x ∈ b x ( ∅ ) β ( x, x ) ◦ ψ | S | ( R , x ) . Supposing that R is consistent over S , we say that R is minimal at ¯ x if ψ ( R , ¯ x ) = 0 .We need one crucial calculation. For i ∈ S , take H i to be the subset of x ∈ b x ( ∅ ) with π i ( x ) = π i ( x ) . For T ⊆ S , take H T = \ i ∈ T H i . We can write any σ in Gal ( K (¯ x ) / Q ) in the form. σ = X i ∈ T σ σ i , where σ i is the unique nontrivial element of this Galois group that fixes q π ( π j (¯ x )) π ( π j (¯ x )) for all j = i in S , and where T σ is a subset of S . We claim that, for x in b x ( ∅ ) ,(2.5) X ∅6 = T ⊆ T σ H T ∋ x ( − | T |− = ( if χ ( x, x )( σ ) = − otherwise. . For take T x to be the maximal T so that x ∈ H T . Then the right hand sideis one if | T x ∩ T σ | is odd. Calling this cardinality m , the left hand side is X ∅6 = T ⊆ T σ ∩ T x ( − | T |− = m X k =1 (cid:18) mk (cid:19) ( − k − = 12 (cid:0) − ( − m (cid:1) , via the binomial theorem. This equals the right hand side, establishing (2.5). ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Proposition 2.5.
Take R to be a set of raw cocycles on X . Take S to be anonempty subset of [ d ] where R is consistent. Choose some ¯ x ∈ X S where ψ (¯ x ) = ψ ( R , ¯ x ) is defined.Suppose that, for any T ( S and any ¯ x ∈ b x ( T ) , R is minimal at ¯ x .Then ψ (¯ x ) maps into N [2] , and its coboundary is zero. That is, it corre-sponds to an element in C ( G Q , N [2]) .Proof. It is clear that ψ (¯ x ) is zero by the minimality assumptions, so ψ (¯ x ) maps into N ( x )[2] . To show it is a coboundary, we calculate d (cid:0) β ( x, x ) ◦ ψ | S | ( x ) (cid:1) ( σ, τ )= (cid:0) σβ ( x, x ) − β ( x, x ) σ (cid:1) ◦ ψ | S | ( x )( τ )= ( σβ ( x, x ) ◦ ψ | S |− ( x )( τ ) if χ ( x , x )( σ ) = − otherwise.Then dψ (¯ x )( σ, τ ) = X χ ( x,x )( σ )= − σβ ( x, x ) ◦ ψ | S |− ( x )( τ ) Taking T σ as above, we can use (2.5) to write this sum as(2.6) X ∅6 = T ⊆ T σ ( − | T |− σ X x ∈ H T β ( x, x ) ◦ ψ | S |−| T | ( x )( τ ) ! . But the inner sum is zero for each T by the minimality hypothesis, so thecoboundary is zero. This gives the proposition (cid:3) In light of the coboundary calculation of this proposition, we see that, if ψ is minimal at ¯ x ∈ X S , it is minimal at any ¯ y ∈ b x ( T ) for any T ⊆ S .We now start comparing sets of governing expansions with sets of rawcocycles. Definition.
Take R to be a set of raw cocycles on X , and take G to be aset of governing expansions on X . Choose a subset S of [ d ] , and choose ¯ x ∈ X S .If R is i a ( G ) -consistent over S , we say that R agrees with G at ¯ x if ψ ( R , ¯ x ) and φ ¯ x ( G ) exist and ψ ( R , ¯ x ) − ι ( R ) ◦ φ ¯ x ( G ) = 0 . If S does not contain i a ( G ) and if R is consistent over S , we say that R agrees with G at ¯ x ∈ X S if it is minimal at ¯ x . Proposition 2.6.
Take R to be a set of raw cocycles on X , and take G tobe a set of governing expansions on X . Choose S ⊆ [ d ] so that R is i a ( G ) -consistent over S , and take ¯ x ∈ X S so that ψ ( R , ¯ x ) and φ ¯ x ( G ) both exist. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 19 Suppose that, for any T ( S and any ¯ x ∈ b x ( T ) , R agrees with G at ¯ x .Then ψ (¯ x ) − ι ◦ φ ¯ x ∈ C ( G Q , N [2]) Proof.
As before, ψ (¯ x ) = 0 by the minimality hypotheses, so we just needto check the cocycle condition. We can rewrite (2.6) as dψ (¯ x )( σ, τ ) = X ∅6 = T ⊆ S χ T, ¯ x ( σ ) · X x ∈ H T β ( x, x ) ◦ ψ | S |−| T | ( x )( τ ) ! . From the hypothesis on ¯ x , we find that this equals ι ◦ X i a T ⊆ S χ T, ¯ x ( σ ) · φ ¯ x S − T ( τ ) = ι ◦ dφ ¯ x ( σ, τ ) . Then ψ (¯ x ) − ι ◦ φ ¯ x has zero coboundary, giving the proposition. (cid:3) Raw expansions for class groups.
Take K/ Q to be an imaginaryquadratic field Q ( √− n ) . Supposing X , . . . , X d are disjoint sets of oddprimes that are unramified in this extension, we define K ( x ) = Q s − n Y i ≤ d π i ( x ) for x ∈ X . Throughout this section, we will presume that, for any i ≤ d ,the value of p mod is the same for all p in X i . We will take N ( x ) to bethe module Q / Z twisted by the quadratic character corresponding to theextension K ( x ) / Q .Choose T a ⊆ [ d ] , and take ∆ a to be a squarefree integer dividing n .From this information, we define a character ψ ( x ) : G Q → N [2] by ψ ( x ) = χ ∆ a + X i ∈ T a χ π i ( x ) . We assume that the field of definition of ψ ( x ) is unramified above K ( x ) for all x . In this case, ψ ( x ) corresponds to an element of the dual classgroup Cl ∨ K ( x )[2] . Proposition 2.7.
Take ψ ( x ) as above, and take K ( x ) ur to be the maximalextension of K ( x ) that is unramified everywhere. Then, for k > , we havethat ψ ( x ) (cid:12)(cid:12) Gal ( Q /K ( x ) ) ∈ k − Cl ∨ K ( x )[2 k ] if and only if, for some ψ k ( x ) ∈ C (cid:0) Gal ( K ( x ) ur / Q ) , N ( x )[2 k ] (cid:1) , we have ψ ( x ) = 2 k − ψ k ( x ) . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Proof.
We see that ψ k ( x ) restricted to the absolute Galois group of K ( x ) is in Cl ∨ K ( x )[2 k ] , so the sufficiency of finding such a ψ k ( x ) is easy. Con-versely, given a map ψ k ( x ) ′ ∈ Cl ∨ K ( x )[2 k ] , we know that the field of definition L of ψ k ( x ) ′ is dihedral over Q , withits unique order k cyclic subgroup corresponding to the intermediate field K ( x ) . To prove the converse, we need to extend the character ψ k ( x ) ′ fromGal ( L/K ( x )) to a cocycle ψ k ( x ) on Gal ( L/ Q ) . Choosing some F in thisGalois group so that we have a coset decompositionGal ( L/ Q ) = Gal ( L/K ( x )) + F · Gal ( L/K ( x )) , and choosing some α ∈ N ( x ) with k − α = ψ ( F ) , we can define such a ψ k ( x ) by setting ψ k ( x )( σ ) = ψ k ( x ) ′ ( σ ) and ψ k ( x )( F · σ ) = α − ψ k ( x ) ′ for all σ ∈ Gal ( L/K ( x )) . We can verify that ψ k ( x ) obeys the cocyclecondition, giving the proposition. (cid:3) In light of this, we defineCl ∨ K ( x )[2 k ] = C (cid:0) Gal ( K ( x ) ur / Q ) , N ( x )[2 k ] (cid:1) . We always have Cl ∨ K ( x )[2 k ] ∼ = Cl ∨ K ( x )[2 k ] ⊕ ( Z / k Z . ) For w a = ( T a , ∆ a ) corresponding to an element of Cl ∨ K ( x )[2 k ] , we define R ( w a ) to be a set of raw cocycles on X so that, for all x ∈ X , ψ ( R , x ) = ψ ( x ) and so that rk ( R )( x ) is the maximal integer k such that ψ ( x ) correspondsto an element of k − Cl ∨ K ( x )[2 k ] , with ψ k ( R , x ) ∈ Cl ∨ K ( x )[2 k ] whenever the left hand side is defined.Now, take w b = ( T b , ∆ b ) , where T b is any subset of [ d ] and ∆ b is apositive squarefree divisor of n (or, if K ( x ) has even discriminant, n ).For any x ∈ X , we define an ideal w b ( x ) of the integers of K ( x ) by Y p | ∆ b P ( p ) · Y i ∈ T b P (cid:0) π i ( x ) (cid:1) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 21 where P ( p ) is the unique prime dividing p in K ( x ) . Taking Cl K ( x )[2] to be the set of ideals with squarefree norm dividing the discriminant of K ( x ) / Q , we see that the mapCl K ( x )[2] → Cl K ( x )[2] is a surjective and has kernel isomorphic to Z / Z . We write k − Cl K ( x )[2 k ] for the preimage of k − Cl K ( x )[2 k ] under this map.For a cocycle ψ k , write L ( ψ k ) for the field of definition of ψ k over K ( x ) .If ψ k ( x ) exists, we see that the Artin symbol (cid:20) L ( ψ k ) /K ( x ) P (cid:21) lies in the order subgroup of Gal ( L ( ψ k ) /K ( x )) at any P dividing thediscriminant of K ( x ) / Q . Identifying this subgroup with Z / Z , we havethe following result. Theorem 2.8.
Take X and n as above, and choose w a and w b as above thatcorrespond to elements of Cl ∨ K ( x )[2] and Cl K ( x )[2] respectively. Choose S ⊆ [ d ] of cardinality at least three, and choose ¯ x ∈ X S . Take G to be aset of governing expansions on X , writing i a = i a ( G ) , and take R to be theset of raw cocycles R ( w a ) . We assume i a is in S .We next assume that w b ( x ) ∈ | S |− Cl K ( x )[2 | S |− ] for all x ∈ b x ( ∅ ) and that there is some i b ∈ S other than i a so that S ∩ T ( w b ) ⊆ { i b } and S ∩ T ( w a ) ⊆ { i a } . Take i ab to equal i a if T ( w b ) does not meet S , otherwise taking i ab to equal i b . For i in S other than i ab , choose ¯ z i in b x ( S − { i } ) .(1) Suppose either that T ( w a ) does not meet S or that T ( w b ) does notmeet S . Assume that, for each i in S other than i ab and each ¯ y in b z i ( S − { i, i ab } ) , we have that R is minimal at ¯ y . Then ψ | S |− ( R , x ) exists for all x ∈ b x ( ∅ ) and X x ∈ b x ( ∅ ) " L (cid:0) ψ | S |− ( R , x ) (cid:1) /K ( x ) w b ( x ) = 0 . (2) Now assume that both T ( w a ) and T ( w b ) meet S . Choose ¯ z in b x ( S −{ i b } ) , and assume that φ ¯ z ( G ) exists. Assume further that, for every i ∈ S other than i b and each ¯ y in b z i ( S − { i, i b } ) , we have that R agrees with G at ¯ y . Then ψ | S |− ( R , x ) exists for all x ∈ b x ( ∅ ) .Furthermore, writing ( p b , p b ) = π i b (¯ x ) , ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS we have X x ∈ b x ( ∅ ) " L (cid:0) ψ | S |− ( R , x ) (cid:1) /K ( x ) w b ( x ) = φ ¯ z ( G ) (cid:0) Frob ( p b ) · Frob ( p b ) (cid:1) . Proof.
For both parts, choose ¯ z in b x ( S − { i ab } ) . For x ∈ b z ( ∅ ) , note that theassumption on w b ( x ) means that " L (cid:0) ψ | S |− ( R , x ) (cid:1) /K ( x ) w b ( x ) depends only on w a , w b , and x , and not on the choice of raw cocycles R .Write b for the norm of the ideal w b ( x ) for any x ∈ b z ( ∅ ) ; our restrictionson T b mean that b does not depend on x . Take p to be a prime divisorof b , and take P to be the prime dividing p in K ( x ) . Write ∆( x ) for thediscriminant of K ( x ) / Q . Assuming ψ | S |− ( R , x ) exists, we can write itlocally at p as χ or χ + χ ∆( x ) , where χ is unramified. We then can say " L (cid:0) ψ | S |− ( R , x ) (cid:1) /K ( x ) P = inv p ( χ ∪ χ b ) . We also have inv p ( χ ∆( x ) ∪ χ b ) = 0 from our requirements on w b , so we find(2.7) " L (cid:0) ψ | S |− ( R , x ) (cid:1) /K ( x ) P = inv p ( ψ | S |− ( x ) ∪ χ b ) . Take x to be the element of b z ( ∅ ) outside of all the sets b z i ( ∅ ) , and write ¯ y i for the element in b z ( S − { i, i ab } ) ∩ b z i ( S − { i, i ab } ) . For the first part,consider ψ = − X x ∈ b z ( ∅ ) −{ x } β ( x, x ) ◦ ψ | S |− ( R , x ) . From Proposition 2.5, we know that this is a cocycle mapping to N ( x ) ,and we find | S |− ψ = ψ ( x ) . From the minimality assumption, we have ψ = − X x ∈ b z ( ∅ ) − b y i ( ∅ ) −{ x } β ( x, x ) ◦ ψ | S |− ( R , x ) for each i ∈ S − { i ab } . From this, we must have that the field of definitionof ψ is unramified at each π i (¯ z i ) for i ∈ S − { i ab } . Then ψ must have ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 23 field of definition unramified above K ( x ) , so some quadratic twist of ψ isunramified above K ( x ) , and ψ | S |− ( R , x ) exists. Then, via (2.7), we find X x ∈ b z ( ∅ ) " L (cid:0) ψ | S |− ( R , x ) (cid:1) /K ( x ) w b ( x ) = X p | b inv p (cid:0) ψ (¯ z ) ∪ χ b (cid:1) . The assumption on w b means the choice of ψ | S |− ( R , x ) does not affectthe value of this sum, so we can take ψ (¯ z ) to be a quadratic character. ByHilbert reciprocity, this equals X p ∤ b inv p (cid:0) ψ (¯ z ) ∪ χ b (cid:1) . But χ b is locally trivial at all primes ramifying in any K ( x ) that do notdivide b , so this is zero. This gives the first part of the theorem.For the second part, we instead take ψ = ι ◦ φ ¯ z − X x ∈ b z ( ∅ ) −{ x } β ( x, x ) ◦ ψ | S |− ( R , x ) . From Proposition 2.6, we see that this is a cocycle mapping to N ( x ) , andwe again find | S |− ψ = ψ ( x ) . Furthermore, we have ψ = ι ◦ φ ¯ y i − X x ∈ b z ( ∅ ) − b y i ( ∅ ) −{ x } β ( x, x ) ◦ ψ | S |− ( R , x ) for each i ∈ S − { i b } , where we are taking φ ¯ y ia = 0 . Then ψ must havefield of definition unramified above K ( x ) . Then ψ | S |− ( R , x ) exists andcan be taken to be a quadratic twist of this ψ . Following the logic of the firstpart, we can ignore the quadratic twist, and we find X x ∈ b z ( ∅ ) " L (cid:0) ψ | S |− ( R , x ) (cid:1) /K ( x ) w b ( x ) = X p | b inv p ( φ ¯ z ∪ χ b ) . Repeating this for the other ¯ z ∈ b x ( S − { i b } ) , we find X x ∈ b x ( ∅ ) " L (cid:0) ψ | S |− ( R , x ) (cid:1) /K ( x ) w b ( x ) = inv p b (cid:0) φ ¯ z ∪ χ p b (cid:1) + inv p b (cid:0) φ ¯ z ∪ χ p b (cid:1) , a synonym for what is claimed. This gives the part and the theorem. (cid:3) Raw expansions for Selmer Groups.
Take E/ Q to be an ellipticcurve with full rational -torsion; that is to say, we have an isomorphismof Galois modules E [2] ∼ = ( Z / Z ) defined over Q . Take N to be the conductor of E , and take X , . . . , X d tobe disjoint sets of odd primes not dividing N . We assume that, for each ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS i ≤ d , the value of p mod is the same for all p ∈ X i . We also assume that E [4] has no order four cyclic subgroup defined over Q .We define E ( x ) = E ( p ····· p d ) where p i = π i ( x ) , with E ( n ) denoting the quadratic twist of E in Q ( √ n ) . The k -Selmer groupof E ( x ) is defined to beSel k ( E ( x )) = ker H (cid:0) G Q , E [2 k ] (cid:1) −→ Y v H (cid:0) Gal ( Q v / Q v ) , E (cid:1)! , the product being over all rational places v . Our main group of study willinstead be the corresponding set of cocycles ker C (cid:0) G Q , E [2 k ] (cid:1) −→ Y v H (cid:0) Gal ( Q v / Q v ) , E (cid:1)! , a group we will denote by Sel k E ( x ) . In our case, if we write Sel k E ( x ) inthe form Im ( E [2]) ⊕ H , we can find a corresponding isomorphismSel k E ( x ) ∼ = ( Z / k Z ) ⊕ H. In particular, Sel E ( x ) equals Sel E ( x ) .Writing E [2] ∼ = ( Z / Z ) e + ( Z / Z ) e , we have a (non-canonical) iso-morphism H ( G Q , E [2]) ∼ = H ( G Q , Z / Z ) × H ( G Q , Z / Z ) . From this, we can write any -Selmer element of E ( x ) as a pair of quadraticcharacters ( χ , χ ) , with the χ i ramified only at bad primes of E ( x ) . As inthe previous section, each χ i corresponds to a choice of divisor ∆ i of N and a choice of subset T i of [ d ] . We will use the letter w to denote a choiceof tuple ( T , T , ∆ , ∆ ) and write w ( x ) for the cocycle in C ( G Q , E ( x )[2]) corresponding to w at x .Taking N ( x ) = E ( x )[2 ∞ ] , we define R ( w ) to be a set of raw cocycles for which ψ ( R , x ) = w ( x ) for all x ∈ X and for which rk ( R )( x ) is maximum of one and the maximal k such that w ( x ) is in k − Sel k E ( x ) , with ψ k ( R , x ) lying in Sel k E ( x ) whenever it is defined for all k ≥ . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 25 There is a natural alternating pairing defined on the Selmer group calledthe Cassels-Tate pairing; Milne’s standard text is our reference for the pair-ing’s construction [16]. Suppose w a ( x ) and w b ( x ) are both -Selmer el-ements at x . Suppose further that rk ( R ( w a ))( x ) is at least k , and take ψ = ψ k ( R ( w a ) , x ) . Take ψ ′ to be any map from G Q to E ( x ) satisfying ψ ′ = ψ , and take ǫ to be a -cochain from G Q to Z / Z satisfying dǫ = dψ ′ ∪ w b ( x ) , where the cup product comes from the natural Weil pairing on E [2] . Finally,for each rational place v , take ψ ◦ v ∈ ker (cid:18) C (cid:0) Gal ( Q v / Q v ) , E [2 k +1 ] (cid:1) → H ( Gal (cid:0) Q v / Q v ) , E (cid:1)(cid:19) satisfying ψ ◦ v = ψ v . Then we can define the Cassels-Tate pairing as (cid:10) ψ k ( R ( w a ) , x ) , w b ( x ) (cid:11) CT = X v inv (cid:0) ( ψ ◦ v − ψ ′ v ) ∪ w b ( x ) + ǫ v (cid:1) . We have one final piece of notation:
Definition.
Given E and X as above, and given S ⊆ [ d ] and ¯ x ∈ X S , wecall ¯ x quadratically consistent if, for all i ∈ S and x ∈ b x ( ∅ ) , we have that π ( π i (¯ x )) π ( π i (¯ x )) is a quadratic residue at and at all the bad primes of E ( x ) besides π i ( x ) .(We also define this for K as in the previous section, where we require theabove to be a quadratic residue at and all ramified primes of K ( x ) / Q besides the π i ( x ) ). Theorem 2.9.
Take E/ Q and X as above. Choose S ⊆ [ d ] of cardinalityat least three, choose some quadratically consistent ¯ x ∈ X S , and choosetuples w a and w b as above corresponding to -Selmer elements of E ( x ) for x ∈ b x ( ∅ ) . Take G to be a set of governing expansions on X , writing i a = i a ( G ) , and take R to be the set of raw cocycles R ( w a ) . We assume i a is in S .We next assume that w b ( x ) ∈ | S |− Sel | S |− E ( x ) for all x ∈ b x ( ∅ ) and that there is some i b ∈ S other than i a so that T ( w a ) ∩ S ⊆ { i a } , T ( w b ) ∩ S ⊆ { i b } , and T ( w a ) ∩ S = T ( w b ) ∩ S = ∅ . Take i ab to equal i a if T ( w b ) does not meet S , otherwise taking i ab to equal i b . For i in S other than i ab , choose ¯ z i in b x ( S − { i } ) . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS (1) Suppose either that T ( w a ) does not meet S or that T ( w b ) does notmeet S . Assume that, for each i in S other than i ab and each ¯ y in b z i ( S − { i, i ab } ) , we have that R is minimal at ¯ y . Then rk ( R )( x ) ≥| S | − for each x ∈ b x ( ∅ ) , and X x ∈ b x ( ∅ ) (cid:10) ψ | S |− ( R , x ) , w b ( x ) (cid:11) CT = 0 . (2) Now assume that T ( w a ) and T ( w b ) both meet S . Choose ¯ z ∈ b x ( S − { i b } ) , and assume that φ ¯ z ( G ) exists. Assume further that, forevery i ∈ S other than i b and each ¯ y in b z i ( S − { i, i b } ) , we have that R agrees with G at ¯ y . Then rk ( R )( x ) ≥ | S | − for each x ∈ b x ( ∅ ) .Furthermore, writing ( p b , p b ) = π i b (¯ x ) , we have X x ∈ b x ( ∅ ) (cid:10) ψ | S |− ( R , x ) , w b ( x ) (cid:11) CT = φ ¯ z ib ( G ) (cid:0) Frob ( p b ) · Frob ( p b ) (cid:1) . Proof.
For both parts, choose ¯ z ∈ b z ( S − { i ab } ) . Take x to be the elementof b z ( ∅ ) outside of all the sets b z i ( ∅ ) , and write ¯ y i for the element in b z ( S −{ i, i ab } ) ∩ b z i ( S − { i, i ab } ) . For the first part, consider ψ = − X x ∈ b z ( ∅ ) − b y i −{ x } β ( x, x ) ◦ ψ | S |− ( R , x ) for i ∈ S − { i ab } . The minimality hypotheses mean that this does notdepend on the choice of i , and Proposition 2.5 implies that it is a cocyclewith values in N ( x ) . We also see that | S |− ψ = w a ( x ) .From quadratic consistency, we know that β ( x, x ) is an isomorphismlocally at ∞ and at each prime that is simultaneously bad for E ( x ) and E ( x ) . Because of this, for each i ∈ S − { i ab } , we can show that ψ obeyslocal conditions at all places other than at the primes in π j (¯ x ) with j ∈ S −{ i } . By varying i , we then find that ψ is a Selmer element, so rk ( R )( x ) ≥| S |− , and we rechoose R at x to make R minimal at all ¯ y ∈ b z ( S −{ i, i ab } ) for each i in S −{ i ab } . Modifying R does not affect the Cassels-Tate pairingby the assumptions on w b , so rechoosing R in this way will not affect thesum we are calculating.Next, choose an ǫ ( x ) and a ψ ′ ( x ) above ψ | S |− ( x ) at all x ∈ b z ( ∅ ) − { x } as in the definition of the Cassels-Tate pairing. Then take ψ ′ ( x ) = − X x ∈ b z ( ∅ ) −{ x } β ( x, x ) ◦ ψ ′ ( x ) and ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 27 ǫ ( x ) = − X x ∈ b z ( ∅ ) −{ x } ǫ ( x ) . Via the coboundary calculation of Proposition 2.5, we find that dǫ ( x ) = dψ ′ ( x ) ∪ w b ( x ) . For the second part, we instead take ψ = ι ◦ φ ¯ y i − X x ∈ b z ( ∅ ) − b y i −{ x } β ( x, x ) ◦ ψ | S |− ( R , x ) . for i ∈ S − { i b } . As before, this is a cocycle with values in N ( x ) thatsatisfies | S |− ψ = w a ( x ) , and the agreement hypotheses mean that it doesnot depend on the choice of i . The φ ¯ y i are locally trivial at the primes of π i (¯ z ) since φ ¯ z exists, so we again find that ψ is a Selmer element for E ( x ) .Then rk ( R )( x ) ≥ | S | − , and we rechoose R at x to make R minimal atall ¯ y ∈ b z ( S − { i, i ab } ) for each i in S − { i ab } .Next, choose an ǫ ( x ) and a ψ ′ ( x ) above ψ | S |− ( x ) at all x ∈ b z ( ∅ ) − { x } as in the definition of the Cassels-Tate pairing, and take ψ ′ ( x ) = − X x ∈ b z ( ∅ ) −{ x } β ( x, x ) ◦ ψ ′ ( x ) and ǫ ( x ) = φ ¯ z ( G ) ∪ χ ( w b )( x ) − X x ∈ b z ( ∅ ) −{ x } ǫ ( x ) . From a coboundary calculation as in Proposition 2.6, we find that dǫ ( x ) = dψ ′ ( x ) ∪ w b ( x ) . For both parts, we know that β ( x, x ) is locally an isomorphism at ∞ and at each prime that is simultaneously bad for E ( x ) and E ( x ) . At suchplaces, and at all simultaneously good places, we set ψ ◦ v ( x ) = − X x ∈ b z ( ∅ ) −{ x } β ( x, x ) ◦ ψ ◦ v ( x ) . At a place of the form π ( π i (¯ z )) or π ( π i (¯ z )) , we instead choose some ver-tex y of each ¯ y ∈ b z ( S − { i } ) to define ψ ◦ v ( y ) = − X x ∈ b y ( ∅ ) −{ y } β ( y, y ) ◦ ψ ◦ v ( y ) . Minimality implies that these choices for ψ ◦ v have the properties required ofthem.For the first part, we see that w b ( x ) is constant on b z ( ∅ ) , while the sums ψ , ψ ◦ v , and ǫ over this all sum to zero. This gives the result on the sum ofCassels-Tate pairings. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS For the second part, we instead find X x ∈ b z ( ∅ ) (cid:10) ψ | S |− ( R , x ) , w b ( x ) (cid:11) CT = X v bad for some E ( x ) inv v (cid:0) φ ¯ z ( G ) ∪ χ ( w b )( x ) (cid:1) . Repeating this for the other ¯ z ∈ b x ( S − { i b } ) then gives the theorem. (cid:3)
3. A
DDITIVE -R ESTRICTIVE SYSTEMS
We now introduce the notion of an additive-restrictive system, a construc-tion that abstracts some of the details for sets of governing expansions andsets of raw cocycles. Take X to be a product of disjoint sets X , . . . , X d ,and take all other notation as at the beginning of Section 2. Definition 3.1. An additive-restrictive system is a sequence of objects ( Y S , Y ◦ S , F S , A S ) indexed by S ⊆ [ d ] so that • For each S ⊆ [ d ] , A S is an abelian group, Y S and Y ◦ S are setssatisfying Y ◦ S ⊆ Y S ⊆ X S , and F S is a function F S : Y S → A S with kernel Y ◦ S . • If S is nonempty, Y S = (cid:8) ¯ x ∈ X S : b x ( T ) ⊂ Y ◦ T for all T ( S (cid:9) . • (Additivity) Choose s ∈ S , and suppose ¯ x , ¯ x , ¯ x are elements of Y S satisfying π [ d ] −{ s } (¯ x ) = π [ d ] −{ s } (¯ x ) = π [ d ] −{ s } (¯ x ) and π s (¯ x ) = ( p , p ) , π s (¯ x ) = ( p , p ) , π s (¯ x ) = ( p , p ) for some p , p , p ∈ X s . Then F S (¯ x ) + F S (¯ x ) = F S (¯ x ) . We will use the letter A to denote an additive-restrictive system, writing Y S ( A ) , F S ( A ) , etc. to denote the data associated with A .The crucial property of additive-restrictive sequences is that we can boundhow quickly the sets Y ◦ S shrink as S increases. We do this with the follow-ing proposition. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 29 Proposition 3.2.
Suppose X = X × · · · × X d is a product of finite sets,and suppose (cid:0) ( Y S , Y ◦ S , F S , A S ) : S ⊂ [ d ] (cid:1) is an additive-restrictive system on X . Write δ for the density of Y ◦∅ in X ,and write | A | for the maximum size of a group A S . Then, for any S ⊆ [ d ] ,the density of Y ◦ S in X S is at least δ | S | | A | − | S | . Proof.
Write δ T for the density of Y ◦ T in X T . For s ∈ S and ¯ x ∈ X S ,define M (¯ x ) = π − d ] −{ s } (cid:0) π [ d ] −{ s } (¯ x ) (cid:1) and consider V = Y ◦ S −{ s } ∩ M (¯ x ) and W = Y ◦ S ∩ M (¯ x ) . We see that W naturally injects into V × V . Furthermore, by the additivityof our additive-restrictive sequence, W takes the form of an equivalencerelation on V . Given ¯ x , ¯ x in V and T a subset of S containing s , write ¯ x ∼ T ¯ x if • ¯ x ∼ T ′ ¯ x for all proper subsets T ′ of T that contain s , and • F T is zero on all elements of b x ( T ) if ¯ x satisfies b x ( S − { s } ) = { ¯ x , ¯ x } . The relation ∼ S splits V into Y s ∈ T ⊆ S (cid:12)(cid:12) A S (cid:12)(cid:12) | S |−| T | ≤ | S |− Y i =0 | A | ( | S |− i ) i = | A | | S |− equivalence classes, and W describes this equivalence relation.Write δ ¯ x for the density of V in X S −{ s } ∩ M (¯ x ) . Then the density of V × V in X S ∩ M (¯ x ) is δ x , and the density of W in this space is then atleast | A | − | S |− · δ x The average of the δ ¯ x is δ S −{ s } , and Y ◦ S is given by the union of the W overall ¯ x , so Cauchy’s inequality gives δ S ≥ | A | − | S |− · δ S −{ s } . Repeating this argument gives δ S ≥ | A | − | S |− (1+ + + + ... ) · δ | S | ∅ = δ | S | | A | − | S | , ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS as claimed. (cid:3) We now turn to constructing additive-restrictive systems. We first do thisfor sets of governing expansions.
Proposition 3.3.
Take G to be a set of governing expansions on a space X = X × · · ·× X d , and choose a nonempty subset S max of [ d ] that contains i a = i a ( G ) . There is then an additive-restrictive system A on X so that, for S ⊆ S max , we have Y S ( A ) = Y S ( G ) . Furthermore, for all S ⊆ [ d ] , this additive restrictive system satisfies (cid:12)(cid:12) A S ( A ) (cid:12)(cid:12) ≤ | S max | +1 . Proof.
We will construct maps F S ( A ) : Y S ( G ) → A S ( A ) as in this propo-sition statement for all S ⊆ S max . We will do this based on the structure of S .First, suppose S is a singleton { j } . Then we take F S (¯ x ) = 0 if and onlyif π ( π j (¯ x )) π ( π j (¯ x )) is a quadratic residue at at and at all primes in π S max −{ j } (¯ x ) . Two bitsencode the residue information at , and one bit encodes it at the remaining | S max | − primes, so we can take Y ◦ S as the kernel of a map to A S ( A ) = ( Z / Z ) | S max | +1 . Suppose instead that | S | > and that S contains i a . Then we want totake F S (¯ x ) = 0 if and only if φ ¯ x ( G ) is a trivial map at the place π i (¯ x ) forall i ∈ S max − S . We know that φ ¯ x is an unramified quadratic character ateach such place, so this information can be encoded at one bit per place in S max − S . Then we can take Y ◦ S as a kernel of a map to A S ( A ) = ( Z / Z ) | S max |−| S | . Outside of these two cases, we take A S to be the trivial group. This definesour additive-restrictive system, and we can verify from the definition ofa set of governing expansions that it satisfies Y S ( A ) = Y S ( G ) for S ⊆ S max . (cid:3) Additive-Restrictive systems for class and Selmer groups.
We nowturn to the construction of an additive-restrictive system which can be usedto control the sizes of class groups and Selmer groups. The construc-tions are similar for Selmer groups and class groups, so we define themat the same time. We first define the data needed to construct the additive-restrictive system. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 31 Definition 3.4.
Take K/ Q to be a quadratic imaginary field, or take E/ Q to be an elliptic curve with full rational two torsion and no rational orderfour cyclic subgroup; the former case will be called the class side , the latterthe Selmer side . We write a basis for E [2] as e , e .Take X , . . . , X d to be disjoints sets of odd primes where K/ Q is unram-ified on the class side, and where E/ Q is good on the Selmer side. Write X for the products of the X i .In the notation of Section 2, we suppose that every ¯ x ∈ X [ d ] is quadrati-cally consistent. We then define the additive-restrictive input as an assign-ment of the following six pieces of interconnected data: (1) A choice of lower pairings: Choose some x ∈ X . On the class side,find the set D ∨ (2) of tuples w a = ( T a , ∆ a ) with w a ( x ) in Cl ∨ K ( x )[4] ;similarly, find the set D (2) of tuples mapping to Cl K ( x )[4] . By quadraticconsistency, we see that these sets do not depend on the choice of x . Write t a for the nontrivial element of the kernel of D ∨ (2) → Cl ∨ K ( x )[4] . and similarly define t b in D (2) . Choose an integer m ≥ and filtrations D ∨ (2) ⊇ D ∨ (3) ⊇ · · · ⊇ D ∨ ( m ) ∋ t a and D (2) ⊇ D (3) ⊇ · · · ⊇ D ( m ) ∋ t b of vector spaces. For k < m , choose a bilinear pairingArt ( k ) : D ∨ ( k ) × D ( k ) → F whose left kernel is D ∨ ( k +1) and whose right kernel is D ( k +1) .On the Selmer side, take D (1) to be the set of tuples mapping to the -Selmer group of E ( x ) . By quadratic consistency, this set does not dependon the choice of x . We write Im ( E [2]) for the image of the -torsion in the -Selmer group, and specifically write t for the image of e in the -Selmergroup. Choose an integer m ≥ and a filtration D (1) ⊇ D (2) ⊇ D (3) ⊇ · · · ⊇ D ( m ) ⊇ Im ( E [2]) of vector spaces. For k < m , choose an alternating pairingCtp ( k ) : D ( k ) × D ( k ) → F whose kernel is D ( k +1) . (2) A choice of basis: On the class side, take n k to be the dimensionof D ∨ ( k ) / h t a i for ≤ k ≤ m . Then choose w a , . . . , w an ∈ D ∨ (2) and w b , . . . , w bn ∈ D (2) so that, for ≤ k ≤ m , the first n k vectors in the firstsequence are a basis for D ∨ ( k ) / h t a i , and the first n k vectors in the secondsequence are a basis for D ( k ) / h t b i . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS On the Selmer side, take n k to be the dimension of D ( k ) / Im ( E [2]) for ≤ k ≤ m . Take w , . . . , w n ∈ D (1) so that, for ≤ k ≤ m , the first n k vectors in this sequence generate D ( k ) / Im ( E [2]) . (3) A choice of variable indices: Choose i b ≤ d , and for j , j ≤ n , choose an element i a ( j , j ) and a set S ( j , j ) containing both i b and i a ( j , j ) . We require these sets to obey different properties on the class andSelmer side.On the class side, for all j ≤ n , we assume that T ( w aj ) and T ( w bj ) donot contain i b . In addition, for all j , j ≤ n , we assume the following: • We assume that S ( j , j ) has cardinality m + 1 . • We assume that S ( j , j ) is disjoint from T ( w aj ) and T ( w bj ) for all j ≤ n other than j and j . • We assume that T ( w bj ) ∩ S ( j , j ) = T ( w aj ) ∩ S ( j , j ) = ∅ ,T ( w aj ) ∩ S ( j , j ) = (cid:8) i a ( j , j ) (cid:9) and S ( j , j ) ⊆ T ( w bj ) ∪ { i b } . On the Selmer side, for all j ≤ n , we assume that T ( w j ) and T ( w j ) do not contain i b . In addition, if j equals j , we assume that S ( j , j ) isthe empty set; and if j is greater than j , we assume that S ( j , j ) equals S ( j , j ) . In addition, for all j < j ≤ n , we assume the following: • We assume that S ( j , j ) has cardinality m + 2 . • We assume that S ( j , j ) is disjoint from T ( w j ) and T ( w j ) for all j ≤ n other than j or j . • We assume that T ( w j ) ∩ S ( j , j ) = T ( w j ) ∩ S ( j , j ) = ∅ ,T ( w j ) ∩ S ( j , j ) = (cid:8) i a ( j , j ) (cid:9) and S ( j , j ) ⊆ T ( w j ) ∪ { i b } . We use the term variable indices to describe the S ( j , j ) because, whenwe actually prove our equidistribution results in Proposition 7.5, we willhave fixed a choice of prime in each X i other than at the i in S ( j , j ) . (4) A choice of raw cocycles: On the class side, we find a set of rawcocycles R ( w aj ) for each j ≤ n , where the set of raw cocycles is as inSection 2.3. On the Selmer side, we find a set of raw cocycles R ( w j ) foreach j ≤ n . (5) A choice of governing expansions: For each distinct i a = i a ( j , j ) marked in the third part of the definition, we take G ( i a ) to be a set of gov-erning expansions over X with i a = i a ( G ( i a )) . For every S of the form S ( j , j ) − { i b , i } for some i ∈ S ( j , j ) other than i a ( j , j ) or i b , and for ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 33 every ¯ x ∈ X S , we assume that the expansion φ ¯ x ( G ( i a ( j , j ))) exists. We also assume it is trivial when restricted to Gal ( Q v / Q v ) for v coming from a certain set of places: • On the class side, we presume that the expansion is trivial at , at ∞ ,at all places in π [ d ] − S (¯ x ) , and at all primes dividing the discriminantof K/ Q . • On the Selmer side, we presume that the expansion is trivial at , at ∞ , at all places in π [ d ] − S (¯ x ) , and at all primes dividing the conductorof E/ Q . (6) A choice of inertia elements: Take M r / Q to be the least numberfield containing L ( ψ k ( R , x )) whenever it exists for any x ∈ X , any R as defined in (4), and any k ≤ m . Take M to be the least number fieldextending M r that also contains the field of definition of each expansionfound in any G ( i a ( j , j )) . M/ Q has ramification degree at most two at anyprime; for each prime p where it ramifies, choose some σ p in Gal ( M/ Q ) sothat { , σ p } is the inertia group of some prime dividing p in M .We will use P to denote an assignment of the additive-restrictive input.To define the additive-restrictive system associated to P , we will first needto understand the role of the choice of inertia elements. Definition.
Suppose we have some choice of additive-restrictive input P ,and choose w in D ∨ (2) on the class side and in D (1) on the Selmer side where R ( w ) has been chosen. Choose S ⊆ [ d ] over which R ( w ) is consistent or i a ( j , j ) -consistent. Suppose ¯ x ∈ X S satisfiesrk ( R ( w ))( x ) ≥ | S | + 1 for all x ∈ b x ( ∅ ) . In the case of consistency, for any i ∈ [ d ] − S , we call R ( w ) acceptablyramified at (¯ x, i ) if X x ∈ b x ( ∅ ) β ( x, x ) ◦ ψ | S | +1 (cid:0) R ( w ) , x (cid:1) ( σ π i (¯ x ) ) = 0 . In the case of i a -consistency, choose i ∈ [ d ] − S , and suppose there issome ¯ z ∈ X S ∪{ i } satisfying ¯ x ∈ b z ( S ) at which φ ¯ z ( G ( i a )) is defined and has π ( π i a (¯ z )) = π ( π i a (¯ z )) . We then call R ( w ) acceptably ramified at (¯ x, i ) if X x ∈ b x ( ∅ ) β ( x, x ) ◦ ψ | S | +1 (cid:0) R ( w ) , x (cid:1) ( σ π i (¯ x ) ) = φ ¯ z ( σ π i (¯ x ) ) . We can now define our additive-restrictive system. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Definition 3.5.
Take P to be an additive-restrictive input as above, andtake all notation as in Definition 3.4. Choose j , j ≤ n ; on the Selmerside, we assume that j is less than j . We will define an additive-restrictivesystem A ( j , j ) = ( Y S , Y ◦ S , F S , A S ) as follows.First, on the class side, take Y ◦∅ ( j , j ) to be the set of x ∈ X on whichthe natural pairings k − Cl ∨ K ( x )[2 k ] × k − Cl K ( x )[2 k ] −→ F agree with the pairings Art ( k ) for each k in the range < k < m . On theSelmer side, take Y ◦∅ to be the set of x on which the natural pairings k − Sel k E ( x ) × k − Sel k E ( x ) −→ F agree with the pairings Ctp ( k ) for each k in the range < k < m .Next, choose S ⊆ [ d ] . We will now define a set Y ◦ S ( j , j ) in Y S ( j , j ) .We have done this already for S empty. Next, if S is not contained in S ( j , j ) , or if S has cardinality greater than | S ( j , j ) |− , we take Y ◦ S ( j , j ) = Y S ( j , j ) .Now suppose S ⊂ S ( j , j ) with cardinality at most | S ( j , j ) | − . Onthe class side, we say that ¯ x ∈ Y S is in Y ◦ S if we have the following: • For j ≤ n | S | +1 other than j , we have that R ( w aj ) is minimal at ¯ x . • We have that R ( w aj ) agrees with G (cid:0) i a ( j , j ) (cid:1) at ¯ x . • For j ≤ n | S | +1 , we have that R ( w aj ) is acceptably ramifeid at (¯ x, i ) for all i in S ( j , j ) − S .On the Selmer side, we say that ¯ x ∈ Y S is in Y ◦ S if we have the following: • For j ≤ n | S | other than j , j , we have that R ( w j ) is minimal at ¯ x . • We have that R ( w j ) agrees with G ( i a ( j , j )) at ¯ x . • If | S | < m , then for j ≤ n | S | other than j , we have that R ( w j ) isacceptably ramified at (¯ x, i ) for all i in S ( j , j ) − S .Suppose ¯ x is in Y S ( j , j ) for some subset S of S ( j , j ) of cardinality atmost | S ( j , j ) | − . Then Proposition 2.5 and 2.6 imply that ψ ( R ( w ) , ¯ x ) or ψ ( R ( w ) , ¯ x ) + φ ¯ x ( G ( i a )) is a cocycle for each w considered in the abovedefinition. Call this cocycle ψ .On the class side, ψ is a quadratic character. The acceptable ramificationconditions prevent ψ from being ramified at any prime in π S (¯ x ) , so it is anunramified character over any K ( x ) with x ∈ b x ( ∅ ) . As rk ( R ( w ))( x ) > | S | for each x ∈ b x ( ∅ ) , and from the local triviality assumptions we made in part(5) of Definition 3.4, we find that ψ is trivial over any K ( x ) at all primeswhere K ( x ) / Q ramifies besides those in π S ( x ) . If ψ is trivial over K ( x ) atall primes in π S ( x ) , we then have that ψ corresponds to an element of D ∨ (2) .We have | S | bits describing the behavior at π S ( x ) , and an element in D ∨ (2) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 35 can be described with n + 1 bits. Finally, acceptable ramification can bedescribed with | S ( j , j ) − S | bits. The conditions of each of these bits areadditive, and we have one set of conditions for each of the n vectors w aj ,so we can take Y ◦ S ( j , j ) to be the kernel of some additive map Y S ( j , j ) → ( Z / Z ) n ( n + m +2) . On the Selmer side, ψ can be thought of as a pair of quadratic characters.The cocycle is unramified at all primes in π S (¯ x ) . Choose some x ∈ b x ( ∅ ) . If ψ is trivial at all the primes in π S ( x ) , we find that ψ is a -Selmer element(again using part (5) of Definition 3.4 if necessary). Each local conditionis described with two bits, and it takes n + 2 bits to describe an elementin D (1) . Finally, acceptable ramification can be described with two bits ateach i in S ( j , j ) − S . This gives a set of n + 2 m + 6 bits to describe theconditions accrued from one w j . Varying j in [ n ] − { j } , we see we cantake Y ◦ S ( j , j ) to be the kernel of some additive map Y S ( j , j ) → ( Z / Z ) ( n − n +2 m +6) . This defines the additive-restrictive sequence associated with ( j , j ) . Proposition 3.6.
Take P to be an additive-restrictive input defined eitherwith respect to an elliptic curve or imaginary quadratic field, and choosesome A ( P )( j , j ) as defined above. Take S = S ( j , j ) , and take ¯ x ∈ X S .Suppose that, for each i in S , there is some ¯ z i ∈ b x ( S − { i } ) so that ¯ z i ∈ Y ◦ S −{ i } (cid:0) A ( P )( j , j ) (cid:1) . Then b x ( ∅ ) is a subset of Y ◦∅ . Furthermore, write ( p b , p b ) for π i b (¯ x ) and i a for i a ( j , j ) . Then, on the class side, we have X x ∈ b x ( ∅ ) " L (cid:0) ψ m ( R ( w aj ) , x ) (cid:1) /K ( x ) w bj ( x ) = ( φ ¯ z ib ( G ( i a )) (cid:0) Frob ( p b ) · Frob ( p b ) (cid:1) if ( j , j ) = ( j , j )0 otherwisefor all j , j ≤ n m . On the Selmer side, we instead have X x ∈ b x ( ∅ ) (cid:10) ψ m ( R ( w j ) , x ) , w j ( x ) (cid:11) CT = ( φ ¯ z ib ( G ( i a )) (cid:0) Frob ( p b ) · Frob ( p b ) (cid:1) if ( j , j ) = ( j , j )0 otherwisefor j < j ≤ n m . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Proof.
Take x to be the element of b x ( ∅ ) not in any ¯ z i . We first need tocheck that the Cassels-Tate pairings or Artin pairings corresponding to x is given by Ctp ( k ) or Art ( k ) for k < m . On the class side, we do this byconsidering the value of these pairings on each ( w aj , w bj ) or, for j = j ,on ( w aj , t b + w bj ) . The value of the pairing at these tuples determines thepairing everywhere by bilinearity. But, given the minimality restrictions onthe R ( w ) , we see that the first part of Theorem 2.8 implies that the Artinpairings for k < m at x equal the sum of the Artin pairings at all othervertices in b x ( ∅ ) . This is enough to give that x is in Y ◦∅ .The pairings at k = m follow similarly except at ( w aj , t b + w bj ) . Atthis tuple, though, the second part of the theorem applies, and we get theclaimed result on the Artin pairing.On the Selmer side, we can follow the same argument, finding the sumof the Cassels-Tate pairing over x in b x ( ∅ ) . We again use pairs of the form ( w j , w j ) and ( w j , t + w j ) ; we can assume that j and j are both notequal to j . We see that it is enough to prove that the Cassels-Tate pairingbehaves as we expect on pairs of this form since the pairing is alternating.Theorem 2.9 then gives us that x is in Y ◦∅ and that the sum of the Cassels-Tate pairing at level m over x ∈ b x ( ∅ ) obeys the given formula. (cid:3)
4. R
AMSEY -T HEORETIC RESULTS
In Proposition 3.6, we found a condition on ¯ x ∈ X S under which the sum X x ∈ b x ( ∅ ) (cid:10) ψ m ( R ( w j ) , x ) , w j ( x ) (cid:11) CT ∈ F was determined by an Artin symbol in the field of definition of some gov-erning expansion, with an analogous form found on the class side. Thisinformation is not enough to determine the value of the pairing at any par-ticular x ∈ b x ( ∅ ) . However, if we have enough choices of ¯ x where we canfind this sum, we can still usually prove that the value of the pairing isforced to be on about half the vertices in Y ◦∅ .The first question is whether there is even one choice of such a ¯ x whosevertices lie in Y ◦∅ . This is a question in Ramsey theory; we can prove thatsuch a ¯ x exists if Y ◦∅ is large enough. This is the r = 2 case of the followingproposition. Proposition 4.1.
Take d ≥ to be an integer, take − d − > δ > , and take X , . . . , X d to be finite sets with cardinality at least n > . Suppose that Y is a subset of X = X × · · · × X d of cardinality at least δ · | X | . Then, for ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 37 any positive r satisfying r ≤ (cid:18) log n δ − (cid:19) / ( d − , there exists a choice of sets Z , . . . , Z d , each of cardinality r , such that Z × · · · × Z d ⊆ Y. Proof.
We can find subsets X ′ i of the X i so | X ′ i | = n and so Y has density atleast δ in X ′ × · · · × X ′ d . Because of this, we can without loss of generalityassume that X , . . . , X d have cardinality exactly n .For any positive integers d and r and any Y ⊂ X , write N ( r, Y ) forthe number of ways of choosing subsets Z i of X i for all i ≤ d , each ofcardinality r , so that Z × · · · × Z d ⊂ Y . Write N d ( n, r, δ ) for the minimumof N ( r, Y ) over all Y of cardinality at least δ ·| X | . To prove the proposition,we will show that for d > , δ > , and n ≥ r ≥ satisfying(4.1) (2 − d − δ ) r d − · nr − ≥ , we have(4.2) N d ( n, r, δ ) ≥ (2 − d − δ ) rd +1 − rr − n rd ( r !) d . The condition of the proposition is stricter than (4.1), so this will be suffi-cient to show the proposition.We prove the claim by induction. Setting d = 1 , we find N ( n, r, δ ) ≥ ( δn − r ) r r ! For r ≤ δn , this gives N ≥ ( δ/ r n r r ! , and this gives us the base case for (4.2).Now consider the case of d > , and choose Y with N ( r, Y ) minimal.Take X thick to be the subset of x ∈ X so that Y x = Y ∩ (cid:0) { x } × X × · · · × X d (cid:1) has density at least δ/ in { x } × X × · · · × X d . X thick has density at least δ/ in X .Take Z to be the set of choices of subsets Z , . . . , Z d , Z i ⊆ X i such thateach Z i has cardinality r . We have | Z | ≤ r !) d − n r ( d − . For z = ( Z , . . . , Z d ) ∈ Z , ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS take n z to be the number of x ∈ X thick such that Y contains { x } × Z × · · · × Z d . Then N d ( n, r, δ ) = N ( r, Y ) ≥ X z ∈ Z n z ≥ r r ! ( n z − r ) r ≥ X z ∈ Z r r ! n r z − r r r ! . We have X z ∈ Z n z ≥ | X thick | · N d − ( n, r, δ/ ≥ δ n (2 − d − δ ) rd − rr − n r ( d − ( r !) d − so P z ∈ Z n z P z ∈ Z ≥ δ n (2 − d − δ ) rd − rr − ≥ n (2 − d − δ ) rd − r − . With Cauchy-Schwarz, we then get N d ( n, r, δ ) ≥ n r ( d − ( r !) d − (cid:18) − r r r ! + 12 r r ! 4 r n r (2 − d − δ ) rd +1 − rr − (cid:19) . But, for r ≥ , r d +1 − rr − ≤ r d , so (4.1) implies N d ( n, r, δ ) ≥ (2 − d − δ ) rd +1 − rr − n rd ( r !) d , as claimed. This is thus true for all d by induction, proving the proposition. (cid:3) As a first consequence of this proposition, we will show that, if a subset Z of X is large enough, then a function from Z to F with “generic dif-ferential” will typically be on about half of Z . The next definition andproposition formalize this notion. Definition 4.2.
Take X , . . . , X d to be disjoint finite nonempty sets, andtake X to be their product. Choose a nonempty subset S of [ d ] of cardinalityat least two, and choose some Z ⊆ X so π [ d ] − S ( Z ) is a point. Taking F tobe a function from Z to F , we define a function dF : (cid:8) ¯ x ∈ X S : b x ( ∅ ) ⊆ Z (cid:9) −→ F by dF (¯ x ) = (P x ∈ b x ( ∅ ) F ( x ) if | b x ( ∅ ) | = 2 | S | otherwise. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 39 Write G S ( Z ) for the image of this map d . In addition, for ǫ > , write G S ( ǫ, Z ) for the set of g ∈ G S ( Z ) expressible in the form g = dF for some F that equals on more than (0 . ǫ ) | Z | or fewer than (0 . − ǫ ) | Z | pointsin Z . Proposition 4.3.
Taking X and Z as in the previous definition, choose δ > so that | Z | ≥ δ · | π S ( X ) | . Suppose | X i | ≥ n for each i ∈ S . Then, for ǫ > , | G S ( ǫ, Z ) || G S ( Z ) | ≤ exp (cid:16) | π S ( X ) | · (cid:16) − δǫ + 2 | S | +2 · n − / | S | (cid:17)(cid:17) . Proof.
Take Z ′ to be a maximal subset of Z so that there is no ¯ z ∈ X S satisfying | b z ( ∅ ) | = 2 | S | and b z ( ∅ ) ⊆ Z ′ . We see that the kernel of the map d : F Z → G S ( Z ) then has size at most | Z ′ | . From applying (4.1) with r = 2 , we also have | Z ′ | ≤ | π S ( X ) | · | S | +2 · N − / | S | . Then we must have | G S ( Z ) | ≥ | Z | · exp (cid:16) −| π S ( X ) | · | S | +2 · N − / | S | (cid:17) On the other hand, from Hoeffding’s inequality, the number of F equal-ing on more than (0 . ǫ ) | Z | or fewer than (0 . − ǫ ) | Z | points in Z isbounded by | Z | +1 exp (cid:0) − ǫ | Z | (cid:1) by Hoeffding’s inequality [19, Theorem 1]. Then G S ( ǫ, Z ) is bounded by | G S ( ǫ, Z ) | ≤ | Z | exp (cid:0) − ǫ | Z | (cid:1) . Taking ratios of these estimates then gives the result. (cid:3)
We run into two issues when we try to apply Proposition 4.3 directly toProposition 3.6. The first is that we do not a priori have any control on theform of Z = Y ◦∅ . We can restrict an element of G S ( π S ( X S )) to G S ( Z ) ,but the preimages of the various G S ( ǫ, Z ) will depend on the choice of Z .Furthermore, in the context of Proposition 3.6, it is not enough that b x ( ∅ ) liein Y ◦∅ to conclude dF (¯ x ) = g (¯ x ) for the relevant F and g ; we must haveinstead that b x ( T ) meet Y ◦ T for each proper T in S .Fortunately, thanks to the structure already found for additive-restrictivesystems, both of these issues can be solved with a little more work. First,we have a regularity condition on Y ◦∅ proved in Proposition 3.6, where wefound that x could be proved to be in this set by finding a nice cube ¯ x ∈ X S with all other vertices in this set. Because of this, we do not need to ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS consider all possible Z . Furthermore, thanks to Proposition 3.2, we can finda minimal density for Y ◦ S in terms of the density of Y ◦∅ , and this is enoughto circumvent the second issue. The end result is the following proposition. Proposition 4.4.
There is an absolute positive constant A so that we havethe following:Take X and S as in the the previous definition. For a ≥ and ǫ > ,define G S ( ǫ, a, X ) to be the set of g ∈ G S (cid:0) π S ( X S ) (cid:1) for which there is some Z ⊆ X , some F : Z → F , some additive restrictive system A on X , and some subset Z S of X S so that • The image of Z under π [ d ] − S is a point. • For each T ⊆ S , we have | A T ( A ) | ≤ a . • If ¯ x is in Z S , then b x ( ∅ ) ⊆ Z and dF (¯ x ) = g (¯ x ) . • We have equalities Z S = \ T ( S (cid:8) ¯ x ∈ X S : b x ( T ) ∩ Y ◦ T ( A ) = ∅ (cid:9) and Z = Y ◦∅ ( A ) . • The function F is on more than . | Z | + ǫ | π S ( X ) | or fewer than . | Z | − ǫ | π S ( X ) | of the points in Z .Write n for min i ∈ S | X i | . Then, if ǫ is less than a − and (4.3) log n ≥ A · | S | log ǫ − , we have | G S ( ǫ, a, X ) || G S ( π S ( X S )) | ≤ exp (cid:0) − | π S ( X ) | · n − / (cid:1) . Proof.
Consider a function g coming from F , Z , and A as in the proposi-tion. For any x ∈ Z , define Z ( x ) as the set of x in Z for which there issome ¯ x ∈ X S with x, x ∈ b x ( ∅ ) such that, if T is a proper subset of S and ¯ y is an element of b x ( T ) that contains the vertex x , then ¯ y is in Y ◦ T . FromProposition 3.2, we see that there is some sequence x , . . . , x r of points in Z so that(4.4) Z ( x j ) − Z ( x j − ) − · · · − Z ( x ) has density at least (0 . · a − ǫ ) | S | ≥ ǫ | S | +1 for j ≥ and so that thecomplement Z − Z ( x r ) − · · · − Z ( x ) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 41 has density at most ǫ/ in W . Each Z ( x j ) is determined by the sequence ofstructures Z ( x j ) ∩ π − S −{ i } ( x j ) as i varies through S ; this info can be specified by | π S ( X ) | · X i ∈ S | X i | ≤ | π S ( X ) | · | S | · n − bits. There are at most ǫ − | S | +1 elements x j , so, given x , . . . , x r the info ofall the Z ( x j ) can be specified with at most ǫ − | S | +1 | π S ( X ) | · | S | · n − bits. Writing Z ′ ( x j ) for the expression (4.4), we find that there must be a j so that F equals on at least | Z ′ ( x j ) | · . ǫ ) vertices in Z ′ ( x j ) .The conditions on Z ′ ( x j ) imply that, if x is in Z ′ ( x j ) , then there is a cube ¯ x ∈ X S with x, x j ∈ b x ( ∅ ) such that dF (¯ x ) = g (¯ x ) . Using the additivityof dF and g we find then that, if ¯ x ∈ X S has b x ( ∅ ) contained in Z ′ ( x j ) ,then dF (¯ x ) = g (¯ x ) . Then Proposition 4.3 implies that the number of g in G S ( ǫ, a, X ) corresponding to this choice of Z ′ ( x j ) is bounded by | G S ( π S ( X S )) | · exp (cid:16) | π S ( X S ) | · (cid:16) − ǫ | S | +1 + 2 | S | +2 · n − / | S | (cid:17)(cid:17) . For sufficient A , we use (4.3) to bound this by | G S ( π S ( X S )) | · exp (cid:16) −| π S ( X S ) | · ǫ | S | +1 (cid:17) . Summing this over all possible choices of the ( x , . . . , x r ) , over all choicesof the Z ( x i ) , and over all the choices of j , we find that the ratio being esti-mated by the proposition is bounded by r | π S ( X S ) | r exp (cid:16) | π S ( X S ) | · (cid:16) − ǫ | S | +1 + ǫ − | S | +1 N − | S | (cid:17)(cid:17) . For sufficient A , this is less than r | π S ( X S ) | r exp (cid:16) −| π S ( X S ) | · ǫ | S | +1 (cid:17) . If A is sufficiently large, we find that n − / > log | π S ( X S ) | / | π S ( X S ) | , andthe ratio is bounded by exp (cid:16) −| π S ( X S ) | · ǫ | S | +1 (cid:17) , which is within the bounds of the proposition for sufficient A . (cid:3) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS
5. P
RIME DIVISORS AS A P OISSON POINT PROCESS
Take N to be a large real number, and take n to be a positive integer cho-sen uniformly from [1 , N ] . Taking p < · · · < p r to be the prime divisorsof n , it is commonly understood that the values log log p i behave more orless like random variables uniformly chosen from the interval [0 , log log N ] .This model breaks down for the prime divisors near the endpoints of the in-terval, but it is otherwise fairly robust (see [11], for example).This is a very convenient model to have. In order for our argument toapply to an integer n , we need there to be some i ≥ √ r with log log p i +1 − log log p i ≥ log log log p i +1 + 12 log log log log N that obeys some technical conditions, so we rely on the fact that most inte-gers have a gap this large. It is far easier to prove that such a gap usuallyexists by working in the corresponding Poisson point process than by di-rectly dealing with the prime factors.In proving the model, the key object to understand is I k ( u ) = Z t ,...,t k ≥ t + ··· + t k ≤ u dt t . . . dt k t k where u > and k ≥ . This integral dates back to Ramanujan, with recentwork done by Soundararajan [25]. We clearly have I k ( u ) ≤ (cid:18)Z u dtt (cid:19) k = (log u ) k . Our first result is a better estimate for this integral.
Lemma 5.1.
For u ≥ and k ≥ , we have (cid:12)(cid:12)(cid:12)(cid:12) I k ( u ) − e − γα Γ(1 + α ) (log u ) k (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) ( α + 1)(log u ) k (log log u ) log u (cid:19) where α = k/ log u , γ is the Euler-Mascheroni constant, and the implicitconstant is absolute. Next, we give three properties a well-behaved sample of points on aninterval should have. We have given these properties the names comfortablespacing , regularity , and extravagant spacing . Definition.
Take
L > , take n to be a positive integer satisfying | L − n | < L / , and take X , . . . , X n to be independent random variables, eachdistributed uniformly on [0 , L ] . For i ≤ n , take U ( i ) to be the i th orderstatistic of this sample set; that is, take U ( i ) to be value of the i th smallest X i . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 43 • For δ > and ≤ L < L , we call the sample δ -comfortablyspaced above L if, for all i < n such that U ( i ) ≥ L , we have U ( i +1) − U ( i ) ≥ δ exp( − U ( i ) ) . • For C > , we call the sample C -regular if, for all i ≤ n , (cid:12)(cid:12) U ( i ) − i (cid:12)(cid:12) < C / · max( i, C ) / . • We call the sample extravagantly spaced if, for some m ≥ √ n , wehave exp( U ( m ) ) ≥ U ( m ) · (log L ) / · m − X i ≥ exp( U ( i ) ) ! . Proposition 5.2.
Take X , . . . , X n to be a sample as in the above definition.(1) For δ > and ≤ L < L , the probability that this sample is not δ -comfortably spaced above L is bounded by O (cid:0) δ · exp( − L ) (cid:1) with absolute implicit constant.(2) There is a positive constant c so that, for C > , the probabilitythat this sample is not C -regular is bounded by O (cid:0) exp( − c · C ) (cid:1) with absolute implicit constant.(3) There is a positive constant c so that the probability that the sampleis not extravagantly spaced is bounded by O (cid:16) exp (cid:16) − c · p log L (cid:17)(cid:17) with absolute implicit constant. Once this result is proved, we can move on to its number theoretic ana-logue. For
N, D positive real numbers and r a positive integer, we define S r ( N, D ) to be the set of squarefree positive integers less than N with ex-actly r prime factors and no prime factors less than D . Definition 5.3.
Take
N > and D > to be real numbers satisfying (log N ) / > log D , and take r to be a positive integer satisfying(5.1) (cid:12)(cid:12)(cid:12)(cid:12) r − log (cid:18) log N log D (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ log (cid:18) log N log D (cid:19) / . For n ∈ S r ( N, D ) , write ( p , . . . , p r ) for the primes dividing n in orderfrom smallest to largest. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS • For δ > and D > D , we call n comfortably spaced above D if,for all i < r such that p i > D , we have D < p i < p i +1 . • For C > , we call n C -regular if, for all i ≤ r , | log log p i − log log D − i | < C / · max( i, C ) / . • We call n extravagantly spaced if, for some m in (cid:0) . · r / , . · r (cid:1) ,we have(5.2) log p m ≥ log (cid:18) log p m log D (cid:19) · (log log log N ) / · m − X i ≥ log p i ! . Theorem 5.4.
Choose N , D , and r as in the previous definition. Choose n uniformly at random from the set S r ( N, D ) .(1) For D > , the probability that n is not comfortably spaced above D is O (cid:0) (log D ) − (cid:1) + O (cid:0) (log N ) − / (cid:1) with absolute implicit constant.(2) There is a positive constant c so that, for C > , the probabilitythat n is not C regular is O (cid:0) exp( − c · C ) (cid:1) + O (cid:0) exp( − c (log log N ) / ) (cid:1) with absolute implicit constant.(3) There is a positive constant c so that the probability that n is notextravagantly spaced is bounded by O (cid:0) exp (cid:0) − c · (log log log N ) / (cid:1)(cid:1) with absolute implicit constant. Proof of Lemma 5.1.
Proof.
The structure of our argument comes largely from [25]. We take abranch of the logarithm that is holomorphic away from the nonpositive realsand which is real on the positive reals. Following [25], we have(5.3) I k ( u ) = 12 πi Z c + i ∞ c − i ∞ e s s (cid:18)Z ∞ e − ts/u t dt (cid:19) k ds for any positive c . We recognize the inner integral as the exponential inte-gral function E ( s/u ) , which can be rewritten for s/u off the negative realaxis as E ( s/u ) = Z ∞ s/u e − z z dz, ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 45 where the integral is along any path that does not cross the negative realaxis [1]. We also have E ( s/u ) = − γ − log s/u + ∞ X n =1 ( − s/u ) n n · n ! . If | s/u | < and s/u is off the negative real axis, we find | E ( s/u ) | ≤ − log | s/u | + A with A some absolute constant. If | s/u | ≥ , we instead find | E ( s/u ) | ≤ | e − s/u | + A. From this, if u > k , we can write Z c + i ∞ c + iR e s s (cid:0) E ( s/u ) (cid:1) k ds = Z −∞ + iRc + iR e s s (cid:0) E ( s/u ) (cid:1) k ds for R > . Choose | c + iR | < u with R > e A . We have (cid:12)(cid:12)(cid:12)(cid:12)Z −∞ + iRc + iR e s s (cid:0) E ( s/u ) (cid:1) k ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z c −∞ e t R (cid:16)(cid:0) e − t/u + A (cid:1) k + (log u ) k (cid:17) dt. Assuming u > k and that u is greater than some constant determined from A , and choosing c = 1 , this is bounded by O (cid:18) R (log u ) k (cid:19) . for some choice of constant A > . Repeating this for negative R , we find (cid:12)(cid:12)(cid:12)(cid:12) I k ( u ) − Z iR − iR e s s (cid:0) E ( s/u ) (cid:1) k ds (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) R (log u ) k (cid:19) for R sufficiently large and | iR | < u .If z has positive real part, we also have z ) = 12 πi Z i ∞ − i ∞ e s s − z ds (see [5, Ch. IX, Misc. Ex. 24]). Then e − γα Γ(1 + α ) = 12 πi Z i ∞ − i ∞ e s s e − α (log s + γ ) ds. For
R > , we have (cid:12)(cid:12)(cid:12)(cid:12)Z i ∞ iR e s s e − α (log s + γ ) ds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z −∞ + iR iR e s s e − α (log s + γ ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z −∞ e s R ds = eR . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Take R = log u and take s on the segment [1 − iR, iR ] . We have ( E ( s/u )) k = (log u ) k (cid:18) − log s + γ + O ( s/u )log u (cid:19) k = (log u ) k exp (cid:18) − k log s + γ log u + O (cid:18) ks/u log u + k log s log u (cid:19)(cid:19) = (log u ) k exp (cid:18) − k log s + γ log u (cid:19) + (log u ) k · O (cid:18) ks/u log u + k log s log u (cid:19) Then, if R = log u and u > k is sufficiently large, I k ( u ) = (log u ) k e − γα Γ(1 + α ) + O (cid:18) (log u ) k · (cid:18) k · log R log u + 1 R (cid:19)(cid:19) , within the bounds of the lemma. If u ≤ k or u is small, the lemma isimplied by I k ( u ) ≤ (log u ) k . (cid:3) Proof of Proposition 5.2.
Proof of (1).
For i, j ≤ n , the probability that X i > L is less than X j andthat the gap X j − X i is uncomfortable is L Z LL Z X i + δ · e − Xi X i dX j dX i = 1 L Z LL δ · e − X i dX i ≤ L δ · e − L . There are n ( n − choices of the pair i, j , and this number is O ( L ) , so theprobability that some pair gives an uncomfortable gap is O (cid:0) δ · exp( − L ) (cid:1) . (cid:3) Proof of (2).
We note it is sufficient to show that there is a positive constant A so that, for C ≥ , the probability that(5.4) (cid:12)(cid:12) U ( i ) − i (cid:12)(cid:12) < A · C / · max( i, C ) / does not hold for some i is bounded by O (exp( − C )) with absolute implicitconstant.If C ≥ L , the proposition is trivial, so we assume C < L . We alsoassume ( L/C ) / is an integer. If it is not, we can rechoose C from theinterval [ C , C ] so that this is the case. For j > , define a sequence α j = min (cid:0) C · j , L (cid:1) . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 47 Take k j to be the number of X i less than α j . We can think of k j as the resultof running k j +1 Bernoulli trials with success rate α j /α j +1 . Then, for any j ,the probability that(5.5) (cid:12)(cid:12)(cid:12)(cid:12) k j − α j α j +1 k j +1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ q ( C + j ) k j +1 is not satisfied by the sample is bounded by exp( − C − j ) by Hoeffding’sinequality. Then the probability that this inequality is not satisfied some-where is bounded by X j ≥ exp( − C − j ) ≤ exp( − C ) . So suppose that (5.5) is satisfied for all j . We then claim that, with theproper choice of constant A , the sample satisfies (5.4) at all i . First, weclaim that there is some positive constant B > so that(5.6) (cid:12)(cid:12)(cid:12) k j − nL α j (cid:12)(cid:12)(cid:12) ≤ B · C / · α / j . for all j . This is clear if α j = L , as in this case k j = n .We can then proceed by induction. Suppose (5.6) holds for all j > m and that we wish to prove it for j = m . Then (cid:12)(cid:12)(cid:12) k m − nL α m (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) k m − α m α m +1 k m +1 (cid:12)(cid:12)(cid:12)(cid:12) + α m α m +1 (cid:12)(cid:12)(cid:12) k m +1 − nL α m +1 (cid:12)(cid:12)(cid:12) ≤ p ( C + m ) k m +1 + (cid:18) mm + 1 (cid:19) · B · C / · α / m +1 = p ( C + m ) k m +1 + mm + 1 · B · C / · α / m To prove the inequality, we then just need(5.7) p ( C + m ) k m +1 ≤ m + 1 B · C / · α / m . The square value of the left hand side of (5.7) has upper bound (cid:0) C + m (cid:1) · (cid:0) C · m + B · C · ( m + 1) (cid:1) , which can be expanded to a sum of four monomials. The square value ofthe right hand side of (5.7) has lower bound B · C · m . For each of the four monomials from the left hand side, we can choose B so that the monomial is bounded by B · C · m ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS if B > B . For example, C · m ≤ B · C · m holds for B ≥ . If B is greater than each B , then (5.7) necessarily holds,finishing the induction step. Then (5.6) holds for all m .We now turn to (5.4). If i ≤ k , we have (cid:12)(cid:12) U ( i ) − i (cid:12)(cid:12) ≤ α + k = O ( C ) , in the bound of the inequality. Now, suppose i > k . Then i is in someinterval ( k j , k j +1 ] . We have (cid:12)(cid:12) U ( i ) − i (cid:12)(cid:12) = max (cid:0) U ( i ) − i, i − U ( i ) (cid:1) ≤ max (cid:0) α j +1 − k j , k j +1 − α j (cid:1) ≤ | α j − α j +1 | + max (cid:0) | α j − k j | , | α j +1 − k j +1 | (cid:1) We can bound α j +1 − α j by O ( j · C ) . Using i ≥ α j , we find α j +1 − α j = O (cid:16) C / · i / (cid:17) . Using (5.6) and the estimate Ln = 1 + O ( L − / ) , we also have max (cid:0) | α j − k j | , | α j +1 − k j +1 | (cid:1) = O (cid:16) C / · i / (cid:17) . Then (5.4) is satisfied for some sufficiently large constant A , giving thepart. (cid:3) Proof of (3).
For the third part, we note that we can assume that L is largerthan some arbitrarily large positive constant L . Define sequences k j = ⌊ − k · n ⌋ and k ′ j = ⌊ . · − k · n ⌋ for j ≥ . Take M to be the maximal M such that k M +1 ≥ √ n . Suppose u > u > · · · > u M +1 . is a sequence of real numbers such that u = L and such that | u j − k j | ≤ k / j if j ≤ M + 1 . We say a sample obeys condition U if U ( k j ) equals u j for all j ≤ M + 1 .For m ≥ √ n in the interval ( k ′ j , k j ] , we say that E m is satisfied if U ( m ) − U ( m − ≥ log 2 + log k j + 12 log log L. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 49 We say further that E ′ m is satisfied if exp( U ( m ) ) ≥ · U ( m ) · (log L ) / · m − X i>k j +1 exp( U ( i ) ) . Finally, we say that R m is satisfied if U ( m ) ≥ U ( k j +1 ) . For m as above, we have P (cid:0) E m (cid:12)(cid:12) U (cid:1) = (cid:18) − log 2 + log k j + log log Lu j − u j +1 (cid:19) k j − k j +1 − ≥ − log 2 + log k j + log log L k j − k / j ! k j − ≥ exp (cid:18) − log 2 − log k j −
12 log log L − o (1) (cid:19) ≥ e − o (1) · k j · √ log L ≥ · k j · √ log L for a sufficiently large choice of L .Next, we note that there is a small positive constant c so that, for suffi-ciently large L , P (cid:0) R m (cid:12)(cid:12) U (cid:1) ≥ − e − c · k j This is an easy consequence of Hoeffding’s inequality. We also see that P (cid:0) E ′ m (cid:12)(cid:12) U , R m , E m (cid:1) ≥ I m − k j +1 (exp(0 . · u j +1 ))(0 . · u j +1 ) m − k j +1 . Via Lemma 5.1, for sufficient L , this can be bounded from below by somesmall positive constant c . Then P (cid:0) E ′ m (cid:12)(cid:12) U (cid:1) ≥ c · k j · √ log L − e − c · k j . For sufficient L , this is bounded by c · k j · √ log L .
We say that T j is satisfied if E m is satisfied for some m in ( k ′ j , k j ] , andwe say T ′ j is satisfied if E ′ m is satisfied for some m in this interval. Then wehave ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS P (cid:0) T j (cid:12)(cid:12) U (cid:1) = O p log k j ! . For sufficient L , this is bounded from above by . . At the same time, wehave P (cid:0) T ′ j (cid:12)(cid:12) U (cid:1) ≥ k j X m>k ′ j P (cid:0) E ′ m (cid:12)(cid:12) U , E m +1 , . . . , E k j (cid:1) · P (cid:0) E m +1 , . . . , E k j (cid:12)(cid:12) U (cid:1) ≥ k j X m>k ′ j P (cid:0) E ′ m (cid:12)(cid:12) U (cid:1) · P (cid:0) T j (cid:12)(cid:12) U (cid:1) ≥ c ( k j − k ′ j )8 · k j · √ log L ≥ c √ log L for a sufficiently small constant c .Given U , the T ′ j are independent events. Therefore, given U , the proba-bility that none of the T ′ j are true for j ≤ M is at most (cid:18) − c √ log L (cid:19) M = O (cid:16) exp (cid:0) − c · p log L (cid:1)(cid:17) for a sufficiently small constant c .Now, if our sample is log L -regular, then the U ( k j ) will all be within k / j of k j for sufficiently large k j . Then the probability that the sample is not log L -regular or that no T ′ j holds is at most O (exp( − c · log L ))+ O (cid:16) exp (cid:0) − c · p log L (cid:1)(cid:17) = O (cid:16) exp (cid:0) − c · p log L (cid:1)(cid:17) . If the sample is log L -regular, we find that exp( U ( k ′ j ) ) ≥ · U ( k j ) · (log L ) / · X i ≤ k j +1 exp( U ( i ) ) is true for all j ≤ M if L is sufficiently large. Assuming this, we find that E ′ m implies that exp( U ( m ) ) ≥ U ( m ) · (log L ) / · m − X i =1 exp( U ( i ) ) ! . Because of this, if L is sufficiently large, and if the sample is log L -regularand satisfies T ′ j for some j ≤ M , we must have that the sample is extrava-gantly spaced. This gives the proposition. (cid:3) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 51 Proof of Theorem 5.4.
For positive x , we define F ( x ) = X p ≤ x p , the sum being over the primes no greater than x .Using the prime number theorem, we know there are constants A, c > so that, for all x ≥ . , (cid:12)(cid:12) F ( x ) − log log x − B (cid:12)(cid:12) ≤ A · e − c √ log x , where B is the Mertens’ constant.With this in mind, suppose that T is a collection of tuples of primes oflength r . We define Grid ( T ) ⊆ R r to be the union [ ( p ,...,p r ) ∈ T Y i ≤ r (cid:20) F ( p i ) − p i − B , F ( p i ) − B (cid:21) . Now suppose V ⊆ R r contains log log T = (cid:8) (log log p , . . . , log log p r ) : ( p , . . . , p r ) ∈ T (cid:9) For ( x , . . . , x r ) ∈ R k , we define τ ( x , . . . , x r )= Y i ≤ r (cid:2) x i − A · exp( − c · e x i / ) , x i + A · exp( − c · e x i / ) (cid:3) . We then define V big = [ x ∈ V τ ( x ) and V small = { x ∈ R ≥ ( − B ) r : τ ( x ) ⊆ V } , where R ≥ ( − B ) is the set of reals ≥ − B .For a proper choice of the constants A and c , we then see that, if T is themaximal set of prime tuples such that log log T is contained in V , we have(5.8) V small ⊆ Grid ( T ) ⊆ V big . This equation is extremely useful, as we haveVol (cid:0)
Grid ( T ) (cid:1) = X ( p ,...,p r ) ∈ T p · · · · · p r . For example, for u a positive real and r a positive integer, take V r ( u ) ⊆ R r to be the set of ( x , . . . , x r ) satisfying e x + · · · + e x r ≤ u. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS We see that exp (cid:0) x + A exp( − c · e x/ ) (cid:1) − exp( x ) ≤ κ for some κ depending on A and c but not x . Then V r ( u ) big is contained in V r ( u + rκ ) , while V r ( u ) small contains V r ( u − rκ ) ∩ R ≥ ( − B ) r . At the same time, we see thatVol ( V r ( u ) ∩ R ≥ ( B ) r ) = I r ( e − B u ) . Thus, for
N, D > and r a positive integer, we have I r (cid:18) log N − rκ exp( F ( D ) − B ) (cid:19) ≤ X p ,...,p r >Dp ····· p r
Take > ǫ > , and suppose N, D are real numberssatisfying log log
N > (1 + ǫ ) log log D > and that r is a positive integer satisfying ≤ ǫ · r ≤ log u ≤ ǫ − · r where u = log N exp( F ( D ) − B ) . Then there are positive constants C, c dependingonly on ǫ such that c · N log N (log u ) r − ( r − < (cid:12)(cid:12) S r ( N, D ) (cid:12)(cid:12) < C · N log N (log u ) r − ( r − for all sufficiently large N . Taking ǫ , N , D , and r as in this proposition, and taking k ≤ r , we define S r, k ( N, D ) to be the subset of S r ( N, D ) of elements n with exactly k prime divisorssmaller than N = exp (cid:16)p log N · exp( F ( D ) − B ) (cid:17) . Proposition 5.6.
Given ǫ , N , D , and r as above, the density of the set [ | r − . k | > r / S r, k ( N, D ) in S r ( N, D ) is bounded by O (cid:0) exp( − c (log log N ) / ) (cid:1) for some c > , with c and the implicit constant depending only on ǫ .Now suppose | . r − k | ≤ r / , and suppose that T and T are col-lections of tuples ( p , . . . , p k ) of distinct primes less than N in increasingorder. Writing S r ( N, D, T ) for the subset of n in S r, k ( N, D ) whose k small-est prime factors ( p , . . . , p k ) lie in T , we have (cid:12)(cid:12) S r, k ( N, D, T ) (cid:12)(cid:12)(cid:12)(cid:12) S r, k ( N, D, T ) (cid:12)(cid:12) = O Vol (cid:0)
Grid ( T ) (cid:1) Vol (cid:0)
Grid ( T ) (cid:1) ! ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 55 with the implicit constant depending only on ǫ .Proof. We have | S r, k ( N, D ) | = X D
depending only on ǫ such that(5.11) c · P | S r ( N, N ) | < | S r ( N/P, N ) | < C · P | S r ( N, N ) | for sufficient N . From this, we find | S r, k ( N, D ) | = O (cid:18) N log N − r − (log log N − F ( D )) r − ( r − k + 1)! k ! (cid:19) for r > k . Hoeffding’s inequality gives the first part of the proposition ifwe remove the case r = k from the union. The case r = k is insignificant,with its contribution necessarily limited by N r , so we have the first part ofthe proposition. The second part of the proposition is just a recast of (5.11)in terms of the corresponding statements for S r, k ( N, D ) , and we have theproposition. (cid:3) With this proposition, parts two and three of Theorem 5.4 are straightfor-ward. First, we restrict to considering S r, k ( N, D ) with | k − . r | ≤ r / .We need not consider other k , as the union of all other S r, k ( N, D ) fits in theerror bound.Take T to be the set of k -tuples of distinct primes from the interval ( D, N ) . We findVol ( Grid ( T )) ≥ c (log log N − log log D ) k . For part (2), take T to be the set of non- C -regular prime tuples in T . Thebiggification of the grid of T consists of samples that are not C − κ regularfor some constant κ > not depending on C . From Proposition 5.2, wethen find that the volume of this bigification is bounded by O (cid:0) exp( − c · C ) · (log log N − log log D ) k (cid:1) . Then Proposition 5.6 gives the part.For part (3), take T to be the set of prime tuples so, for m > k / , log p m ≤ log (cid:18) log p m log D (cid:19) · (log log log N ) / · m − X i ≥ log p i ! . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS The bigification of this grid consists of tuples ( x , . . . , x k ) so, for m > k / , e x m ≤ Ax m · (log log log N ) / · m − X i =1 e x i ! for some absolute constant A . Repeating part (3) of the proof Proposition5.2 to take account of the A , we find that the volume of this bigification is O (cid:0) exp (cid:0) − c · (log log log N ) / ) · (log log N − log log D ) k (cid:1) . Then Proposition 5.6 gives the part.For the first part of the theorem, we opt to start from scratch. The numberof uncomfortably spaced examples is bounded by N X p>D p X q>p | S r − ( N/pq, D ) | We can restrict this to the case that p < N / and p ≥ N / . The formercase has size boundable by O | S r − ( N, D ) | · N X p>D p X q>p pq ! = O (cid:0) | S r ( N, D ) | · (log D ) − (cid:1) The sum over p ≥ N / can be bounded by O ( N/ log N ) , so we have theresult if we can show (log log N − F ( D ) + B ) r − ( r − ≥ p log N for all sufficiently large N . This is a simple consequence of (5.1) and therestriction log D < (log N ) / , and we have the theorem. (cid:3)
6. E
QUIDISTRIBUTION OF L EGENDRE SYMBOLS
From elementary work of R´edei and Reichardt [22], we know that therank of the -class group of a quadratic field with discriminant ∆ can bedetermined from the kernel of a matrix of Legendre symbols (cid:16) dp (cid:17) , where d varies over the divisors of ∆ and p varies over the odd prime divisors of ∆ . For quadratic twists of elliptic curves E with full two torsion and norational cyclic subgroup of order four, we can also give the -Selmer rankas the kernel of a certain matrix of Legendre symbols.In [26], under the assumption that the associated Legendre symbol ma-trices were uniformly distributed among all posibilites, Swinnerton-Dyerfound the distribution of -Selmer ranks among the set of all twists. Kanethen proved that the actual distribution of Selmer ranks agreed with the dis-tribution found by Swinnerton-Dyer [15]. In contrast to the work of Fouvry ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 57 and Kl¨uners on -class groups [8] and the work of Heath-Brown on thecongruent number problem [12], Kane’s work relied on the fact that thedistribution over Legendre symbol matrices had already been found, thusstreamlining the argument in a way that hadn’t previously been possible.However, in line with the prior papers, Kane’s eventual argument was thatthe moments of the -Selmer groups were consistent only with the claimeddistribution.As an alternative to Kane’s approach, we can prove Swinnerton-Dyer’sassumption is correct. If we set up our sets of integers correctly, we canprove that the matrices of Legendre symbols are essentially equidistributed;this is the content of Proposition 6.3 and Theorem 6.4. Both of these re-sults concern points chosen from a product space of increasing intervalsof primes; we will make the translation to from arbitrary integers to suchproduct spaces in Section 6.3.Our first task is to concretely define what we mean by matrices of Le-gendre symbols. Definition 6.1.
Take P to be an arbitrary set of prime numbers, and take P = {− } ∪ P . Choosing r > , take M to be some subset of (cid:8) { i, j } : i, j ∈ [ r ] } and take M P to be some subset of [ r ] × V . Also take a to be an arbitraryfunction from M ∪ M P to ± .Take X , . . . , X r to be disjoint sets of odd primes not meeting P , andwrite X for the product of the X i . We then define X ( a ) to be the set of ( x , . . . , x r ) in X satisfying (cid:18) x i x j (cid:19) = a (cid:0) { i, j } (cid:1) for all i < j with { i, j } ∈ M and (cid:18) dx j (cid:19) = a (cid:0) ( i, d ) (cid:1) for all ( i, d ) ∈ M P . Our goal is to find situations where the order of | X ( a ) | is well approxi-mated by −| M P ∪ M | ·| X | . To do this unconditionally, we need to account forthe possibility of Siegel zeros in the L -functions of the associated quadraticcharacters. We use the following definition. Definition 6.2.
For c > , take Sieg ( c ) to be the set of squarefree integers d so that the quadratic character χ d associated with Q ( √ d ) / Q has Dirichlet L -function satisfying L ( χ d , s ) = 0 for some ≥ s ≥ − c (log 2 d ) − . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS We can order Sieg ( c ) by increasing magnitude, getting a sequence d , d , . . . .By Landau’s theorem (see [13, Theorem 5.28]), we can choose an absolute c that is sufficiently small so that d i < (cid:12)(cid:12) d i +1 (cid:12)(cid:12) for all i ≥ . Fix such a choice of c for this entire paper. We call a given d Siegel-less if it is outside the sequence of such d i We can now state our first result.
Proposition 6.3.
Choose positive constants c , c , c , c , c , c , c , c . Wepresume that c > , that c > , and that > c + c log 22 + 1 c + c c . Then there is a constant A depending only on the choice of these constantsso that we have the following:Choose sets X , . . . , X r and a function a with associated set P as inDefinition 6.1, and choose a sequence of real numbers A < t < t < t ′ < t < t ′ < · · · < t r < t ′ r such that X i is a subset of ( t i , t ′ i ) for each i ≤ r . Choose ≤ k ≤ r suchthat, if M P contains some element of the form ( i, d ) , we have i > k . For i > k , we assume that X i = (cid:26) t i < p < t ′ i : (cid:18) dp (cid:19) = a (cid:0) ( i, d ) (cid:1) for all ( i, d ) ∈ M P (cid:27) . For D P any product of elements in P and D X any product over i of atmost one element from each X i , we assume that D P D X is Siegel-less if | D P D X | > t .We also assume that we have the following:(1) We assume that all primes in P are less than t ′ .(2) We assume that t ′ > r c and t ′ k < exp( t ′ c ) . (3) We assume that, for ≤ i ≤ r , we have | X i | ≥ c i · t ′ i (log t ′ i ) c and | P | ≤ log t ′ i − i. (4) If k = r , we assume that t ′ k +1 > exp (cid:0) (log t ′ ) c (cid:1) , exp (cid:0) t c (cid:1) . (5) We assume that k < c log t ′ and that, for any i ≤ r and any j satisfying r ≥ j ≥ i − c log t ′ i , we have exp (cid:0) (log t ′ i ) c (cid:1) < t ′ j . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 59 Then (cid:12)(cid:12) | X ( a ) | − −| M | | X | (cid:12)(cid:12) ≤ t ′ − c · −| M | | X | . The lower bounds assumed of t in the above proposition are essentialfor the proposition to be correct; we do not have sufficiently strong controlon the Legendre symbols involving only small primes to give the equidis-tribution result we want. However, there is a combinatorial trick that allowsus to circumvent this bad behavior: Definition.
Given X = X × · · · × X r and a as in Definition 6.1, and givena permutation σ : [ r ] → [ r ] , we take X ( σ, a ) = ( X σ (1) × X σ (2) × · · · × X σ ( r ) )( a ) . Given k ≤ r , we take P ( k ) to be the set of permutations of [ r ] that arethe identity outside of [ k ] .There are two key points to make about this definition. First, when matri-ces of Legendre symbols are used to find -Selmer groups or -class groups,the order of the prime factors of the quadratic twist or discriminant do noteffect the eventual rank. Because of this, the Selmer or class structure of apoint in X ( σ, a ) is not affected by the choice of σ .Second, the application of σ to X has the effect of mixing the bad cornerof the Legendre symbol matrix in with the rest of the matrix. As a result,the average of | X ( σ, a ) | over all choices of σ is almost independent of thechoice of a , as we detail in the next theorem. Theorem 6.4.
Choose positive constants c , . . . , c . We presume that c > , that c > , that > c + c log 22 + 1 c + c c , and that c log 2 + 2 c + c < and c + c < c . Then there is a constant
A > depending only on the choice of theseconstants so that we have the following:Choose a sequence of real numbers t < t ′ < t < t ′ < · · · < t r < t ′ r , and a positive number t > A , and take X i to be the set of primes in theinterval ( t i , t ′ i ) . Write X for the product of the X i , and take a to be as inDefinition 6.1. Choose integers k , k , k satisfying ≤ k ≤ k < k ≤ r and t ′ k +1 > t . We will write t ′ for t ′ k +1 . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS For D P any product of elements in P and D X any product over i of atmost one element from each X i , we assume that D P D X is Siegel-less if | D P D X | > t . We also assume k > A .We also make the following assumptions:(1) We assume that all primes in P are less than t ′ .(2) We assume that t ′ > r c and t ′ k < exp( t ′ c ) . (3) For i > k , we assume that | X i | ≥ | P | + c i · k c · t ′ i (log t ′ i ) c and | P | ≤ log t ′ i − i. (4) If k = r , we assume that t ′ k +1 > exp (cid:0) (log t ′ ) c (cid:1) , exp (cid:0) t c (cid:1) (5) We assume that k − k < c log t ′ and that, for any k < i ≤ r andany j satisfying r ≥ j ≥ i − c log t ′ i , we have exp (cid:0) (log t ′ i ) c (cid:1) < t ′ j . (6) We assume that c log k > | P | + k and c log k > log k . Then, for any choice of the subsets M and M p , we have X a ∈ F M ∪ M p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −| M ∪ M P | · k ! · | X | − X σ ∈ P ( k ) (cid:12)(cid:12) X ( σ, a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16)(cid:16) k − c + t ′− c (cid:17) · k ! · | X | (cid:17) . Equidistribution via Chebotarev and the Large Sieve.
There aretwo main methods to predict the distribution of (cid:16) dp (cid:17) over a given set of p . First, if d is small relative to the set of p , then we can use the Cheb-otarev density theorem to predict the distribution. On the other hand, if d issimilarly sized to the set of p , we can use the large sieve results of Jutila topredict the distribution of these symbols for most d [14]. The key to provingthis proposition is combining these two tools properly.We start with the form of the Chebotarev density theorem that we will beusing. For the proof of Proposition 6.3, we only need to apply this proposi-tion with L = Q . However, the full power of this proposition, including thecumbersome form of the error term, will be necessary to prove our resultsfor k -Selmer groups and class groups. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 61 Proposition 6.5.
Suppose M/ Q is a Galois extension and G = Gal ( M/ Q ) is a -group. Suppose M equals the composition KL , where L/ Q is Galoisof degree d and K/ Q is an elementary abelian extension, and where thediscriminant d L of L/ Q and the discriminant d K of K/ Q are relativelyprime. Take d K to be the maximal absolute value of the discriminant of aquadratic subfield K of K .Take F : G → [ − , to be a class function of G with average over G equal to zero. Then there is an absolute constant c > such that X p ≤ x F (cid:18)(cid:20) M/ Q p (cid:21)(cid:19) · log p = O x β · | G | + x · | G | · exp − cd − log x √ log x + 3 d log (cid:12)(cid:12) d K d L (cid:12)(cid:12) ! (cid:0) d log (cid:12)(cid:12) xd K d L (cid:12)(cid:12)(cid:1) ! for x ≥ , where β is the maximal real zero of any Artin L -function definedfor G , the term being ignored if no such zero exists. The implicit constanthere is absolute.Proof. Take ρ to be a nontrivial irreducible representation of G . As it is a p -group, G is nilpotent and is hence a monomial group, so the Artin con-jecture is true for ρ . That is, the Artin L -function L ( ρ, s ) is entire. Therepresentations ρ ⊗ ρ and ρ ⊗ ρ also satisfy the Artin conjecture, so L ( ρ ⊗ ρ, s ) is entire except for a simple pole at s = 1 , and L ( ρ ⊗ ρ, s ) is entire unless ρ is isomorphic to ρ .Then [13, Theorem 5.10] applies for L ( ρ, s ) . We also see that [13,(5.48)]holds for this L function by the argument given in [13] after thisequation. Then Theorem 5.13 of [13] applies for this L -function. We notethat ρ is defined on Gal ( K L/ Q ) for some quadratic extension K / Q inside K , so its degree is bounded by d and the conductor of L ( ρ, s ) is boundedby the discriminant of K L/ Q , which is bounded by d dK · d L . Then [13, Theorem 5.13] gives X p ≤ x χ ρ (cid:18)(cid:20) M/ Q p (cid:21)(cid:19) · log p = O x β + x · exp − cd − log x √ log x + 3 d log (cid:12)(cid:12) d K d L (cid:12)(cid:12) ! (cid:0) d log (cid:12)(cid:12) xd K d L (cid:12)(cid:12)(cid:1) ! . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Now, we can write F in the form P ρ a ρ χ ρ , the sum being indexed by thenontrivial irreducible representations of G . Then X ρ (cid:12)(cid:12) a ρ (cid:12)(cid:12) = X ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | G | X g ∈ G F ( g ) · χ ρ ( g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X ρ | G | X g ∈ G F ( g ) · F ( g ) ! / · | G | X g ∈ G χ ρ ( g ) χ ρ ( g ) ! / ≤ X ρ ≤ | G | . We then get the proposition. (cid:3)
We now give the form of the large sieve we use.
Proposition 6.6.
Take X and X to be disjoint sets of odd primes withupper bounds t ′ and t ′ respectively. Then, for any ǫ > , we have X x ∈ X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ X (cid:18) x x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:16) t ′ · t ′ / ǫ + t ′ · t ′ / ǫ (cid:17) , with the implicit constant depending only on the choice of ǫ .Proof. By [14, Lemma 3], we have X x ∈ X X x ∈ X c ( x ) (cid:18) x x (cid:19)! = O (cid:18) t ′ · | X | + t ′ / · t ′ log t ′ (cid:19) for any function c : X → ± . Then, for c : X → ± , Cauchy’s theoremgives X x ∈ X x ∈ X c ( x ) c ( x ) (cid:18) x x (cid:19) = O (cid:18) t ′ · t ′ / + t ′ / · t ′ log t ′ (cid:19) . Choosing a , a ∈ ± , we apply the above estimate to the subsets of X , X where x ≡ a (4) and x ≡ a (4) . For each of the four possibilities of ( a , a ) , we have that (cid:16) x x (cid:17) (cid:16) x x (cid:17) is constant by quadratic reciprocity, andwe deduce the proposition. (cid:3) Proof of Proposition 6.3.
We will show that, subject to the assumptions ofthe proposition, we have (cid:12)(cid:12) | X ( a ) | − −| M | | X | (cid:12)(cid:12) ≤ r · t ′ − c − c · −| M | A | X | . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 63 The bound on t ′ shows that this implies the proposition. We proceed byinduction on r . The statement is obvious for r = 1 , where M is alwaysempty.Now, suppose we wish to show it for X × · · · × X r , once we know theresult for every product of length r − . To this end, for x ∈ X , take X i ( a, x ) to be the subset of elements x i in X i satisfying (cid:18) x x i (cid:19) = a ( { , i } ) should { , i } lie in M .If { , i } is in M for i ≤ k , we apply Proposition 6.6 to say that X x ∈ X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x i ∈ X i (cid:18) x x i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:16) t ′ i · t ′ / ǫ (cid:17) . Then, for any ǫ > , the bounds on the size of the X i then force X x ∈ X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x i ∈ X i (cid:18) x x i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < t ′ − + c c + ǫ · | X | · | X i | for sufficiently large A . Choosing constants c a , c b > with c a + c b < − c c , we expect that, for all x ∈ X besides at most k · t ′− c a · | X | exceptions,we have (cid:12)(cid:12) | X i ( a, x ) | − . | X i | (cid:12)(cid:12) < t ′− c b · | X i | for all { , i } ∈ M . Write X bad for the set of exceptional x . We will choose c a > c log 2 + c + 1 c and c b > c + 1 c . Given the conditions on the constants, it is always possible to find such c a , c b .Meanwhile, suppose { , i } is in M with i > k . We apply Proposition6.5 to the field M generated by √ x and by all √ d with d in P . Take F : Gal ( M/ Q ) → [ − , +1] to equal − −| P |− for σ corresponding tothe Frobenius class of the elements of X i ( a, x ) , and to otherwise equal − −| P |− . We are interested in bounding X p ≤ t ′ i F (cid:18)(cid:20) M/ Q p (cid:21)(cid:19) · log p. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Choose a constant c c > . By Siegel’s theorem [13, Theorem 5.28], we canchoose a large enough constant A so (1 − β ) > t − c c if β is an exceptional real zero of the L function corresponding to some χ D with | D | < t . Then, when applying Proposition 6.5, we find x β < t ′ i exp (cid:0) − (log t ′ i ) − ccc (cid:1) . From log | d K | = O (cid:0) (log t ′ ) (cid:1) , we can also bound exp − cd − log x √ log x + 3 d log (cid:12)(cid:12) d K d L (cid:12)(cid:12) ! ≤ exp (cid:0) − (log t ′ i ) / ǫ (cid:1) for some constant ǫ > . From | G | ≤ t ′ , we then find that we can write X p ≤ t ′ i F (cid:18)(cid:20) M/ Q p (cid:21)(cid:19) · log p ≤ t ′ i exp (cid:0) − (log t ′ i ) / ǫ (cid:1) for sufficient A . By reweighting this series, we can also show that X p ≤ t ′ i F (cid:18)(cid:20) M/ Q p (cid:21)(cid:19) s ≤ t ′ i exp (cid:0) − (log t ′ i ) / ǫ (cid:1) . Then, for i ≥ k , we always find (cid:12)(cid:12) | X i ( a, x ) | − . | X i | (cid:12)(cid:12) < t ′− · | X i | . Write X bad ( a ) for the subset of X ( a ) with x in X bad . Our first task is tobound this set. Choose x ∈ X bad ( a ) , and add it to P , shifting its conditionsfrom M to M P . Consider the product X × · · · × X k × X k +1 ( a, x ) × · · · × X r ( a, x ) . This product has length r − . Once we shift up k , it obeys all the conditionsof the proposition, so the induction step tells us that the subset of X ( a ) starting with x ′ has size at most −| M | + k +1 | X || X | . Then X bad ( a ) has size bounded by k +1 · t ′ − c a · −| M | | X | . On the good side, we instead look at the product X ( a, x ) × · · · × X r ( a, x ) . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 65 From this and the induction step, we find that the subset of X ( a ) startingwith a good x has size at most −| M | | X || X | · (1 + ( r − · t ′ − c − c ) · (1 + t ′ − c b ) k · (1 + t ′ − ) r and at least −| M | | X || X | · (1 − ( r − · t ′ − c − c ) · (1 − t ′ − c b ) k · (1 − t ′ − ) r . We have k < c log t ′ , so the term (1 + t ′ − c b ) k gives error fitting into t ′ − c − c from our lower bound on c b . The term (1 + t ′ − ) r gives errorfitting into this whenever > c + 2 c − , which is always satisfied.Next, we see that the contribution from X bad ( a ) fits into the error boundwhenever k · t ′ − c a fits into t ′ − c − c . From k < c log t ′ and the bounds on c a , we find that this is also the case. This gives the proposition. (cid:3) Combinatorial approaches to small primes.
We now wish to proveTheorem 6.4 from Proposition 6.3. This transition is entirely combinatorial:the key is the following proposition.
Proposition 6.7.
Choose X , P , M , and M P as in Definition 6.1. We as-sume M and M P are maximal given r and P .Choose integers ≤ k ≤ k ≤ k ≤ r so that | P | + k +1 · k < k . For σ a permutation of [ r ] and a as in Definition 6.1, take X C ( σ, a ) to bethe set of x = ( x , . . . , x r ) in X so that (cid:18) dx j (cid:19) = a (cid:0) ( σ − ( j ) , d ) (cid:1) for all ( j, d ) ∈ [ k ] × P and (cid:18) x i x j (cid:19) = a (cid:0) { σ − ( i ) , σ − ( j ) } (cid:1) whenever i, j ≤ k and σ − ( i ) ≤ σ − ( j ) and either i ≤ k or j ≤ k . Write m C for the number of Legendre symbol conditions specified; that is, m C = k | P | + 12 ( k − k ) + k ( k − k ) . Then, for any x ∈ X , we have X a ∈ F M ∪ M P (cid:18) − m C · k ! − (cid:8) σ ∈ P ( r ) : x ∈ X C ( σ, a ) (cid:9)(cid:19) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS ≤ | P | + k +1 · k k · − m C + | M ∪ M P | · k ! . Proof.
Write W ( a ) for (cid:8) σ ∈ P ( r ) : x ∈ X C ( σ, a ) (cid:9) . We see that theaverage size of W ( a ) over all a is − m C · r ! , as the condition that x is in X C ( σ, a ) for a given x and σ is given by m C binary conditions on a .We now consider the average of | W ( a ) | . We see that | W ( a ) | is thenumber of permutation pairs ( σ , σ ) so that x is in X C ( σ , a ) and X C ( σ , a ) .Write W ( σ , σ ) for the set of a so that x is in both of these sets.The maximal number of conditions on an a in W ( σ , σ ) is m C ; a lowerbound on the number of conditions depends on σ and σ . Take d to be thenumber of i ∈ [ r ] so that σ − ( i ) and σ − ( i ) are both at most k . Then wesee that W ( σ , σ ) is determined by at least m c − d ( | P | + k ) conditions. So | W ( σ , σ ) | ≤ − m c + d ( | P | + k )+ | M ∪ M P | At the same time, the number of ways to choose a permutation π of [ k ] so that (cid:12)(cid:12)(cid:12)(cid:12) π (cid:0) [ k ] (cid:1) ∩ [ k ] (cid:12)(cid:12)(cid:12)(cid:12) ≥ d is bounded by the number of ways to choose two cardinality d subsets from [ k ] and a bijection between these sets and a bijection between their com-plements in [ k ] . This is bounded by d ! · (cid:18) k d (cid:19) · ( k − d )! ≤ (cid:18) k k (cid:19) d · k ! Then the mean value of | W ( a ) | is bounded by X d ≥ − m c + d ( | P | + k ) (cid:18) k k (cid:19) d · k ! . This is a geometric sum; combining this with our calculation of the meanof | W ( a ) | then gives the proposition. (cid:3) Proof of Theorem 6.4.
Without loss of generality, we may assume that M and M P are both maximal as in Proposition 6.7. We also define m C and X C ( σ, a ) as in that proposition, and we assume that X , . . . , X k are sin-gletons x , . . . , x k . ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 67 We prove the theorem by bounding X a (cid:12)(cid:12)(cid:12)(cid:12) k ! | X | − m C · X σ ∈ P ( k ) | X C ( σ, a ) | (cid:12)(cid:12)(cid:12)(cid:12) + X σ ∈ P ( k ) X a (cid:12)(cid:12)(cid:12)(cid:12) m C · | X C ( σ, a ) | − | M ∪ M P | | X ( σ, a ) | (cid:12)(cid:12)(cid:12)(cid:12) . The former we can bound via the previous proposition by k − c · | X | · | M ∪ M P | · k ! For the second sum, fix a σ and a choice of a outside of the values referencedin the definition of X C ( σ, a ) . There are then m c choices of a , and these a partition X into sets X C ( σ, a ) = { x }×· · ·×{ x k }× X k +1 ( a ) ×· · ·× X k ( a ) × X k +1 ×· · ·× X r , with X i ( a ) the subset of X i consistent with the choice of P and x , . . . , x k .Given an k < i ≤ k , the union of all X C ( σ, a ) for which | X i ( a ) | ≤ | P | + k · k c · | X i | has order at most k − c | X | . Because of this, we can restrict the sum to beover only ( σ, a ) that do not satisfy this inequality at all such i , introducingan error with magnitude bounded by k k − c · | X | · | M ∪ M P | · k ! Once restricted, each summand can be bounded by Proposition 6.3 to beless than t ′− c · m C | X | , giving the theorem. (cid:3) Boxes of integers.
The results of this section, like the results of thefirst half of this paper, all apply to points in the product spaces of sets ofprimes. In this section, we finally give the definitions and results that allowus to move from the set of positive integers less than a certain bound to sucha product space. As before, S r ( N, D ) denotes the set of squarefree integersless than N with exactly r prime factors, of which all are greater than D . Definition 6.8.
Take N ≥ D ≥ D ≥ to be real numbers, and take r to bea positive integer satisfying (5.1). Let W be a subset of elements S r ( N, D ) that are comfortably spaced above D (cf. Definition 5.3).Let k ≤ r be a nonnnegative integer, and choose a sequence of increasingprimes D < p < · · · < p k < D ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Take D < t k +1 < t k +2 < · · · < t r to be an increasing sequence of real numbers. For i > k , define t ′ i = (cid:18) e i − k · log D (cid:19) · t i . Take X = X × · · · × X r , where X i = { p i } for i ≤ k and X i is the set of primes in the interval ( t i , t ′ i ) for i > k .If t ′ i < t i +1 for all r > i > k , we see that there is a natural bijection from X to a subset of S r ( N, D ) ; abusing notation, we write X for this subset too.We call X a box meeting W if X ∩ W is nonempty.The restriction to comfortably spaced W means that, if X ∩ W is nonempty,then we automatically have that the X i are disjoint sets and none of themcontain any prime below D . This is very convenient. Proposition 6.9.
Take N ≥ D ≥ D ≥ with log log N ≥ D and r satisfying (5.1) , and take W to be a subset of S r ( N, D ) that is com-fortably spaced above D . Suppose V is any other subset of S r ( N, D ) , andsuppose there are constants δ, ǫ > such that | W | > (1 − ǫ ) · (cid:12)(cid:12) S r ( N, D ) (cid:12)(cid:12) and so that, for any box X meeting W , we have ( δ − ǫ ) · (cid:12)(cid:12) B X (cid:12)(cid:12) < (cid:12)(cid:12) V ∩ B X (cid:12)(cid:12) < ( δ + ǫ ) · (cid:12)(cid:12) B X (cid:12)(cid:12) . Then (cid:12)(cid:12) V (cid:12)(cid:12) = δ (cid:12)(cid:12) S r ( N, D ) (cid:12)(cid:12) + O (cid:18) (cid:0) ǫ + (log D ) − (cid:1) · (cid:12)(cid:12) S r ( N, D ) (cid:12)(cid:12)(cid:19) with absolute implicit constant.Proof. Take D k to be the space of tuples t = ( p , . . . , p k , t k +1 , . . . , t r ) corresponding to boxes meeting W ; we write the corresponding box as X ( t ) . Consider Z D k (cid:12)(cid:12) V ∩ X ( t ) (cid:12)(cid:12) · dp . . . dp k dt k +1 . . . dt r t k +1 . . . t r , where the measure corresponding to dp . . . dp k is one on every prime tupleand zero otherwise. If n ∈ W has exactly k prime factors less than N andcorresponds to the tuple ( q , . . . , q r ) , then n is in X ( t ) if ( q , . . . , q k ) = ( p , . . . , p k ) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 69 and, for i > k , t i ≤ p i ≤ t i (cid:18) e i − k · log D (cid:19) . Then, if(6.1) r Y i = k +1 (cid:18) e i − k · log D (cid:19) n < N, we have that the measure of the subset of D k corresponding to boxes con-taining n is r Y i = k +1 log (cid:18) e i − k · log D (cid:19) . If n is outside W but in S r ( N, D ) with exactly k prime factors below N , orif n is in W but does not satisfy (6.1), then the measure of boxes containing n is bounded by this product. Any n not satisfying (6.1) is in the range N · (cid:0) − A · log( D ) − (cid:1) ≤ n ≤ N where A is some positive constant.Taking H r ( N, D ) as in Section 5.3, and using (5.10) together with Propo-sition 5.5, we find r ! H r ( N, D ) − r ! H r (cid:0) (1 − c ) N, D (cid:1) = O (cid:18) c + (log log N ) log N (cid:19) · (cid:12)(cid:12) S r ( N, D ) (cid:12)(cid:12) with implicit constant absolute whenever c is in (0 , . From this, the num-ber of n not satisfying (6.1) is bounded by O ( | S r ( N, D ) | / log D ) .We then see that X k ≥ r Y i = k +1 log (cid:18) e i − k log D (cid:19) − Z D k (cid:12)(cid:12) V ∩ B ( t ) (cid:12)(cid:12) · dp . . . dp k dt k +1 . . . dt r t k +1 . . . t r is at least as large as (cid:12)(cid:12) V ∩ W (cid:12)(cid:12) − O (cid:0) (log D ) − · (cid:12)(cid:12) S r ( N, D ) (cid:12)(cid:12)(cid:1) and is no larger than | V | . The estimates on (cid:12)(cid:12) V ∩ X ( t ) (cid:12)(cid:12) relative to (cid:12)(cid:12) X ( t ) (cid:12)(cid:12) then give the proposition. (cid:3) In this proposition, W should be considered to be some “nice” subset of S r ( N, D ) . We have already given three different notions of niceness withcomfortable spacing, regularity, and extravagant spacing. Since our mainresults rely on using an effective, unconditional form of the Chebotarevdensity theorem, there is one more form of not-niceness that we must avoid.Whenever possible, we must avoid L functions that have Siegel zeros. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS Proposition 6.10.
Take d , d , . . . to be a potentially infinite sequence ofdistinct squarefree integers satisfying d i < (cid:12)(cid:12) d i +1 (cid:12)(cid:12) . Take d ′ i to be the product of the primes dividing d i that are greater than D ,and take d ′ to be the subset of the d ′ i for which | d i | is greater than D . Take N ≥ D ≥ D ≥ with log log N ≥ D and r satisfying (5.1) , anddefine V = [ X ∩ d ′ · Z = ∅ X. Here, the union is over all boxes of S r ( N, D ) that contain some element n divisible by an element of d ′ . We assume log D > D log D . Then | V | = O (cid:18) D · (cid:12)(cid:12) S r ( N, D ) (cid:12)(cid:12)(cid:19) Proof.
Choose d ′ i in S r ( N, D ) , writing it as a product p · · · · · p m . Supposesome element of X is divisible by d ′ i . Taking n in X , we see that there areprime factors q , . . . , q m of n such that q i = p i if p i < D and q i < p i < q i otherwise . If d ′ i < N / , there is then an absolute constant A so that the number of n sharing a box with a multiple of d ′ i is bounded by A m · Y p i ≤ D p − i · Y p i >D (log p i ) − ·| S r ( N, D ) | = O (cid:18)(cid:18) d ′ i (cid:19) · (cid:12)(cid:12) S r ( N, D ) (cid:12)(cid:12)(cid:19) . We can also bound the contribution from d ′ i ≥ N / by O ( N/ log N ) .We remove the first elements from the sequence d , d , . . . , renumberingso that | d | > D . We then get | d i | > D i , so | d ′ i | > D i − . Then thecontribution from the d ′ i with d ′ i < N / is O | S r ( N, D ) (cid:12)(cid:12) · X i> i log D ! , within the bound. The contribution from d ′ i > N / is O ( N/ log N ) , whichis also within the bound. This proves the proposition. (cid:3) Definition.
We call a box
Siegel-free above D if it is not contained in theset V defined in the above proposition with respect to the sequence definedin Definition 6.2. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 71 In addition, we will call a box C -regular if it contains some C -regularelement of S r ( N, D ) , and we will call it extravagantly spaced if it containssome extravagantly spaced element.In line with the previous two proposition, we see that we can ignore boxesthat are not Siegel-free above D so long as D is sufficiently large. Definition.
Choose absolute constants c , c > , and choose r , N , D satisfying (5.1) and D ≤ log log log N. Taking X to be a (comfortably spaced) box of S r ( N, D ) with D = D (log log N ) c , we call X acceptable if it is C -regular for C = c · log log log N and if it is Siegel-free above D .We now have all the tools to reprove Kane’s results on -Selmer groupsfrom the Markov chain analysis of Swinnerton-Dyer. To reprove the re-sults of Fouvry-Kl¨uners on -class groups of imaginary quadratic fields,we would repeat this argument starting from the Markov chain analysis ofGerth in [9]. Corollary 6.11.
There is an absolute c > so that we have the following:Take E/ Q to be any elliptic curve with full rational -torsion and norational cyclic subgroup of order four, and take P Alt ( j | n ) and R E, ( n ) asin the introduction. Take R to be the set of squarefree integers. Then, forany n ≥ and N > , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) [ N ] ∩ R E, ( n ) (cid:12)(cid:12) | [ N ] ∩ R | − . n →∞ P Alt ( n | n + n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) N ) c (cid:19) , with the implicit constant depending only on the choice of E .Proof. By applying Theorem 5.4 and Proposition 6.10 to Proposition 6.9,we find that it suffices to prove this result on acceptable boxes of twists in S r ( N, D ) with D larger than the largest bad prime of E .We will apply Theorem 6.4 to our acceptable box with k the minimalinteger so t ′ k +1 is larger than D . We take t = D , so the Siegel-lesscondition holds. Choose k minimal so that t ′ k +1 is larger than exp( D c ) ,and take k = r . Finally, take P to be the set of all primes less than D and − , and take M and M P maximal. We need to check that, for sufficiently ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS large N and some appropriate choice of c , . . . , c , the six conditions ofTheorem 6.4 hold.(1) The first condition always holds.(2) The second condition holds for sufficient N if c > c .(3) The third condition holds for sufficient N if c > c log 2 + c c and c is sufficiently small relative to the other constants.(4) The fourth condition holds for sufficient N .(5) The fifth condition holds for sufficient N if c < c and c is suffi-ciently small relative to the other constants.(6) The sixth condition holds for sufficient N if c , c > c and c issufficiently small relative to the other constants.It is a pleasantly mundane exercise to prove that there are positive constants c , . . . , c that satisfy all the inequalities stated above and in Theorem 6.4.Then, considered up to permutation, the Legendre symbol matrices foundin our acceptable box are equidistributed with error within the bound of thecorollary. Since the -Selmer rank depends only on the permutation class,we can now apply Swinnerton-Dyer’s work in [26]. This paper does notgive error estimates, but we can find them with just a little extra work onthe Markov chain described in [26, (20)].There is some A, ǫ > so we have the following: choose k = 0 , ,choose n > , and consider the Markov chain Y described by [26, (20)]with initial state n + k . Under this Markov chain, if T is the first passagetime of our Markov chain to state k , we can bound the expected value of (1 + ǫ ) T by A n . Similarly, if we start another Markov chain X initiallyequaling the stationary, and if T is the minimal time when Y T meets X T ,we find that we can bound the expected value of (1 + ǫ ) T by A n +1 . Then bythe logic of [18, Theorem 1.8.3], we find that there is some constant C so,in the notation of the final section of [26], (cid:12)(cid:12) Q ( d, M, CM ) − α d (cid:12)(cid:12) = O (cid:0) exp( − M ) (cid:1) . Plugging this estimate into the final equation of [26] then shows that, amongall Legendre symbol matrices corresponding to a twist with r prime factors,the proportion corresponding to rank d is . n →∞ P Alt ( j | n + j ) withmaximal error O (cid:0) exp( − cr ) (cid:1) for some constant c > , easily within ourerror term. This gives the corollary. (cid:3) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 73
7. P
ROOFS OF THE MAIN THEOREMS
In the previous section, we reduced distributional questions over the square-free integers to distributional questions over acceptable boxes. In this sec-tion, we extend this logic to more and more specialized product spaces. Ourgoal is to reduce to product spaces on which a combination of Proposition3.6, Proposition 4.4, and the Chebotarev density theorem suffice to provethe equidistribution results we want for k -Selmer groups and class groups.This will be enough to prove Theorems 1.1 and 1.4.With Proposition 3.6, the notions of k -Selmer groups and k -class groupshave become essentially interchangeable. In this section, we will state allour results and arguments on the Selmer side; straightforward adjustmentsto the argument would give the results on the class side.We begin by stating the explicit form of Theorem 1.1 that we will provein this section. Theorem 7.1.
There is an absolute constant c > so that, for any ellipticcurve E/ Q with full -torsion and no rational cyclic subgroup of order four,there is a choice of A > so that, for any choice of N > , any choice of m ≥ , and any sequence n ≥ n ≥ · · · ≥ n m +1 of nonnegative integersof the same parity, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ N ] ∩ m +1 \ k =1 R E, k ( n k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − P Alt ( n m +1 | n m ) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ N ] ∩ m \ k =1 R E, k ( n k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ AN · (log log log log N ) − cm m whenever the latter expression is well defined and positive. From this theorem, we can derive the following explicit form of Corollary1.2.
Corollary 7.2.
Take c to be a positive constant less than log 2log 6 , and take E/ Q to be an elliptic curve as in the previous theorem. Then there is some N > depending on E and c so that, for all N > N , we have (7.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:8) d ∈ [ N ] : corank Sel ∞ E ( d ) ≥ (cid:9)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N (cid:0) log log log log log N (cid:1) c . Proof.
We consider a Markov process whose states are the nonnegative in-tegers. At each step, we take the transition probability from state n to state j to be P Alt ( j | n ) . In this Markov chain, we note that the probability ofstepping to after is . ; the probability of stepping to either or afterany other even state is at least two thirds; and the probability of after anyother odd state is at least . . (All these facts follow from the formula for ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS P Alt ( j | n ) given in [12]). Then, independent of the initial probability dis-tribution, the chance that the process is in a state other than or after m steps is bounded by O (2 − m ) .Choose c < c ′ < log 2log 6 , and take m = (cid:22) c ′ log 2 log log log log log log N (cid:23) . We can assume that this is positive. From Corollary 6.11 and the formulasfrom [12], we see the proportion of d ∈ X N such that E ( d ) has -Selmerrank exceeding m + 2 is bounded by O (cid:16) − c m (cid:17) for some constant c > ,in the range of the corollary’s estimate for sufficient N .We see that the set being bounded in (7.1) is contained in [ N ] ∩ [ n m ≥ R E, m ( n m ) = [ N ] ∩ [ n ≥···≥ n m ≥ m \ k =1 R E, k ( n k ) For sufficient N and some constant c > , we also have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ N ] ∩ m \ k =1 R E, k ( n k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m − Y k =1 P Alt ( n k +1 | n k ) · (cid:12)(cid:12)(cid:12)(cid:12) [ N ] ∩ R E, ( n ) (cid:12)(cid:12)(cid:12)(cid:12) + AmN · (log log log log N ) − c m m for any sequence n ≥ · · · ≥ n m ≥ . By summing this over all paths with n ≤ m and using our Markov chain result, we find that the set in (7.1) hasmaximal size O (cid:0) − m N (cid:1) + Am m +1 N · (log log log log N ) − c m m , which is within the bound of the corollary for sufficiently large N . (cid:3) We now proceed to the proof of Theorem 7.1. To prove the result, werecast it in increasingly specialized situations. The first and easiest of theserecasts is to move from an equidistribution result on integers less than N toboxes in S r ( N, D ) . Proposition 7.3.
There is an absolute constant c > so that, for any choiceof E/ Q as above, there is some A > so that we have the following:Take D one greater than the largest bad prime of E . Choose a positivereal N > , and take D = D (log log N ) / . Choose r satisfying (5.1) , and let X be any box of some S r ( N, D ) with this D that is extravagantly spaced, Siegel free above D , and √ log log log N ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 75 regular. Then, for any choice of m ≥ and any sequence n ≥ · · · ≥ n m +1 of nonnegative integers of the same parity, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ∩ m +1 \ k =1 R E, k ( n k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − P Alt ( n m +1 | n m ) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ∩ m \ k =1 R E, k ( n k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A | X | · (log log log log N ) − cm m whenever the right hand side is defined and positive.Proof that Proposition 7.3 implies Theorem 7.1. With this proposition, The-orem 7.1 is a consequence of applying Theorem 5.4 and Proposition 6.10 toProposition 6.9. (cid:3) As x in X varies, the tuples w = ( T , T , ∆ , ∆ ) corresponding to -Selmer elements change. Our next step is to restrict our attention to sets X ( a ) , where we no longer have this problem. This reduction is technicallycumbersome, as some choices of a will prevent us from finding sets ofvariable indices as in part (3) of Definition 3.4. We begin with the notationwe will need. Definition.
Take E , X , N , and m as in Proposition 7.3, and assume theextravagant spacing of X is between indices k gap and k gap + 1 . Take P tobe the union of the prime numbers less than D with {− } . In the contextof Definition 6.1, take M and M P maximal, and let a be any function in F M ∪ M P .Under these circumstances, any ¯ x ∈ X [ r ] entirely contained in X ( a ) isquadratically consistent, so we can define additive-restrictive input as inDefinition 3.4. Take Ctp (1) , . . . , Ctp ( m − to be a choice of lower pairingsas in part (1) of this definition, choose a basis w , . . . , w n and the integer n m as in part (2), and choose variable indices as in part (3). We assume i b > k gap ; writing S pre-gap for the union of the S ( j , j ) − { i b } , we assume S pre-gap ⊆ (cid:2) . k gap , k gap (cid:3) . Take P − pre-gap to be an element of Q i ∈ [ k gap ] − S pre-gap X i . We assume that a isconsistent with the choice of P − pre-gap .Then all of the data we have chosen so far will be called inital data forProposition 7.4 .We will write X i ( a, P − pre-gap ) for the subset of X i consistent with a and the data of P − pre-gap , and take X ( a, P − pre-gap ) for the subset of X ( a ) equaling P − pre-gap on [ k gap ] − S pre-gap .Finally, given a choice of sequence pairings Ctp (1) , . . . , Ctp ( k ) , take X ( a, P − pre-gap , k ) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS for the subset of i ( a, P − pre-gap ) whose first k Cassels-Tate pairings agree withthe given sequence.
Proposition 7.4.
There is a constant c > so we have the following:Choose initial data for Proposition 7.4 as above. Writing n max = (cid:22)r cm m log log log log log N (cid:23) , we assume n max is defined, positive, and greater than n . We also assumethat we have (7.2) (cid:12)(cid:12) X i ( a, P − pre-gap ) (cid:12)(cid:12) > − k pre-gap · | X i | . for i ∈ S pre-gap .Finally, take Ctp ( m ) to be any n m × n m alternating matrix with coefficentsin F . Then there is some constant A > depending only on E so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( a, P − pre-gap , m ) (cid:12)(cid:12) − − nm ( nm − · (cid:12)(cid:12) X ( a, P − pre-gap , m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A · (cid:12)(cid:12) X ( a, P − pre-gap ) (cid:12)(cid:12) · (log log log log N ) − cm · m . Proof that Proposition 7.4 implies Proposition 7.3.
This implication wouldbe easy if we could prove the above Proposition for arbitrary choices of a , P − pre-gap , and the pairnigs. However, there are three kinds of bad ( a, P − pre-gap ) to consider. First, we need to avoid the case where n is not less than n max .Second, we need to avoid a such that, for some choice of pairings, we can-not find variable indices suitable for the initial data. Finally, we need toavoid ( a, P − pre-gap ) for which (7.2) does not hold for some i ∈ S pre-gap . Weclaim that the union of X ( a, P − pre-gap ) over all three kinds of bad ( a, P − pre-gap ) fits into the error term of Proposition 7.3.We first claim that the union of X ( a ) for which n m ≥ n max fits into theerror term of Proposition 7.3. This is a consequence of the argument ofCorollary 6.11 and the formulas in [12].Next, consider the set of a for which, for some choice of pairings Ctp ( k ) and a basis, there is no choice as in the lemma for the variable indices S ( j , j ) . We claim the union of the X ( a ) over the set of a for which thisholds also fits in this error bound.First, we claim that the proportion of a for which there are -Selmerelements w , w so that either w or w is non-torsion and(7.3) (cid:12)(cid:12) ( T ( w ) + T ( w )) ∩ [0 . k gap , k gap ] (cid:12)(cid:12) > (0 .
25 + 2 − n max ) · k gap has density at most O (cid:0) (15 / r + exp (cid:0) − n max · k gap (cid:1)(cid:1) in the space F M ∪ M P . Here, T + T denotes the symmetric difference. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 77 Call a generic if there is no non-torsion -Selmer tuple w of X ( a ) forwhich T ( w ) , T ( w ) , and [ r ] are not linearly independent sets with respectto symmetric difference, and if there are no pair of non-torsion -Selmertuples ( w , w ) with w + w also non-torsion, but where T ( w ) , T ( w ) , T ( w ) , T ( w ) , and [ r ] are not linearly independent. From Lemmas 4-6 of[26], we see that the proportion of a that are not generic due to the conditionon w is bounded by O (cid:0) | P | · (3 / r (cid:1) . For the condition on ( w , w ) , we can use Lemma 7 from [26] after notingthat the condition u ′ = u ′′ can be weakend to u ′ /u ′′ ∈ X S with no changein the argument. Then, from this lemma, the proportion of non-generic a isbounded by A | P | · (15 / r for some absolute A > .Now suppose w is a generic tuple as above. From genericity, we canprove that the local conditions at the r primes coming from X are inde-pendent, and we find that the proportion of a so that w is a -Selmer tuplefor X ( a ) is bounded by O (4 − r ) . Similarly, if ( w , w ) is generic as above,the probability that w and w are both -Selmer for X ( a ) is bounded by O (16 − r ) .Then Hoeffding’s inequality is sufficient to complete the estimate of thedensity of a in F M ∪ M P not satisfying (7.3) for some w , w .For any a other than those in this set, it is easy to find sets of variableindices if n max is larger than some constant determined by E . First, choosesome i b > k gap , and add torsion to the basis as necessary so i b is not in any T i ( w j ) . Then each S ( j , j ) − { i b , i a ( j , j ) } can be taken to be any subsetof size m inside of T ( w j ) ∩ (cid:0) [ r ] − T ( w j ) (cid:1) ∩ \ j = j (cid:18) [ r ] − (cid:0) T ( w j ) ∪ T ( w j ) (cid:1)(cid:19) . The assumptions on a give that this intersection has density about − n onthe integers in the interval [0 . k gap , k gap ] , which will be larger than m forsufficient n max . We can find i a similarly.If k < . k gap , we see that permutations of the first k indices do notchange whether (7.3) holds for a given a . Then, from Theorem 6.4, wefind that our argument implies that the union of X ( a ) over all a for whichit may be impossible to find a set of variable indices fits into the error ofProposition 7.3.Next, we claim that the union of X ( a, P − pre-gap ) over all ( a, P − pre-gap ) forwhich (7.2) does not hold for some i fits into the error of Proposition 7.3.We will work in the context of Proposition 6.3. To do this, add the primes ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS p , . . . , p k of the box to the set P ; taking X i ( a, P ) to be the subset of X i consistent with P and the choice of a , we will attempt to apply the argumentof the proposition to X ( a, P ) × · · · × X r ( a, P ) . This will only work if no X i ( a, P ) is smaller than t ′ ) c ′ · | X i | for somechoice of the constant c ′ . For a good choice of constants, outside a set ofchoices of a over which the union of the X ( a ) fits into the error of Proposi-tion 7.3, we always have X i ( a, P ) ≥ t ′ ) c ′ · | X i | . Suppose we have such an a . Then a choice of P − pre-gap for which (7.2) doesnot hold would be exceptional in the sense of the proof of Proposition 6.3.Per that proof, the union of all such exceptional sets fits into the error ofProposition 7.3.Finally, we note that there are at most mn sequences of pairingsCtp ( k ) . Writing X aP − for X ( a, P − pre-gap ) , the claim of the proposition thenimplies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X aP − ∩ m +1 \ k =1 R E, k ( n k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − P Alt ( n m +1 | n m ) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X aP − ∩ m \ k =1 R E, k ( n k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A · mn · | X aP − | · (log log log log N ) − cm · m . A computation shows that the sum of this error over all a and P − pre-gap is thenwithin the error of Proposition 7.3. This gives the lemma. (cid:3) Now that we have a set of variable indices, the next structure to add isa set of governing expansions as in part (5) of Definition 3.4. The require-ments on these governing expansions are quite stringent, making this stepthe most interesting part of the reduction of Theorem 7.1. We first neednotation for the extra structure.
Definition.
Choose initial data for Proposition 7.4 that obeys the conditionsof Proposition 7.4. Choose subsets Z i of X i for each i in S pre-gap . Foreach set S ( j , j ) of variable indices, choose a set of governing expansions G ( i a ( j , j )) on the product Z pre-gap of the Z i . For any set S of the form S ( j , j ) − { i b } and any ¯ x ∈ (cid:0) Z pre-gap (cid:1) S , we assume φ ¯ x ( G ( i a ( j , j ))) exists.For x ∈ Z pre-gap , write L ( x ) for the composition of all quadratic fieldsramified only at ∞ , the places of P , and the places of P − pre-gap . Write ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 79 M ( j , j ) for the composition of the fields of definition for the set of φ ¯ x with ¯ x ∈ (cid:0) Z pre-gap (cid:1) S ( j ,j ) −{ i b } . Also write M ◦ ( j , j ) for the composition ofthe fields of definition for the set of φ ¯ x with ¯ x ∈ (cid:0) Z pre-gap (cid:1) S for some propersubset S of S ( j , j ) − { i b } .We assume that, for each S ( j , j ) , the field M ◦ ( j , j ) / Q splits com-pletely at all primes in P , in P − pre-gap , and in any Z i with i outside S ( j , j ) −{ i b } .Finally, take M to be the composition of any L ( x ) with the set of M ( j , j ) ,and take M ◦ to be the composition of any L ( x ) with the set of M ◦ ( j , j ) .We write X i ( M ◦ ) to be the subset of primes p in X i so p is consistent with the choice of a and P − pre-gap and the prime p splits completely in each M ◦ ( j , j ) . Note that X i ( M ◦ ) is described alternatively as the subset of X i mapping under theFrobenius map to one specific central element of Gal ( M ◦ / Q ) . Finally, take Z = { P − pre-gap } × Z pre-gap × Y i>k gap X i ( M ◦ ) . Proposition 7.5.
There is an absolute constant c > so we have the fol-lowing:Choose initial data for this proposition as above. Taking M = (cid:4) (log log log log N ) / m +1) (cid:5) , we assume that M is well defined and positive, and that each Z i has cardi-nality M .Then there is a constant A > depending only on E so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ∩ X ( a, P − pre-gap , m ) (cid:12)(cid:12) − − nm ( nm − · (cid:12)(cid:12) Z ∩ X ( a, P − pre-gap , m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A · (cid:12)(cid:12) Z ∩ X ( a, P − pre-gap ) (cid:12)(cid:12) · (log log log log N ) − cm · m . Proof that Proposition 7.5 implies Proposition 7.4.
Choose initial data forProposition 7.4 obeying the conditions of Proposition 7.4. Write V pre-gap forthe subset of Q i ∈ S pre-gap X i consistent with P − pre-gap and the conditions of a .Take R = (cid:4) exp exp (cid:0) . k gap (cid:1)(cid:5) . We can assume R is positive. We also assume that m < log log log log log N, as Proposition 7.4 is otherwise vaccuous. ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS We will choose t ≥ and, for each i ∈ S pre-gap , we will choose sequencesof subsets Z i , . . . , Z ti ⊆ X i ( a, P − pre-gap ) , with each set of cardinality M . We take Z ℓ pre-gap = Y i ∈ S pre-gap Z ℓi . We assume that these subsets obey the following conditions: • For ℓ = ℓ ′ , we have that Z ℓ pre-gap and Z ℓ ′ pre-gap intersect at at most onepoint. • Each Z ℓ pre-gap is a subset of V pre-gap , and any point in V pre-gap is in atmost R of the Z ℓ pre-gap . • The set Z ℓ pre-gap can be used as initial data for Proposition 7.5.Furthermore, we assume that the sequence of Z ℓ pre-gap cannot be extendedunder these requirements to a sequence of t + 1 subgrids.Write X pre-gap = Y i ∈ S pre-gap X i ( a, P − pre-gap ) . Take V badpre-gap to be the set of points in V pre-gap that are consistent with thechoice of a and P − pre-gap and that are in fewer than R of the Z ℓ pre-gap . Write δ for the density of V badpre-gap in X pre-gap . By a greedy algorithm, we can choosea subset W of V badpre-gap of density at least δ/RM m +1 such that no point in W is in more than two of the Z ℓ pre-gap .By adjoining splitting behavior at the primes in P − pre-gap to the system con-structed in Proposition 3.3, we can then define an additive-restrictive systemon X pre-gap with Y ◦∅ = W and where, if ¯ x ∈ Y ◦ S pre-gap , then the governing ex-pansions defined at ¯ x are as required for Proposition 7.5. The maximalsize of the abelian groups in this additive-restrictive system is bounded by k gap + | P | . Then, by Proposition 3.2, the density of Y ◦ S pre-gap in X pre-gap × X pre-gap is at least (cid:18) δ k gap | P | · RM m +1 (cid:19) | S pre-gap | . We note | S pre-gap | ≤ ( m + 1) n . In addition, for sufficently large N , wealways have | X i ( a, P − pre-gap ) | > exp exp(0 . · k gap ) for i ∈ S pre-gap . Applying Proposition 4.1 and the assumptions on t , we thenhave M m > exp(0 . · k gap )( m + 1)3 ( m +1) n m · (exp(0 . · k gap ) + log δ − ) ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 81 for sufficiently large N . We can then bound δ by exp( − e . k gap ) for suffi-ciently large N . Then, following the logic of Proposition 6.3, we see thatthe subset of x ∈ X ( a, P − pre-gap ) for which π S pre-gap ( x ) is in V badpre-gap fits easilyinto the error term of Proposition 7.4.We associate grids Z ℓ pre-gap with fields M ℓ and M ℓ ◦ and a supergrid Z ℓ asabove. For x ∈ X ( a, P − pre-gap ) with π S pre-gap ( x ) outside of V badpre-gap , write Λ( x ) to be the number of ℓ ≤ t for which x is in Z ℓ . Write d ML for the degreeof M ℓ over some L ( x ) with x ∈ Z ℓ pre-gap ; from Proposition 2.4, we find thisdegree does not depend on ℓ or x . For i > k gap , write X i ( L ( x )) for thesubset of X i ( a, P pre-gap ) consistent with the choice of x . From the Cheb-otarev density theorem as presented in Proposition 6.5 and the definition ofextravagant spacing, we then have | X i ( M ℓ ◦ ) | = d − ML · | X i ( L ( x )) | (cid:0) O (cid:0) e − k gap (cid:1)(cid:1) for i > k gap . Following Proposition 6.3 then gives that the subset of Y i>k gap X i ( M ℓ ◦ ) consistent with a has order d − ( r − k gap ) ML · | X ( a, P − pre-gap ) ∩ π − S pre-gap ( x ) | · (cid:0) O (cid:0) e − k gap (cid:1)(cid:1) . From this, we calculate that Λ( x ) has average value d − ( r − k gap ) ML R · (cid:0) O (cid:0) e − k gap (cid:1)(cid:1) . Similarly, from the requirements on Z ℓ ∩ Z ℓ ′ and Proposition 2.4, we see M ℓ ◦ M ℓ ′ ◦ has degree d ML over L ( x ) for Z ℓ and Z ℓ ′ distinct grids containing x . Then the average square value of Λ( x ) is (cid:16) d − r − k gap ) ML ( R − R ) + d − ( r − k gap ) ML R (cid:17) · (cid:0) O (cid:0) e − k gap (cid:1)(cid:1) = d − r − k gap ) ML · R (cid:0) O (cid:0) e − k gap (cid:1)(cid:1) . Then, outside a set of density O (cid:0) e − . k gap (cid:1) in the domain of Λ , we find that Λ( x ) over the mean value of Λ is within e − . k gap of . The effect of the setof low density fits into the error term of Proposition 7.4, and the variancebetween the Λ( x ) also fits into the error of this proposition. Then, to proveProposition 7.4, it is enough to prove Proposition 7.5 for each grid Z ℓ . (cid:3) Proof of Proposition 7.5.
Take F to be a nonzero multiplicative characterof the vector space of n m dimensional alternating matrices with coefficients ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS in F . For x ∈ Z ∩ X ( a, P − pre-gap , m − , write CT ( x ) for the Cassels-Tatepairing on D ( m ) . To prove the proposition, it is enough to prove that X x ∈ Z ∩ X ( a, P − pre-gap , m − F ( CT ( x ))= O (cid:0)(cid:12)(cid:12) Z ∩ X ( a, P − pre-gap ) (cid:12)(cid:12) · (log log log log N ) − cm · m (cid:1) for each F .Choose an F , and take j < j ≤ n so that F depends on the value ofCT ( x ) j j , and take S = S ( j , j ) . From Proposition 2.4, we find that thereis a natural bijectionGal ( M ( j , j ) M ◦ /M ◦ ) ∼ = G S −{ i b } ( π S −{ i b } ( Z )) of F vector spaces, with our notation as in Definition 4.2. For σ in thisGalois group, we take X i b ( σ ) to be the subset of X i b ( M ◦ ) mapping underFrobenius to σ . From the Chebotarev density theorem, we find | X i b ( σ ) | = 2 − ( M − m +1 · | X i b ( M ◦ ) | · (cid:0) O (cid:0) e − k gap (cid:1)(cid:1) . Choose x i ∈ X i ( M ◦ ) for i above k gap besides i b such that the set of x i isconsistent with a , writing this tuple as P − post-gap . From Proposition 6.6 andPropoosition 6.3, we see that, outside a negligible set of choices of P − post-gap ,if we write X i b ( P − post-gap ) for the subset of X i b consistent with a , we have(7.4) (cid:12)(cid:12) X i b ( σ ) ∩ X i b ( P − post-gap ) (cid:12)(cid:12) = 2 − ( M − m +1 · (cid:12)(cid:12) X i b ( M ◦ ) ∩ X i b ( P − post-gap ) (cid:12)(cid:12) · (cid:0) O (cid:0) e − k gap (cid:1)(cid:1) for each σ .On the grid Z AR = Z pre-gap × (cid:0) X i b ( M ◦ ) ∩ X i b ( P − post-gap ) (cid:1) , we can find full additive-restrictive input as in Definition 3.4. The corre-sponding additive-restrictive system has abelian groups with orders boundedby n max ( n max +2 m +6) . We now apply Proposition 4.4 to the additve-restrictivesystem A ( P )( j , j ) . By Propositions 3.6 and 4.4, if(7.5) ǫ < − n max ( n max +2 m +6) and(7.6) log M ≥ A · m +2 log ǫ − , then there is a choice of σ , . . . , σ M in Gal ( M ( j , j ) M ◦ /M ◦ ) so, for any σ in this Galois group and any choice of Z ′ AR = Z pre-gap × { x , . . . , x M } with x i ∈ X i b ( σ + σ i ) ∩ X i b ( P − post-gap ) for all i ≤ M, ∞ -SELMER GROUPS AND ∞ -CLASS GROUPS 83 we have X x ∈ Z ′ AR F ( CT ( x )) ≤ ǫ · | Z ′ AR | . From the estimate (7.4), we see that Z AR can be split into grids Z ′ AR withleftovers fitting into the error term of the proposition, so we have equidis-tribution on Z AR too.For an appropriate constant c ′ > , we find that ǫ = (log log log log N ) − c ′ ( m +1)6 m satisfies both (7.5) and (7.6) for ǫ sufficiently small. This gives the proposi-tion, hence the proposition, hence the proposition, hence the theorem, hencethe corollary. (cid:3) R EFERENCES [1] Milton Abramowitz and Irene A Stegun,
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