The Cohen-Lenstra heuristics, moments and p j -ranks of some groups
aa r X i v : . [ m a t h . N T ] M a r THE COHEN-LENSTRA HEURISTICS, MOMENTS AND p j -RANKS OF SOME GROUPS CHRISTOPHE DELAUNAY AND FR´ED´ERIC JOUHET
Abstract.
This article deals with the coherence of the model given by theCohen-Lenstra heuristic philosophy for class groups and also for their gen-eralizations to Tate-Shafarevich groups. More precisely, our first goal is toextend a previous result due to E. Fouvry and J. Kl¨uners which proves thata conjecture provided by the Cohen-Lenstra philosophy implies another suchconjecture. As a consequence of our work, we can deduce, for example, a con-jecture for the probability laws of p j -ranks of Selmer groups of elliptic curves.This is compatible with some theoretical works and other classical conjectures. Introduction and notation
The Cohen-Lenstra heuristics and their generalizations to Tate-Shafarevich groupsare models for formulating conjectures related to class groups of number fields andTate-Shafarevich groups of elliptic curves varying in some natural families. Thisarticle deals with the coherence of the model. More precisely, our aim is to provethat a conjecture provided by the Cohen-Lenstra philosophy implies another suchconjecture. This work actually extends and generalizes an earlier one by E. Fouvryand J. Kl¨uners in [FK06] which deals with class groups; we will follow their pre-sentation and adapt the main techniques of their proofs.We will use the following notation. The letter d will denote a fundamental dis-criminant and C ℓ ( K d ) the class group associated to the quadratic number field K d = Q ( √ d ). The letter p will always denote a prime number. If G is a finiteabelian group, the p j -rank of G is defined by rk p j ( G ) = dim F p p j − G/p j G . For anyreal valued function f defined over isomorphism classes of finite abelian groups, wesay that f (C ℓ ( K d )) has average value c ± ∈ R if X < ± d
1. Secondly, we will obtain analogous resultsconcerning heuristics on Tate-Shafarevich groups and on Selmer groups of ellipticcurves ([Del01, Del07, DJ12]).We recall for ( a, q ) ∈ C with | q | < k ∈ Z the q -shifted factorial( a ; q ) k := k = 0(1 − a ) . . . (1 − aq k − ) if k > / (1 − aq − ) . . . (1 − aq k ) if k < , and ( a ; q ) ∞ := lim k → + ∞ ( a ; q ) k . Note that (1 /p ; 1 /p ) k = η k ( p ), where η k is thefunction defined in [CL84] and used in [FK06]. We will also use the q -binomialcoefficient (cid:20) nk (cid:21) q := ( q ; q ) n ( q ; q ) k ( q ; q ) n − k ∈ N [ q ] . A partition λ := ( λ > λ > · · · ) of a nonnegative integer n is a finite decreasingsequence of nonnegative integers whose sum is equal to n . If λ is a partition of n , wewrite | λ | = n and the notation λ = 1 m m · · · ℓ m ℓ means that m i is the multiplicityof the integer i in λ (hence, we have n = λ + λ + · · · = | λ | = m +2 m + · · · + ℓm ℓ ). If µ := ( µ > µ > · · · ) is a second integer partition, then we define ( λ | µ ) := P i λ i µ i (we will often use the statistics ( λ | λ ) = P i λ i which must not be mistaken for | λ | = ( P i λ i ) ). Finally, the notation µ ⊆ λ means that µ i λ i for all i > p -group G has type λ = 1 m · · · ℓ m ℓ if G ≃ ( Z /p Z ) m ⊕ · · · ⊕ (cid:0) Z /p ℓ Z (cid:1) m ℓ . If λ = 1 m · · · ℓ m ℓ is an integer partition, we denote by λ ′ := ( λ ′ > λ ′ > · · · ) itsconjugate defined by λ ′ k = P ℓj = k m j for all k . We have | λ | = | λ ′ | .As in [DJ12], we denote by C λ,µ ( p ) the number of subgroups of type µ in a finiteabelian p -group of type λ , which can be expressed by(1) C λ, µ ( p ) = p P i > µ ′ i +1 ( λ ′ i − µ ′ i ) Y i > (cid:20) λ ′ i − µ ′ i +1 λ ′ i − µ ′ i (cid:21) p , showing that it is a polynomial in the variable p , with positive integral coefficients.Using the Cohen-Lenstra philosophy ([CL84]) and a combinatorial analysis, weobtained the following conjecture ([DJ12, Conjecture 1]). Conjecture 1.
For any positive integer ℓ , let λ = 1 m m · · · ℓ m ℓ be an integerpartition, and assume that p > . As d is varying over the set of fundamental neg-ative discriminants, the average of | C ℓ ( K d )[ p ] | m | C ℓ ( K d )[ p ] | m · · · | C ℓ ( K d )[ p ℓ ] | m ℓ is equal to X µ ⊆ λ C λ,µ ( p ) , In order to simplify the notations, we will exclude p = 2 for class groups. However, one canobtain directly the case p = 2 by replacing C ℓ ( K d ) by C ℓ ( K d ) all along the article. HE COHEN-LENSTRA HEURISTICS, MOMENTS AND p j -RANKS OF SOME GROUPS 3 where the sum is over all integer partitions µ ⊆ λ .Similarly, as d is varying over the set of fundamental positive discriminants, theaverage of | C ℓ ( K d )[ p ] | m | C ℓ ( K d )[ p ] | m · · · | C ℓ ( K d )[ p ℓ ] | m ℓ is equal to X µ ⊆ λ C λ,µ ( p ) p −| µ | . Concerning the probability laws of p j -ranks rk p j (C ℓ ( d )), the following conjecturecomes naturally from [Del11, Corollary 11]. Conjecture 2.
Let ℓ be a positive integer and µ := µ > µ > · · · > µ ℓ > apartition of length ℓ ( µ ) ℓ (i.e., µ ℓ +1 = 0 ). Assume that p > . Then, as d is varying over the set of fundamental negative discriminants, the probability that rk p j (C ℓ ( K d )) = µ j for all j ℓ is equal to (1 /p µ ℓ +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 . Moreover, as d is varying over the set of fundamental positive discriminants, theprobability that rk p j (C ℓ ( K d )) = µ j for all j ℓ is equal to (1 /p µ ℓ +2 ; 1 /p ) ∞ p µ + ··· + µ ℓ +( µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 . Very few is known about these conjectures. Davenport and Heilbronn ([DH71])proved Conjecture 1 for p = 3 and λ = 1 . In [FK07], the authors proved Conjec-ture 1 for p = 2 (replacing C ℓ ( K d ) by C ℓ ( K d ) ) and any λ = 1 m .The conjectures mentioned in the introduction coming from the seminal work of[CL84] and studied in [FK07, FK06] correspond to Conjecture 1 for λ = 1 m andConjecture 2 for ℓ = 1. More precisely, if λ = 1 m , Conjecture 1 says that theaverage of | C ℓ ( K d )[ p ] | m for class groups of imaginary (resp. real) quadratic fieldsis equal to (with the notations of [DJ12]) M ( x n ) = n X k =0 (cid:20) nk (cid:21) p resp. M ( x n ) = 1 p n X k =0 (cid:20) nk (cid:21) p ! Those correspond to N ( n, p ) (resp. M n ( p )) used in [FK07, FK06]. Fouvry andKl¨uners proved in [FK06] that if Conjecture 1 is true with λ = 1 m for all m , thenConjecture 2 is true with ℓ = 1 for all µ >
0. We will adapt their proof anduse a result in [DJ12] to simplify one of its step in order to obtain the followinggeneralization.
Theorem 1.
Assume that p > . Let ℓ be a positive integer and assume that forany λ = 1 m m · · · ℓ m ℓ , as d is varying over the set of fundamental negative dis-criminants, the average of | C ℓ ( K d )[ p ] | m | C ℓ ( K d )[ p ] | m · · · | C ℓ ( K d )[ p ℓ ] | m ℓ is equalto P µ ⊆ λ C λ,µ ( p ) . Then for any µ > µ > · · · > µ ℓ > , as d is varying over theset of fundamental negative discriminants, the probability that rk p j (C ℓ ( K d )) = µ j for all j ℓ is equal to (1 /p µ ℓ +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 . Furthermore, assume that for any positive integer ℓ and any λ = 1 m m · · · ℓ m ℓ ,as d is varying over the set of fundamental positive discriminants, the averageof | C ℓ ( K d )[ p ] | m | C ℓ ( K d )[ p ] | m · · · | C ℓ ( K d )[ p ℓ ] | m ℓ is equal to P µ ⊆ λ C λ,µ ( p ) p −| µ | .Then for any µ > µ > · · · > µ ℓ , as d is varying over the set of fundamental CHRISTOPHE DELAUNAY AND FR´ED´ERIC JOUHET positive discriminants, the probability that rk p j (C ℓ ( K d )) = µ j for all j ℓ isequal to (1 /p µ ℓ +2 ; 1 /p ) ∞ p µ + ··· + µ ℓ +( µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 . One can also adapt the Cohen-Lenstra heuristics for Tate-Shafarevich groups ofelliptic curves. If E is an elliptic curve defined over Q , we denote by X ( E ) its Tate-Shafarevich group. In this context, we assume in this article that X ( E )[ p ∞ ] isfinite for all elliptic curves E/ Q (which is a classical conjecture). In that case, X ( E )[ p ∞ ] is a group of type S (i.e. it is endowed with a bilinear, alternating, nondegenerate pairing β : X ( E )[ p ∞ ] × X ( E )[ p ∞ ] → Q / Z ). Let F u be the family ofelliptic curves E defined over Q with rank u , ordered by their conductor (denotedby N ( E )). If f is a real valued function defined over isomorphism classes of groupsof type S (see [Del01, Del07]), then we say that f ( X ( E )) has average value c ∈ R for E varying over F u if X E ∈ F u N ( E ) Let ℓ be a positive integer, let λ = 1 m m · · · ℓ m ℓ be a partition,and let u be a nonnegative integer. As E/ Q , ordered by conductors, is varying over F u , the average of | X ( E )[ p ] | m | X ( E )[ p ] | m · · · | X ( E )[ p ℓ ] | m ℓ is equal to X µ ⊆ λ C λ,µ ( p ) p −| µ | (2 u − . Concerning the probability laws of p j -ranks rk p j ( X ( E )), we have the following([Del11]). Conjecture 4. Let ℓ be a positive integer, let µ = µ > µ > · · · > µ ℓ > be aninteger partition of length ℓ ( µ ) ℓ and let u be a nonnegative integer. As E/ Q ,ordered by conductors, is varying over F u , the probability that rk p j ( X ( E )) = 2 µ j for all j ℓ is equal to (1 /p u +2 µ ℓ +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ )+(2 u − µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 . As previously, very few is known about these conjectures. Bhargava and Shankar[BS10b, BS10a] obtained several results about the average of | S ( E ) p | over all ellipticcurves E/ Q , where S ( E ) p is the p -Selmer group of E/ Q . Their results, togetherwith a strong form of the rank conjecture (asserting that the rank of E is 0 or 1 withprobability 1 / > λ = 1 and p = 2 , 3. Heath-Brown [HB93, HB94],then Swinnerton-Dyer [SD08] and Kane [Kan11] also obtained results about S ( E ) when E is varying over some families of quadratic twists. Their results, togetherwith a strong rank conjecture, imply Conjecture 3 for λ = 1 and p = 2 for somefamilies of quadratic twists. Furthermore, Conjecture 3 is compatible with theconjecture of Poonen and Rains ([PR12]). This notion is different from the previously mentioned groups of type λ , where λ is a partition. One can replace in our discussion Q by any other number field K . HE COHEN-LENSTRA HEURISTICS, MOMENTS AND p j -RANKS OF SOME GROUPS 5 In this context of elliptic curves, we will prove the following result. Theorem 2. Let u be a nonnegative integer, and let ℓ be a positive integer. As-sume that for any λ = 1 m m · · · ℓ m ℓ , as E/ Q , ordered by conductors, is vary-ing over F u , the average of | X ( E )[ p ] | m | X ( E )[ p ] | m · · · | X ( E )[ p ℓ ] | m ℓ is equal to P µ ⊆ λ C λ,µ ( p ) p −| µ | (2 u − . Then for any µ > µ > · · · > µ ℓ , as E/ Q is varyingover F u , the probability that rk p j ( X ( E )) = 2 µ j for all j ℓ is equal to (1 /p u +2 µ ℓ +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ )+(2 u − µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 . Remarks. 1- In Theorem 1 (resp. Theorem 2) one can replace class groups (resp. Tate-Shafarevich groups) by finite abelian groups (resp. groups of type S) varying insome families. In particular, we can obtain similar results for Selmer groups ofelliptic curves (see section 5).2- From the laws of the p j -ranks for all j = 1 , . . . , ℓ as in theorems 1 and 2, onecan also deduce the probability that the p ℓ -rank is equal to some fixed value fora single ℓ . In this case, we recover the results and conjectures from [Coh85] and[Del11].3- One can also ask if it is possible to deduce the moments from the probabilitylaws of the p j -ranks for all j = 1 , . . . , ℓ . For this, it seems that we need to know anerror term for the probability laws and this error term is not given by the heuristicphilosophy. However, the theoretical results of [FK07, HB93, SD08, Kan11] concernthe p j -ranks of the groups studied (with an explicit error term) from which themoments are deduced. 2. An auxiliary analytic tool In this section, we prove a generalization of [FK06, Lemma 6] which will beuseful later. Lemma 3. Let a ∈ C with | a | > and g ( z ) = P r > w r z r be an entire functionsatisfying the following properties: • there exists an absolute constant C > and α ∈ R such that for all r ∈ N , | w r | Ca − r / αr ; • for all m ∈ N , g ( a m ) = 0 .We denote by ω ∈ N ∪ {∞} the vanishing order of g at z = 0 . Then, if ω > α − / ,we have g ≡ (i.e. ω = ∞ ).Proof. Let k be a nonnegative integer. Taking | z | = | a | k , a direct computationshows that | g ( z ) | C ′ | a | ( k + α ) / , where C ′ = C P r ∈ Z | a | − ( r − α ) / . Assume that g 0. Then [FK06, Lemma 6] givessup | z | = | a | k | g ( z ) | ≫ | a | k ( k +1) / kω . Hence, we must have ( k + α ) > k ( k + 1) + 2 kω for all k ∈ N , which implies ω α − / (cid:3) Corollary 4. Let ℓ ∈ N ∗ , let a ∈ C with | a | > and let g ( z ) = P r w r z r z r · · · z r ℓ ℓ with z = ( z , z , · · · , z ℓ ) ∈ C ℓ and where the sum is over all integer partitions r = r > r > · · · > r ℓ > . We assume that: • | w r | C a − ( r | r ) / α | r | for some absolute constant C and α < / ; • g ( a m , a m , . . . , a m ℓ ) = 0 for all nonnegative integers m , m , . . . , m ℓ . CHRISTOPHE DELAUNAY AND FR´ED´ERIC JOUHET If α < / , then g ≡ and w r = 0 for all r . If α ∈ [1 / , / and w , ,..., = 0 ,then g ≡ (therefore w r = 0 for all r ).Proof. If ℓ = 1, this is proved by the above lemma, since we have ω > α − / ℓ > 2, we fix ( m , . . . , m ℓ ) ∈ N ℓ − and set f ( z ) = X r z r X r > r > ··· > r ℓ w r ,r ,...,r ℓ a r m · · · a r ℓ m ℓ . Then f ( z ) satisfies the condition of the previous lemma since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X r > ... > r ℓ > w r ,r ,...,r ℓ a r m · · · a r ℓ m ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ m ,...,m ℓ a − ( r | r ) / αr . With the conditions of the corollary, we deduce that f ( z ) = 0, therefore for anyfixed r > 0, we must have X r > r > ··· > r ℓ > w r ,r ,...,r ℓ a r m · · · a r ℓ m ℓ = 0 , for all m , . . . , m ℓ . We use the fact that when r is fixed, X r > r > ··· > r ℓ > w r ,r ,...,r ℓ z r · · · z r ℓ r is a polynomial to conclude. (cid:3) Class groups of number fields We will actually prove a more general result displayed in Theorem 5 (whichclearly implies Theorem 1). If K is a number field, we denote by C ℓ ( K ) its classgroup. Let K be a fixed set of number fields ordered by the absolute value of theirdiscriminant disc( K ). If f is a real valued function defined over isomorphism classesof finite abelian groups, then, as before, we say that f (C ℓ ( K )) has average value c ∈ R for K varying over K if X K ∈ K | disc( K ) | Let u be a nonnegative integer and let ℓ be a positive integer. Assumethat for every integer partition λ = 1 m m · · · ℓ m ℓ , as K is varying over K , theaverage of | C ℓ ( K )[ p ] | m · · · | C ℓ ( K )[ p ℓ ] | m ℓ is equal to P µ ⊆ λ C λ,µ ( p ) p − u | µ | . Then forany µ > µ > · · · > µ ℓ , as K is varying over K , the probability that rk p j (C ℓ ( K )) = µ j for all j ℓ , is equal to (1 /p u + µ ℓ +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ + u ( µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 . We follow and generalize the proof of [FK06]. First, we will need the followingproposition. Proposition 6. Let u ∈ N and ℓ ∈ N . For all λ = 1 m m · · · ℓ m ℓ we have (2) X µ ⊆ λ C λ,µ ( p ) p − u | µ | = O p,ℓ ( p ( λ ′ | λ ′ ) / ) . HE COHEN-LENSTRA HEURISTICS, MOMENTS AND p j -RANKS OF SOME GROUPS 7 Proof. Set C λ := P µ ⊆ λ C λ,µ ( p ) p − u | µ | . The equality (cid:20) nk (cid:21) p = ( p ; p ) n ( p ; p ) k ( p ; p ) n − k = p k ( n − k ) (1 /p ; 1 /p ) n (1 /p ; 1 /p ) k (1 /p ; 1 /p ) n − k , and (1 /p ; 1 /p ) ∞ (1 /p ; 1 /p ) k 1, together with the expression (1) of the coeffi-cients C λ,µ ( p ), imply that C λ C X µ ⊆ λ p P i µ ′ i ( λ ′ i − µ ′ i ) C p P i λ ′ i / X µ ⊆ λ , for some constant C depending only on p and ℓ (we used µ ′ i ( λ ′ i − µ ′ i ) λ ′ i / µ ′ i λ ′ i , noting that µ ⊆ λ if and only if µ ′ ⊆ λ ′ ). Now, since ℓ isfixed, the number of sub-partitions µ ⊆ λ is certainly bounded by the product( λ + 1)( λ + 1) · · · ( λ ℓ + 1). By the arithmetico-geometric mean inequality, weobtain X µ ⊆ λ (cid:18) | λ | ℓ (cid:19) ℓ = O ℓ ( | λ | ℓ ) . Finally, we have C λ = O p,ℓ ( p ( λ ′ | λ ′ ) / | λ | ℓ ) = O p,ℓ ( p ( λ ′ | λ ′ ) / ) . (cid:3) Remark. As it can be seen in the proof of the above proposition, we have themore precise upper bound C λ = O p,ℓ ( p ( λ ′ | λ ′ ) / | λ | ℓ ). Nevertheless, the upper boundgiven in the proposition will be sufficient for our application. Proof of Theorem 5. For X > r := r > r > · · · > r ℓ > 0, set N ( X ) := |{ K ∈ K : | disc( K ) | X }| , and N ( X, r ) := |{ K ∈ K : | disc( K ) | X and rk p i (C ℓ ( K )) = r i for all i = 1 , . . . , ℓ }| . For every λ = 1 m m · · · ℓ m ℓ , the assertion of Theorem 5 implies that(3) X r N ( X, r ) N ( X ) p m r + m ( r + r )+ ··· + m ℓ ( r + r + ··· + r ℓ ) = X µ ⊆ λ C λ,µ ( p ) p −| µ | u + o λ (1) , where the sum on the left is over all integer partitions r = r > r > · · · > r ℓ > m r + m ( r + r ) + · · · + m ℓ ( r + r + · · · + r ℓ ) = ( λ ′ | r ). Hence,equation (3) can be written as(4) X r N ( X, r ) N ( X ) p ( λ ′ | r ) = X µ ⊆ λ C λ,µ ( p ) p −| µ | u + o λ (1) , ( X → ∞ ) . For each integer partition r , the sequence n N ( n, r ) /N ( n ) is a real sequence inthe compact set [0 , d r ∈ [0 , 1] and aninfinite subset M of N such that N ( m, r ) /N ( m ) → d r ( m ∈ M , m → ∞ ) . Replacing m i by 2 m i + 1, we see from (3) and Proposition 6 that(5) N ( X, r ) N ( X ) ≪ λ p − (2 m +1) r − (2 m +1)( r + r ) −···− (2 m ℓ +1)( r + ··· + r ℓ ) , uniformly in X and r , from which we deduce that X r N ( m, r ) N ( m ) p ( λ ′ | r ) = O λ (1) . CHRISTOPHE DELAUNAY AND FR´ED´ERIC JOUHET Hence by Lebesgue’s Dominated Convergence Theorem we have X r d r p ( λ ′ | r ) = X µ ⊆ λ C λ,µ ( p ) p −| µ | u . If we consider the infinite multi-dimensional system( S ) X r x r p ( λ ′ | r ) = X µ ⊆ λ C λ,µ ( p ) p −| µ | u for all λ = 1 m · · · ℓ m ℓ , where the unknowns are x r > d r is a solution of ( S ). We already know that x r = (1 /p µ ℓ + u +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ + u ( µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 is a solution (see [DJ12], Theorem 14 or the equality just before this theorem).Therefore we need to prove that there exists at most one solution to the system.Let ( x r ) r be a solution of ( S ). Since λ λ ′ is a bijection, the system is equivalentto X r x r p ( λ | r ) = C λ ′ for all λ, where C λ ′ ( p ) = P µ ⊆ λ ′ C λ ′ ,µ ( p ) p −| µ | u = O ( p ( λ | λ ) / ) by Proposition 6. From x r > x r = O ( p − ( λ | r )+( λ | λ ) / ), so when λ = r , we get0 x r c p − ( r | r ) / , for some absolute constant c . Now, if ( x ′ r ) r is another solution of ( S ), then setting w r = x r − x ′ r , we have a function(6) f ( z ) = f ( z , z , . . . , z ℓ ) = X r w r z r z r · · · z r ℓ ℓ satisfying f ( z ) = 0 if z = p m , z = p m , . . . , z ℓ = p m ℓ for all m , m , . . . , m ℓ ∈ N ,and | w r | c p − ( r | r ) / . Thus we can apply Corollary 4 to conclude that x r = x ′ r . So, as X → ∞ , thesequence N ( X, r ) /N ( X ) has only one limit point, which is d r = x r . (cid:3) Corollary 7. Let u be a nonnegative integer and let ℓ be a positive integer. Assumethat for every integer partition λ = 1 m m · · · ℓ m ℓ , as K is varying over K , theaverage of | C ℓ ( K )[ p ] | m · · · | C ℓ ( K )[ p ℓ ] | m ℓ is equal to P µ ⊆ λ C λ,µ ( p ) p − u | µ | . Thenfor k ∈ N , as K is varying over K , the probability that rk p ℓ (C ℓ ( K )) = k is equal to (1 /p u + k +1 ; 1 /p ) ∞ (1 /p ; 1 /p ) k p ℓk ( u + k ) Q /p,ℓ, (cid:18) p k + u − (cid:19) , where Q q,ℓ, ( x ) := X n > ( − n x ℓn q n ( n +1)(2 ℓ +1) / − n (1 − xq n +1 )( q ; q ) n ( xq n +1 ; q ) ∞ . The series Q q,ℓ,k ( x ) was defined by Andrews (see [And74]). The formula of theabove corollary is the u -probability that the p ℓ -rank of a finite abelian p -group isequal to k , as obtained in [Coh85] (note that we use the definition of u -averagesand u -probabilities of this article). Proof. We define as before N ( X ) and N ( X, r ). Moreover, set N ( X, ℓ, k ) = |{ K ∈ K : | disc( K ) | X and rk p ℓ (C ℓ ( K )) = k }| . We have(7) N ( X, ℓ, k ) N ( X ) = X µ > µ > ··· > µ ℓ − > k N ( X, µ ) N ( X ) , HE COHEN-LENSTRA HEURISTICS, MOMENTS AND p j -RANKS OF SOME GROUPS 9 where the sum is over integer partitions µ = µ > µ > · · · > µ ℓ − > µ ℓ = k . Bythe assumptions and Theorem 5, we havelim X →∞ N ( X, µ ) N ( X ) = (1 /p u + µ ℓ +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ + u ( µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 . In equation (7) we take the limit as X → ∞ and use Lebesgue’s DominatedConvergence Theorem (with equation (5)) to obtain that the probability thatrk p ℓ (C ℓ ( K )) = k is equal to X µ > ··· > µ ℓ − > µ ℓ = k (1 /p u + µ ℓ +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ + u ( µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 = (1 /p u + k +1 ; 1 /p ) ∞ p ℓk ( u + k ) (1 /p ; 1 /p ) k X µ > ··· > µ ℓ − > (1 /p ) µ + ··· + µ ℓ − +( u +2 k )( µ + ··· + µ ℓ − ) Q ℓ − j =1 (1 /p ; 1 /p ) µ j − µ j +1 , the equality being derived by shifting all indices by k . Now, from [Del11, Proposi-tion 13] (or [And74, Eq. (2.5)]), the last sum is exactly Q /p,ℓ, (cid:18) p k + u − (cid:19) , as expected (note that Q q, , ( x ) = 1). (cid:3) Tate-Shafarevich groups of elliptic curves In this section, we prove Theorem 2 and we follow the previous one. We justneed an upper bound for the coefficients P µ ⊆ λ C λ,µ ( p ) p −| µ | (2 u − , which is givenin the following result. Proposition 8. Let u ∈ N and ℓ ∈ N ∗ . For all λ = 1 m m · · · ℓ m ℓ , we have X µ ⊆ λ C λ,µ ( p ) p −| µ | (2 u − = O p,ℓ ( p ( λ ′ | λ ′ ) ) . Proof. We have X µ ⊆ λ C λ,µ ( p ) p −| µ | (2 u − = X µ ⊆ λ p P i µ ′ i +1 ( λ ′ i − µ ′ i )) Y i (cid:20) λ ′ i − µ ′ i +1 λ ′ i − µ ′ i (cid:21) p p −| µ | (2 u − p | λ | X µ ⊆ λ p P i µ ′ i +1 ( λ ′ i − µ ′ i )) Y i (cid:20) λ ′ i − µ ′ i +1 λ ′ i − µ ′ i (cid:21) p . Using the same method as in the proof of Proposition 6, we obtain(8) X µ ⊆ λ C λ,µ ( p ) p −| µ | (2 u − = O p,ℓ ( p ( λ ′ | λ ′ ) / | λ | | λ | ℓ ) , which implies the upper bound given in the proposition. (cid:3) Proof of Theorem 2. For r := r > r > · · · > r ℓ > 0, we denote by 2 r the integerpartition 2 r := 2 r > r > · · · > r ℓ . For X > 1, set N ( X ) := |{ E ∈ F : N E X }| , and N ( X, r ) := |{ E ∈ F : N E X and rk p i ( X ( E )) = r i for all i = 1 , . . . , ℓ }| . Note that since X ( E ) is a group of type S then rk p j ( X ( E )) must be even. Henceif one of the r j ’s is odd, then N ( X, r ) = 0 for all X . So, for any λ = 1 m m · · · ℓ m ℓ ,the assertion of Theorem 2 implies that(9) X r N ( X, r ) N ( X ) p ( λ ′ | r ) = X µ ⊆ λ C λ,µ ( p ) p −| µ | (2 u − + o λ (1) , where the sum is over all integer partition r = r > · · · > r ℓ . As before, we willprove that the sequence N ( X, r ) /N ( X ) has only one limit point as X → ∞ . Weare led to consider the system( T ) X r x r p ( λ ′ | r ) = X µ ⊆ λ C λ,µ ( p ) p −| µ | (2 u − for all λ = 1 m · · · ℓ m ℓ , where the unknowns are x r > 0. If e r is a limit point of N ( X, r ) /N ( X ), then e r is a solution of ( T ). We already know that x r = (1 /p u +2 µ ℓ +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ )+(2 u − µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 is a solution (see [DJ12], remark after Theorem 14). If x r is a solution of ( T ), thenwe have X r x r p λ | r ) = X µ ⊆ λ C λ ′ ,µ ( p ) p −| µ | (2 u − for all λ, where P µ ⊆ λ C λ ′ ,µ ( p ) p −| µ | (2 u − = O ( p ( λ | λ ) ), from which we deduce x r ≪ p ( r | r ) − r | r ) ≪ p − ( r | r ) = ( p ) − ( r | r ) / . Therefore Corollary 4 with a = p gives the unicity. (cid:3) Corollary 9. Let u be a nonnegative integer, and let ℓ be a positive integer. As-sume that for any λ = 1 m m · · · ℓ m ℓ , as E/ Q , ordered by conductors, is vary-ing over F u , the average of | X ( E )[ p ] | m | X ( E )[ p ] | m · · · | X ( E )[ p ℓ ] | m ℓ is equal to P µ ⊆ λ C λ,µ ( p ) p −| µ | (2 u − . Then for k ∈ N , as E/ Q is varying over F u , the proba-bility that rk p ℓ ( X ( E )) = 2 k is equal to (1 /p u +2 k +1 ; 1 /p ) ∞ (1 /p ; 1 p ) k p ℓk (2 u +2 k − Q /p ,ℓ, (1 /p k +2 u − ) . Proof. We proceed as for the proof of Corollary 7. (cid:3) The formula in the above corollary is the u -probability that the p ℓ -rank of a finiteabelian p -group of type S is equal to 2 k , as obtained in [Del11].5. Selmer groups of elliptic curves In the proof of Theorem 2, it is essential to use the fact that X ( E ) is a groupof type S and that rk p j ( X ( E )) is even, since on the left hand side of (9) the suminvolves partitions with only even parts µ j . Nevertheless, one can ask what shouldbe the p j -rank probability laws for other families of groups, if we assume that theirmoments are given as in Conjecture 3. This question can be naturally asked inparticular for Selmer groups of elliptic curves (or more precisely for the p -primaryparts of the Selmer groups). If E is an elliptic curve defined over Q , then we denoteby S ( E ) the p -primary part of its Selmer group. It is the inductive limit of the p n -Selmer group S ( E ) p n of E : S ( E ) = lim −→ S ( E ) p n . We have the exact sequence0 → E ( Q ) ⊗ Q p / Z p → S ( E ) → X ( E )[ p ∞ ] → , HE COHEN-LENSTRA HEURISTICS, MOMENTS AND p j -RANKS OF SOME GROUPS 11 which can be seen as the limit of0 → E ( Q ) /p n E ( Q ) → S ( E ) p n → X ( E )[ p n ] → . We assume for simplicity that E ( Q ) tors is trivial: it is not a restriction since we areconsidering averaging over elliptic curves and on average elliptic curves have trivialrational torsion. The Selmer group S ( E ) can be an infinite group, nevertheless itssubgroup of p n -torsion points is finite and we have S ( E )[ p n ] = S ( E ) p n . We define the p j -rank of S ( E ) by rk p j ( S ( E )) = rk p j ( S ( E )[ p j ]). Note that we haverk p j ( S ( E )) = rk p j ( S ( E )[ p k ]) for all k > j .Since X [ p ∞ ] is finite by assumption, we have S ( E ) ≃ ( Q p / Z p ) r ( E ) , where r ( E ) isthe rank of the Mordell-Weil group E ( Q ) andrk p j ( S ( E )) = r ( E ) for p j large enough.Furthermore, we have rk p j ( S ( E )) ≡ r ( E ) (mod 2) , so the parities of rk p j ( S ( E )) are determined by the parity of r ( E ).If ℓ is a positive integer and λ = 1 m m · · · ℓ m ℓ is an integer partition, then | S ( E )[ p ] | m | S ( E )[ p ] | m · · · | S ( E )[ p ℓ ] | m ℓ is meaningful and we can consider the average value of this function as E is varyingover a family of elliptic curves. The works of [PR12] suggests that the p -Selmergroups should behave in a “global” way independently of the rank of E (except forthe parity of the p -ranks). From [DJ12], we can extract the following conjecture. Conjecture 5. Let ℓ be a positive integer, and let λ = 1 m m · · · ℓ m ℓ be an integerpartition. As E/ Q , ordered by conductors, is varying over all elliptic curves, theaverage of | S ( E )[ p ] | m | S ( E )[ p ] | m · · · | S ( E )[ p ℓ ] | m ℓ is equal to X µ ⊆ λ C λ,µ ( p ) p | µ | . If ℓ = 1, this conjecture is originally due to Poonen and Rains in [PR12], wherethey use a completely different model for Selmer group. Proposition 10. Let ℓ be a positive integer and let δ ∈ { , } . Assume that for anypartition λ = 1 m m · · · ℓ m ℓ , the average of | S ( E )[ p ] | m | S ( E )[ p ] | m · · · | S ( E )[ p ℓ ] | m ℓ is equal to P µ ⊆ λ C λ,µ ( p ) p | µ | as E/ Q , ordered by conductor, is varying over a fam-ily F of elliptic curves, and assuming that the even (resp. odd) rank elliptic curvesin F contribute in a ratio α (resp. − α ). Then, for all µ > µ · · · > µ ℓ , as E/ Q is varying over F , the probability that rk p j ( S ( E )) = 2 µ j + δ for all j ℓ isequal to ( δ (1 − α ) + α (1 − δ )) (1 /p δ +2 µ ℓ +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ )+(2 δ − µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 . Proof. For X > r = r > r > · · · > r ℓ > 0, set as before N ( X ) := |{ E ∈ F : N E X }| , and N ( X, r ) := |{ E ∈ F : N E X and rk p i ( S ( E )) = r i for all i = 1 , . . . , ℓ }| . Let λ = 1 m · · · ℓ m ℓ be an integer partition. Since the rk p j ( S ( E ))’s have all thesame parity for j ∈ N , and by the assumptions of the theorem, we have X r N ( X, r ) N ( X ) p ( λ ′ | r ) = α X µ ⊆ λ C λ,µ p | µ | + o λ (1) , and X r N ( X, r + 1) N ( X ) p ( λ ′ | r +1) = (1 − α ) X µ ⊆ λ C λ,µ p | µ | + o λ (1) . For δ ∈ { , } , set e r + δ = (1 /p δ +2 µ ℓ +1 ; 1 /p ) ∞ p µ + ··· + µ ℓ )+(2 δ − µ + ··· + µ ℓ ) Q ℓj =1 (1 /p ; 1 /p ) µ j − µ j +1 . Thus for δ = 0, we recover e r which was defined in the previous section, where wealready saw that X r αe r p ( λ ′ | r ) = α X µ ⊆ λ C λ,µ ( p ) p | µ | , that αe r is the only solution of the above system, and that N ( X, r ) /N ( X ) → e r as X → ∞ .Now, by the same arguments as before, we obtain that there exists an uniquesolution to the system( T ’) X r (cid:16) x r +1 p ( λ ′ | r +1) (cid:17) = (1 − α ) X µ ⊆ λ C λ,µ ( p ) p | µ | , for all λ = 1 m · · · ℓ m ℓ , where the unknown are x r +1 . Furthermore x r +1 = (1 − α ) e r +1 is a solutionof ( T ’). Indeed, by [DJ12, Remark after Theorem 14] X r e r +1 p ( λ ′ | r +1) = p | λ ′ | X r e r +1 p ( λ ′ | r ) = p | λ | X µ ⊆ λ C λ,µ ( p ) p −| µ | , and moreover by [DJ12, Theorem 1], we have p | λ | X µ ⊆ λ C λ,µ ( p ) p −| µ | = X µ ⊆ λ C λ,µ ( p ) p | µ | . Finally N ( X, r + 1) /N ( X ) → e r +1 as X → ∞ . (cid:3) Now, adapting the proof of Corollary 7, we have the following result. Corollary 11. Let ℓ be a positive integer and set δ ∈ { , } . Assume that for every λ = 1 m m · · · ℓ m ℓ the average of | S ( E )[ p ] | m | S ( E )[ p ] | m · · · | S ( E )[ p ℓ ] | m ℓ is equalto P µ ⊆ λ C λ,µ ( p ) p | µ | , as E/ Q , ordered by conductor, is varying over a family F of elliptic curves, and assuming that the even (resp. odd) rank elliptic curves in F contribute in a ratio α (resp. − α ). Then, for k ∈ N , the probability that rk p ℓ ( S ( E )) = 2 k + δ is equal to ( δ (1 − α ) + α (1 − δ )) (1 /p k +2 δ +1 ; 1 /p ) ∞ (1 /p ; 1 /p ) k p ℓk (2 k +2 δ − Q /p ,ℓ, (1 /p k +2 δ − ) . The value of α can be of course = 1 / 2. Furthermore, even in the case of a family ofquadratic twists of an elliptic curve E defined over a number field K , it is possibleto have α = 1 / K = Q ). HE COHEN-LENSTRA HEURISTICS, MOMENTS AND p j -RANKS OF SOME GROUPS 13 If we consider the family of all elliptic curves, then a general conjecture states that α = 1 / 2, which leads to the following. Conjecture 6. Let ℓ be a positive integer, set k ∈ N and δ ∈ { , } . Then, as E/ Q , ordered by conductor, is varying over all elliptic curves, the probability that rk p ℓ ( S ( E )) = 2 k + δ is equal to f ( p, ℓ, k + δ ) := 12 (1 /p k +2 δ +1 ; 1 /p ) ∞ (1 /p ; 1 /p ) k p ℓk (2 k +2 δ − Q /p ,ℓ, (1 /p k +2 δ − ) . For ℓ = 1, we recover the conjectural distribution X Sel p of [PR12] and the proveddistribution of Sel in [HB93, HB94, Kan11, SD08, KMR13b] for some families ofquadratic twists of an elliptic curve.The conjectural distribution on the p j -rank of the Selmer groups given above is ofcourse compatible with the rank conjecture. Indeed, note that we have Q q, ∞ , ( x ) =1 / ( xq ; q ) ∞ from which we easily deduce thatlim ℓ →∞ f ( p, ℓ, k + δ ) = (cid:26) k > , / k = 0 . Since for ℓ large enough, rk p ℓ ( S ( E )) = r ( E ), we recover the fact that on averagehalf elliptic curves should have rank 0 and half elliptic curves should have rank 1.On the other hand, if we assume Conjecture 5 for ℓ = 1 for infinitely many primes p with α = 1 / 2, then we also recover the previous distribution for the rank of E ( Q ),since lim p →∞ f ( p, , k + δ ) = (cid:26) k > , / k = 0 . We give some numerical approximations for the function f ( p, ℓ, k + δ ) for p = 2 , , ℓ and of 2 k + δ in the following tables. k + δ \ ℓ k + δ \ ℓ · − p = 2 p = 32 k + δ \ ℓ · − · − p = 5 Remark on the uniqueness of the solution In our study related to Tate-Shafarevich group, we were led to consider and todiscuss the unicity of the solution of the following infinite multi-dimensional system( U ) X r x r p ( λ | r ) = X µ ⊆ λ ′ C λ ′ ,µ ( p ) p −| µ | (2 u − for all λ = 1 m · · · ℓ m ℓ , where the unknowns are x r > 0. We only considered solution ( x r ) r such that x r = 0if in r = r > r > · · · > r ℓ at least one of the r j ’s has not the same parity as r .In that case, the term p ( λ | r ) involved in the sum is of the form p ( λ | r + δ ) , and thefactor 2 allowed to have an asymptotic 0 x r + δ ≪ p ( − r | r ) / which implied theunicity of the solution. One can ask about the unicity of the solution without the assumption that the partitions involved in the system have parts with the sameparity.Set µ ∈ R , and for a partition r we define y r ( µ ) = µe r if r is even,(1 − µ ) e r if r is odd,0 otherwise.Then y r ( µ ) is a solution of equation ( U ). If 0 x r is a solution of ( U ) then,using (8), we see that for any fixed α > 1, we have0 x r ≪ p − ( r | r ) / α | r | for all r . 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