AVERAGES OF GROUPS INVOLVING p l -RANK AND COMBINATORIAL IDENTITIES. Christophe Delaunay

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1 AVERAGES OF GROUPS INVOLVING l -RANK AND COMBINATORIAL IDENTITIES by Christohe Delaunay Abstract We obtain averages of secific functions defined over isomorhism classes of some tye of finite abelian grous These averages are concerned with miscellaneous questions about the l -ranks of these grous We aly a classical heuristic rincile to deduce from the averages recise redictions for the behavior of class grous of number fields and of Tate-Shafarevich grous of ellitic curves Furthermore, the comutations of these averages, which comes with an algebraic asect, can also be reinterreted with a combinatorial oint of view This allows us to recover and to obtain some combinatorial identities and to roose for them a natural algebraic context Introduction and notations This article is concerned with the averages, in some sense, of functions defined over finite abelian grous and grous of tye S A finite abelian grou G is called a grou of tye S if it is endowed with a non-degenerate, bilinear, alternating airing: β : G G Q/Z A grou of tye S has the form G = H H where H is a finite abelian grou Recirocally, each grou G, of the form H H has a unique structure of grou of tye S u to isomorhisms of such grous ie isomorhisms reserving the underlying airings In articular, if G is a grou of tye S then Aut s G, the number of automorhisms of G that resect the underlying airing β, does not deend on β see, for examle, [Del0] for these facts Due to the above, the airings associated to grous of tye S will never be secified in the sequel For the rest of this aer, the letters H, H, H 2 are used for isomorhism classes of finite abelian grous and the letter G for isomorhism classes of grous of tye S Let h be a comlex-valued function defined on isomorhism classes of finite abelian grous and let u 0 In [CL84], H Cohen and H Lenstra defined a certain u-average, M u h, of the function h in articular, if h is the characteristic function of a roerty

2 2 CHRISTOPHE DELAUNAY P, M u h is called the u-robability of P We write ζh, s = n s n Hn hh AutH, where Hn means that the sum is over all isomorhism classes of finite abelian grous H of order n It is roved in [CL84] that if h is non-negative, if ζh, s converges for Rs > 0 and can be continued to a meromorhic function in Rs 0 with a unique ole of order at most the function ζh, s could be holomorhic in Rs 0 at s = 0 then ζh, s M u h = lim s u ζ, s H Cohen and H Lenstra introduced such averages as a heuristic model in order to give redictions for the behavior of the class grous of number fields varying in certain natural families In articular, they roved that ζ, s = n = j n s Hn ζs + j = AutH rime j s+j where ζ is the classical Riemann function The reason of writing the above equality as an Euler roduct is that in general we comute u-averages of functions restricted to the -arts of abelian grous and we recover global averages by taking the roduct of all the -comonents The aim of [Del0] was to adat the Cohen-Lenstra s model in order to give redictions for the behavior of Tate-Shafarevich grous of ellitic curves varying in certain natural families Recall that, if they are finite as it is classically conjectured, Tate-Shafarevich grous are grous of tye S One can define, for u 0, the u-average in the sense of grous of tye S, M S ug, for a function g defined on isomorhism classes of grous of tye S We let ζ S g, s = n n s G S n gg Aut S G = n n 2s G S n 2, gg Aut S G, where G S n means that the sum is over all isomorhism classes of grous of tye S, G of order n the sum being emty if n is not a square We have from [Del0], that if g is non-negative, if ζ S g, s converges for Rs > 0 and can be continued to a meromorhic function in Rs 0 with a unique ole of order at most the function ζ S g, s could be holomorhic in Rs 0 at s = 0 then M S ug = lim s u ζ S g, s ζ S, s The reason of taking s instead of s is exlained in [Del0] See also [Del07] for a recision of the heuristics for Tate-Shafarevich grous of ellitic curves of rank 0

3 AVERAGES OF GROUPS INVOLVING l -RANK AND COMBINATORIAL IDENTITIES 3 It can be roved that ζ S, s = n = j n s G S n Aut S G ζ2s + 2j + = 2s+2j+ As for finite abelian grous, we will seak of the u-robability of P if g is the characteristic function of a roerty P of grous of tye S It will always be clear in the context whether we are concerned with average in the sense of finite abelian grous or in the sense of grous of tye S Comutations of u-averages can be viewed as algebraic interretations of combinatorial identities In that context, there is a generic tool that can hel for going from u-averages of finite abelian grous to u-averages of grous of tye S Indeed, let be a rime and assume that H and G are in fact -grous ie the order of H and G are a -ower If H = n and H Z/Z λ Z/ 2 Z λ 2 Z/ s Z λs where λ + 2λ sλ s = n is a artition of n, then, we denote µ, µ 2,, µ s by j µ = λ + λ λ s ; µ 2 = λ λ s ; µ s = λ s Thus, we also have µ + µ µ s = n and the number of automorhisms of H is given by see for examle [Hal38] AutH = µ2 +µ µ2 s, λ j where q a is defined by j s q a = q q 2 q a with a N Now, if G H H is a grou of tye S, we have Aut S G = 2µ2 +µ µ2 s +n j s 2 λ j Whenever a certain equality involving AutH holds, we can usually consider it as a formal identity in the variable q = / To obtain the analogue for -grous of tye S and involving Aut S G, one can aly, at convenient ste, the following rule: first substitute 2 for and then multily by n : we will use roughly this rule several times in this aer even if not exlicitly mentioned

4 4 CHRISTOPHE DELAUNAY The l -rank, r lh, of a finite abelian grou H is the number of cyclic comonents of H of order divisible by l It is defined by r lh = dim Z/Z l H/ l H With the decomosition, it is not difficult to see that r lh = µ l Furthermore, if G H H is a grou of tye S then r lg = 2r lh = 2µ l The first goal of this aer is to obtain some u-averages for finite abelian grous and for grous of tye S These u-averages are concerned with several questions related to l -ranks The first art deals with the u-average of the number of l -torsion elements We aly the heuristic rincile to give redictions for the average number of l -torsion elements in class grous of number fields and in Tate-Shafarevich grous of ellitic curves of rank 0 and in natural families These comutations were motivated by questions of M Bhargava see the works of M Bhargava and A Shankar [BS0b], [BS0a] In the second art, we study related questions about the robability laws of l -ranks This is done by generalizing and adating a work of H Cohen [Coh85] Both in the first and second art, our calculations are, in fact, closed to some combinatorial questions and we investigate a little bit this observation We recover, obtain or use some combinatorial identities and our work gives, in a way, a natural algebraic context for them 2 Average number of l -torsion elements Let λ a artition of n By this we mean that λ = λ, λ 2, and that we have n = λ + 2λ sλ s where s = sλ is the largest integer with λ s 0 Then, we define µ j j s by µ j = s k=j λ k Let H be the finite abelian -grou of order n associated to this artition λ: H = Z/Z λ Z/ 2 Z λ 2 Z/ s Z λs Let T l be the function defined by T l H = H[ l ] = {x H : l x = 0}, so that T l H is the number of l -torsion elements of H We can easily see, with the notations above, that we have T l H = µ+µ2+ +µ l = rh+r 2 H+ +r l H 2 The classical weight We begin this art with the following lemma: Lemma Let l be a ositive integer with l n, we have T l H T l H = AutH n l / n l H n

5 AVERAGES OF GROUPS INVOLVING l -RANK AND COMBINATORIAL IDENTITIES 5 Proof The roof of the lemma is an adatation of the roof of the Theorem 64 of [CL84] For H and H 2 two finite abelian grous, we denote by Hom inj H, H 2 the subset of injective homomorhisms in HomH, H 2 If l n, we have H[ l ] Z/Z λ Z/ l Z λ l Z/ l Z λ l+ Z/ l Z λ s Z/Z λ Z/ l Z λ l Z/ l Z µ l On the one hand, if we denote by ordx the order of a grou element x, we have Hom inj Z/ l Z, H[ l ] = {x Z/ l Z µ l, ordx = l } λ+ +l λ l = lµ l λ+2λ2+ +l λ l µ l = µ+ +µ l µ l = T l H T l H The second equality comes from the fact that the number of elements of order l in Z/ l Z b is bl / b On the other hand, Hom inj Z/ l Z, H[ l ] = Hom inj Z/ l Z, H H n = AutZ/ l Z {H subgrou of H, H Z/ l Z} Finally, Proosition 4 of [CL84] gives AutZ/ l {H subgrou of H, H Z/ l Z} Z = AutH n l / n l We deduce from the lemma: Theorem 2 For finite abelian grous, we have x n T l H AutH = xl+ x x <, j 0 j H n and for grous of tye S x 2n T l G Aut S G = x2l+ l+ G S 2n j 0 x < x2 2j+ Proof We rove the theorem by induction on l For l = 0, we have [CL84] x n AutH = x n /n, / n H n and the results come from the following formal identity due to Euler: q n x n = q n xq j j

6 6 CHRISTOPHE DELAUNAY Writing T l = T l T l + T l and using an induction argument, we have x n T l H = x l x n /n + x + + x l AutH / n x H n j j = x + x + + x l, j j which is exactly the result required for finite abelian grous For grous of tye S, we substitute by 2 and then x by x 2 / We deduce from the theorem that we have ζt l, s = ζ, s + s + 2s + + ζ S T l, s = ζ S, s sl, Rs > 0, + 2s+ + 22s s+l, Rs >, where T l H res T l G is the number of l -torsion elements in the -art of H res G So, Corollary 3 For finite abelian grous, the u-average of the number of l -torsion elements is M u T l = + u + + ul For grous of tye S, the u-average of the number of l -torsion elements is M S ut l = + 2u + + 2u l We now aly the heuristic argument for Tate-Shafarevich grous of ellitic curves Let E be some natural family of rank 0 ellitic curves E and suose that the behavior of the -art of XE is not biased for this family The average number of l -torsion elements in XE should be l Recently, M Bhargava and A Shankar [BS0b], [BS0a] obtained results on the Selmer grous of ellitic curves that give, in articular, very strong evidence towards the revious rediction for the family of all rank 0 ellitic curves Our rediction also imlies that the average number of elements in XE of order exactly n should be n since, by the heuristic rincile, the -arts should behave indeendently In their work, M Bhargava and A Shankar obtained the same rediction with a totally different aroach One can also find in [PR0] some general theoretical results on Selmer grous For a family of rank ellitic curves, the average should be l

7 AVERAGES OF GROUPS INVOLVING l -RANK AND COMBINATORIAL IDENTITIES 7 Of course we can deduce similar conjectures for the -art of class grous of quadratic fields for 2 to our knowledge these redictions have not been done before This imlies that for a rime 3 the average number of l -torsion elements in the class grous of imaginary quadratic fields should be l+ In the case of real quadratic fields, the average number of l -torsion elements in the class grous should be l exactly as in equation 2 just due to the fact that u = 2u if and only if u = As exlained in the introduction, Theorem 2 can been seen as a formal identity letting q = / and then we deduce the following corollary: Corollary 4 We have x n λ=n q µ2 +µ µ2 s µ+ +µ l q λ q λs = j xq j + x + + xl where λ=n means that the sum is over all artitions λ of n Using the substitution x xq and l + l, one can re-write the identity above as 3 x µ+µ2+ +µs q µ2 +µ µ2 s +µ l+µ l+ + +µ s = q λ q λs xq j xql j λ where the sum is over all artitions λ and where s = sλ deends on the artition This equality should be comared with an identity due to Andrews [And74] Letting k 2 and l k be fixed integers, we define the series Q q,k,l x by Q q,k,l x = n x kn q 2k+nn+ 2 ln x l q 2n+l q n j n+ xqj In articular, for x =, Jacobi s identity gives Q q,k,l = n n 0,±l mod 2k+ q n We can now state the results of Andrews [And74, Theorem and equation 25]: Theorem 5 Andrews With the notations above: 4 λ sλ k x µ+µ2+ +µs q µ2 +µ µ2 s +µ l+ +µ s q λ q λ q λs = Q q,k,l x Equation 4 with x = and the roduct formula are known as the full Andrews- Gordon identities They are very famous and dee combinatorial generalizations of the Rogers-Ramanujan identities which are equation 4 for k = 2, l =, 2, x = and the roduct formula see also [And98] We remark that through equation 3

8 8 CHRISTOPHE DELAUNAY we have in fact roved, with a comletely algebraic asect, Andrews s identity in the very secial and easier case k = since Q q,,l x = j xq j xql Note that we could have directly obtained Theorem 2 using it Nevertheless, we focus on the fact that our aroach gives a natural algebraic interretation of these identities 22 The weight w k This art is just concerned with the combinatorial asects of u-averages and not with consequences on the the heuristics In their article, Cohen and Lenstra defined in fact a more general k, u-average in which the weight / AutH has to be relaced by w k H = {surjective homomorhisms Zk H} H k Aut H = k k r H Aut G, with the convention that / a = 0 whenever a < 0 Of course, one immediately sees that lim w kh = k + AutH Dulicating the roof of Lemma, we obtain that for k, n two ositive integers and for l N with l n we have k n l n l+k w k HT l H T l H = H n Furthermore, for l = : x n H n Hence, an induction argument gives: k T Hw k H = + k x k j= n l x j Proosition 6 For k, l, we have x n k T l Hw k H = x j H n j= xl+ x k x + xl+ k From this formula we could easily recover the Dirichlet series that would be associated to T l and w k H Letting q = /, x xq and substituting l for l +, we obtain:

9 AVERAGES OF GROUPS INVOLVING l -RANK AND COMBINATORIAL IDENTITIES 9 Corollary 7 We have the following formal identity 5 λ µ k q k x µ+µ2+ +µs q k µ j s q q µ 2 +µ µ2 s +µ l+µ l+ + +µ s = λ j k j=0 xq j+ q l x l xq k+ + x l q k+l We recall that, in the formula above, s = sλ deends on λ Of course, letting k, we recover equation 3 Now, if we let l in equation 5 we obtain exactly one of Hall s formula [Hal38] Hence, equation 5 which seems to be reviously unknown can be seen as one of its generalization One could robably obtain equation 5 with some classical combinatorial tools, however, as before, our equation occurs naturally in an algebraic way In an other direction, Stembridge [Ste90] gave some refinements of Hall s formula mainly with sλ bounded by a fixed m but still with l =, which allowed him to recover Andrews identity in the case l = for all k note that the cases l = and l = are closely related, and therefore the Rogers-Ramanujan identities Stembridge s method uses also some algebraic asects but with a different aroach It should be interesting to obtain a Hall like formula with general l through Stembridge s aroach ie with l and sλ bounded 3 Probabilities on l -ranks The first aim of this section is to comute the robability laws of l -ranks of finite abelian grous and grous of tye S More recisely, let k, l 0 be two integers Having fixed l + non-negative integers r k r k+ r k+l we are interested in the u-robability and related questions that a -grou H res a -grou of tye S, G satisfies r l+j H = r k+j res r l+j G = 2r k+j for all 0 j l We remark that the case l = 0 gives the robability laws for the k -rank and the case k = gives the robability that a grou has its j -ranks fixed for all j l As this is a generalization and an adatation of Theorem 24 of [Coh85] which is for finite abelian grous and l = 0, we will just give the main stes of the roof We also remark that the formula we are going to give for grous of tye S cannot be obtained directly from [Coh85] and the rule given in the introduction That is one of the reasons that we kee an indeterminate x all along the comutations and need to use the full general Andrew s identity For convenience, we use the following notation: of order n, we write n = eh if H is a finite abelian grou

10 0 CHRISTOPHE DELAUNAY Suose H is a -grou with r k+j H = r k+j for 0 j l Then, we can write H H H 2, with H Z/ i Z λi i k+l where the λ i are subject to certain conditions and H 2 Z/ i Z λi l i s for some s and λ l + λ l+ + + λ s = r k+l For H, we first remark that the conditions on the k+j -ranks mean that λ, λ 2,, λ k are not restricted but that λ k,, λ k+l have to be be fixed by the following l equations: λ k = r k r k+ ; λ k+ = r k+ r k+2 ; λ k+l = r k+l r k+l We let λ be the restricted artition λ = λ,, λ k and µ = λ + + λ k ; µ k = λ k Some technical calculations lead to the following equalities and k+l j= k µ 2 j = k eh = j= j= k+l µ j + j=k k µ 2 j + 2r k r k+l Furthermore, one can see that j= r j + k r k k + l r k+l µ j + k+l k+l k r k r k+l 2 + rj 2 2r k+l r j + lrk+l 2 j=k AutH = AutH AutH 2 2reH We now fix an integer a k and we remark that j=k H/ a H = r ka µ+ µ2+ + µ k µ a+ µ a++ + µ k Finally, we can state, with the convention that Q q,,l x = :

11 AVERAGES OF GROUPS INVOLVING l -RANK AND COMBINATORIAL IDENTITIES Theorem 8 Let k, r k r k+ r k+l be l + non-negative integers and a k For finite abelian grous, we have x n H n r k H=r k r k+l H=r k+l H/ a H Aut H r ka For grous of tye S, we have x 2n G S n 2 r k G=2r k r k+l G=2r k+l Proof We let We have So, 2r ka = r ka = r ka r k+l G/ a G Aut S G = r k+l x kr k+r k+ + +r k+l j= kr2 k +r2 k+ + +r2 k+l = x 2kr k+r k+ + +r k+l r k+l j= x j Q x2 2kr2 k +r2 k+ + +r2 k+l 2 := H, H 2 as above H = r ka / rk+l ST H as above r k+l 2j+ r k+l H/ a H x eh Aut H k+l j=k,k,a Q 2,k,a k+l j=k x 2r k r j r j+ x 2 4r k 2 r j r j+ x eh x eh2 2reH µ+ + µ k µ a+ + µ k AutH AutH 2 2reH xeh µ+ + µk µa+ + µk AutH J l Z r H 2 Z r /J x eh2 H 2 r

12 2 CHRISTOPHE DELAUNAY We have used the fact see [Coh85] that for a given H 2, the number of J l Z r such that H 2 Z r /J is equal to H 2 r / r / AutH 2, and we denoted S = T = H as above J l Z r H 2 Z r /J 2reH x eh µ+ + µ k µ a+ + µ k ; AutH x eh2 H 2 r Still following [Coh85], we have for T : T = lr+αr x lr+α α 0 J l Z r l Z r /J = α α / α+r Finally, we obtain = r ka / rk+l λ,,λ k = lr2 x lr α 0 = / lr2 x lr r x α j= x k rk+rk+rk++ +rk+l k+l j=k x j / r / α rk+l j= x j / rj r j+ k r2 k +r2 k +r2 k+ + +r2 k+l 2r k µ + + µ k x µ+ + µ k µ2 + + µ2 k and a little calculation gives Theorem 8 k j= / λ j µ+ + µk µa+ + µk We are now able to give two direct alications of Theorem 8 related to the comutations of averages and to conjectures on class grous and on Tate-Shafarevich grous via the heuristic rincile, 3 The case l = 0 This is the original goal of [Coh85] for finite abelian grous On the one hand, we recover Theorem 24 of [Coh85] with a slightly different notation: n n s Hn r k =r H/ a H Aut H j r+ r j+s = Q krs+r ra,k,a 2r+s ζ, s

13 AVERAGES OF GROUPS INVOLVING l -RANK AND COMBINATORIAL IDENTITIES 3 If we let a =, then we obtain that the u-robability that a finite abelian grou has k -rank equal to r is j r+ 6 j+u Q kru+r,k, 2r+u r Note that Q,k, = Q,k,k, which allows us to recover the 0 and -robabilities as a roduct formula as in [Coh85, Corollary 3] and in fact, with our arameter x, the formula with a > is useless for the 0 and -robability we have in mind On the other hand, we have for grous of tye S for simlicity we only give the Dirichlet function for a = j r+ 2s+2j+ n n 2s G S n 2 r k =2r Aut S G = From the formula, we deduce: 2 r kr2s+2r+ Q 2,k, 4r+2s ζ S, s Proosition 9 For grous of tye S, the u-robability that a grou of tye S has k -rank equal to 2r is j r+ 2 r 2u+2j kr2u+2r Q 2,k, 4r+2u 3 Once again, we aly the heuristic argument for Tate-Shafaverich grous of ellitic curves Let E be a natural family of rank 0 ellitic curves E such that the behavior of the -art of XE is not biased The robability that the k -rank of XE is equal to 2r should be P 0 k, r = P k, r = j r+ 2 r 2j kr2r Q 2,k, 4r 3 For a family of rank ellitic curves, the robability should be j r+ 2j+ 2 r kr2r+ Q 2,k, 4r The form of the two formulas above in articular the fact that the owers of in the index and in the argument of Q have different arities suggests that it is robably hoeless to find a roduct formula for the u-robabilities related to Tate-Shafarevich grous unlike the case of class grous of quadratic fields Here are few numerical aroximations of values of P 0 k, r and P k, r

14 4 CHRISTOPHE DELAUNAY P i k, r = 2 = 3 = 5 P 0 2, P 0 2, P 0 2, P 0 3, P 0 3, P 2, P 2, P 3, The case k = We must have a = Let us fix l integers r r 2 r l r l 0 Then Theorem 8 gives x eh 7 Aut H = x r r+ +rl l r2 + +r2 x j= j l H r H=r r l H=r l r l j= r j r j+ It is clear that the factor x r+ +r l r2 r2 l / l j= / r j r j+ aears in each term of the sum in the left hand side Naturally, this factor also aears in the right hand side Corollary 0 We have for finite abelian grous n n s Hn r H=r r l H=r l and for grous of tye S n n 2s G S n 2 r G=2r r l G=2r l Aut H = Aut S G = ζ, s j r l + r2 + +r2 l +sr+ +r l ζ S, s j r l + 2r2 + +r2 l +2s+r+ +r l s+j l r l j= r j r j+ 2s+2j+ l 2 r l 2 r j r j+ j=, We deduce from the revious the u-robability laws:

15 AVERAGES OF GROUPS INVOLVING l -RANK AND COMBINATORIAL IDENTITIES 5 Corollary Let l and r r 2 r l r l 0 For finite abelian grous, the u-robability that a finite abelian grou has j -rank equal to r j for all j l is u+j j r l + l r2 + +r2 l +ur+ +r l r j r j+ r l j= For grous of tye S, the u-robability that a grou of tye S has j -rank equal to 2r j for all j l is 2u+2j j r l + l 2r2 + +r2 l +2u r+ +r l 2 r l 2 r j r j+ Once again, we could deduce, from the heuristic rincile, conjectures concerning class grous and Tate-Shafarevich grous 33 Examles and identities We obtain from the first and second arts some combinatorial identities and we only focus on this asect The following formula seems to be reviously unknown: j= Corollary 2 We have 0 r l r Proof We aly q r2 + +r2 l +u r+ +r l j r l + l q rl q rj r j+ j= q u+j = + q u + + q ul 0 r l r u-robr H = r, r 2H = r 2, r lh = r l T l H = M u T l Then, we use Corollary 3 and Corollary and substitue / q In articular, if u is an integer, the formula in Corollary 2 becomes 8 0 r l r qr 2 + +r2 l +u r+ +r l = l q rl +u q rl q rj r j+ j= + q u + + q ul q The formula 8 and the formula in Corollary 2 are nearly of the same tye as the ones in [Hal38] and in [Ste90] They do not seem however to be direct consequences

16 6 CHRISTOPHE DELAUNAY of it There are neither secial cases of the generalizations of Stembridge s formulas given in [IJZ06] and in [JZ05] Finally, we conclude with a last remark Let k and r k r k+ r k+l be l + non-negative integers Theorem 8 can be re-written as 9 λ µ k =r k µ k+ =r k+ µ k+l =r k+l x µ+µ2+ +µs q µ2 +µ µ2 s q λ q λ2 q λs q µ µ2 µs+µa+µa++ +µs = We fix r k+l = r and write q ar xkr k+r k+ + +r k+l q kr2 k +r2 k+ + +r2 k+l Qq,l, xq 2rk k rk+l j= k+l xqj q rk+l j=k q rj r j+ λ µ k+l =r X = r k,r k+, r k+l λ r k r k+ r k+l r µ k =r k µ k+l =r k+l µ k+l =r and we aly equation 9 with convenient arameters to the left hand side and to the inner sum of right hand side in fact, we also take r = 0 since the general case r > 0 is a direct consequence of it Then, we use the substitution x x/q, we reindex r k, r k+,, r k+l by r, r 2,, r l and we obtain directly the following general recursion formula on the Q q,k,a : Proosition 3 We have for l, k, r 0 and a k 0 Q q,k+l,a x = x kr+r2+ +r l q kr2 +r r2 l +kr+r2+ +r Q l q ar q,k,a xq 2r l r r l 0 q rl j= q r j r j+ When l =, formula 0 is exactly one of the inuts needed by Andrews for roving equation 4 Formula 0 is robably also a consequence of consecutive alications of the secial case l = The questions of using or obtaining other classical combinatorial identities with this algebraic asect will be discussed elsewhere X, 4 Acknowledgements This work was suorted by the French Agence Nationale our la Recherche, the ANR roject no 07-BLAN-0248 ALGOL The author would like to thank M Bhargava for his questions about the heuristics and F Jouhet for many discussions about the combinatorial identities occurring in this aer

17 AVERAGES OF GROUPS INVOLVING l -RANK AND COMBINATORIAL IDENTITIES 7 References [And74] George E Andrews An analytic generalization of the Rogers-Ramanujan identities for odd moduli Proc Nat Acad Sci USA, 7: , 974 [And98] George E Andrews The theory of artitions Cambridge Mathematical Library Cambridge University Press, Cambridge, 998 Rerint of the 976 original [BS0a] M Bhargava and A Shankar Binary quartic forms having bounded invariants, and the boundedness of the average rank of ellitic curves Prerint available on arxiv:006002, 200 [BS0b] M Bhargava and A Shankar Ternary cubic forms having bounded invariants, and the existence of a ositive roortion of ellitic curves having rank 0 Prerint available on arxiv: , 200 [CL84] H Cohen and H W Lenstra, Jr Heuristics on class grous In Number theory New York, 982, volume 052 of Lecture Notes in Math, ages Sringer, Berlin, 984 [Coh85] Henri Cohen On the k -rank of finite abelian grous and Andrews generalizations of the Rogers-Ramanujan identities Nederl Akad Wetensch Indag Math, 474: , 985 [Del0] Christohe Delaunay Heuristics on Tate-Shafarevitch grous of ellitic curves defined over Q Exeriment Math, 02:9 96, 200 [Del07] Christohe Delaunay Heuristics on class grous and on Tate-Shafarevich grous: the magic of the Cohen-Lenstra heuristics In Ranks of ellitic curves and random matrix theory, volume 34 of London Math Soc Lecture Note Ser, ages Cambridge Univ Press, Cambridge, 2007 [Hal38] P Hall A artition formula connected with Abelian grous Comment Math Helv, :26 29, 938 [IJZ06] Masao Ishikawa, Frédéric Jouhet, and Jiang Zeng A generalization of Kawanaka s identity for Hall-Littlewood olynomials and alications J Algebraic Combin, 234:395 42, 2006 [JZ05] Frédéric Jouhet and Jiang Zeng New identities for Hall-Littlewood olynomials and alications Ramanujan J, 0:89 2, 2005 [PR0] B Poonen and E Rains Random maximal isotroic subsaces and selmer grous Prerint available on arxiv: , 200 [Ste90] John R Stembridge Hall-Littlewood functions, lane artitions, and the Rogers- Ramanujan identities Trans Amer Math Soc, 392: , 990 December 5, 200 Christohe Delaunay, Université de Lyon, CNRS, Université Lyon, Institut Camille Jordan, 43, boulevard du novembre 98, F Villeurbanne Cedex, France delaunay@mathuniv-lyonfr Url : htt://mathuniv-lyonfr/~delaunay