THE COHEN-LENSTRA HEURISTICS, MOMENTS AND p^{j}-RANKS OF SOME GROUPS

CHRISTOPHE DELAUNAY AND FRÉDÉRIC JOUHET

Abstract. This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to E. Fouvry and J. Klüners which proves that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture 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.

1. Introduction and notation

The Cohen-Lenstra heuristics and their generalizations to Tate-Shafarevich groups are models for formulating conjectures related to class groups of number fields and Tate-Shafarevich groups of elliptic curves varying in some natural families. This article deals with the coherence of the model. More precisely, our aim is to prove that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. This work actually extends and generalizes an earlier one by E. Fouvry and J. Klüners in [FK06] which deals with class groups; we will follow their presentation and adapt the main techniques of their proofs.

We will use the following notation. The letter d will denote a fundamental discriminant and Cℓ(K_{d}) the class group associated to the quadratic number field K_{d} = ℚ(\sqrt{d}). The letter p will always denote a prime number. If G is a finite abelian group, the p^{j}-rank of G is defined by rk_{p^{j}}(G) = dim_{𝔽_{p}} p^{j−1}G/p^{j}G. For any real valued function f defined over isomorphism classes of finite abelian groups, we say that f(Cℓ(K_{d})) has average value c_{±} ∈ ℝ if

\frac{∑_{0<±d<X} f(Cℓ(K_{d}))}{∑_{0<±d<X} 1} = c_{±} + o(1), as X → ∞.

If f is the characteristic function of some property, then c_{±} is called the probability of the property (or the density of the set of the class groups satisfying it). The Cohen-Lenstra heuristics allow one to formulate conjectures for the average values of f(Cℓ(K_{d})_{odd}) for “reasonable” functions f, where Cℓ(K_{d})_{odd} denotes the odd part of Cℓ(K_{d}). Concerning the 2-part of class groups, genus theory enables one to determine the structure of Cℓ(K_{d})[2]. However Gerth ([Ger84], [Ger87]) succeeded in generalizing the Cohen-Lenstra heuristics for the 4-part of class groups, and also obtained conjectures for the average values of f(Cℓ(K_{d})^{2}) for reasonable functions f.

In particular, the above heuristics give on the one hand a prediction for the average value of the function f(p^{k rk_{p}(Cℓ(K_{d})^{2})}) for all k ∈ ℕ, and on the other hand a prediction for the probability that rk_{p}(Cℓ(K_{d})^{2}) = r for all r ∈ ℕ (note that for odd p, rk_{p}(Cℓ(K_{d})^{2}) = rk_{p}(Cℓ(K_{d}))). In [FK06], the authors proved that the former prediction implies the latter. The aim of this article is to generalize the results of [FK06] in two directions. First, we will consider higher moments (including for example the average values of f(p^{k rk_{p^{j}} (Cℓ(K_{d})^{2})}) for all positive integers (k, j)) and probability laws of p^{j}-ranks for j ⩾ 1. Secondly, we will obtain analogous results concerning heuristics on Tate-Shafarevich groups and on Selmer groups of elliptic curves ([Del01, Del07, DJ12]).

Date: April 2, 2013.

We recall for (a, q) ∈ ℂ^{2} with |q| < 1 and k ∈ ℤ the q-shifted factorial

(a; q)_{k} := \{ \begin{matrix}1 if k = 0 \\ (1 − a) … (1 − aq^{k−1}) if k > 0 \\ 1/(1 − aq^{−1}) … (1 − aq^{k}) if k < 0, \end{matrix}

and (a; q)_{∞} := lim_{k→+∞}(a; q)_{k}. Note that (1/p; 1/p)_{k} = η_{k}(p), where η_{k} is the function defined in [CL84] and used in [FK06]. We will also use the q-binomial coefficient

\left[{n \atop k}\right]_{q} := \frac{(q; q)_{n}}{(q; q)_{k}(q; q)_{n−k}} ∈ ℕ[q].

A partition λ := (λ_{1} ⩾ λ_{2} ⩾ ⋯) of a nonnegative integer n is a finite decreasing sequence of nonnegative integers whose sum is equal to n. If λ is a partition of n, we write |λ| = n and the notation λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} means that m_{i} is the multiplicity of the integer i in λ (hence, we have n = λ_{1}+ λ_{2}+⋯ = |λ| = m_{1}+2m_{2}+⋯+ℓm_{ℓ}). If μ := (μ_{1} ⩾ μ_{2} ⩾ ⋯) is a second integer partition, then we define (λ|μ) := ∑_{i} λ_{i}μ_{i} (we will often use the statistics (λ|λ) = ∑_{i} λ_{i}^{2} which must not be mistaken for |λ|^{2} = (∑_{i} λ_{i})^{2}). Finally, the notation μ ⊆ λ means that μ_{i} ⩽ λ_{i} for all i ⩾ 1. Recall that a finite abelian p-group G has type λ = 1^{m_{1}} ⋯ ℓ^{m_{ℓ}} if

G ≃ (ℤ/pℤ)^{m_{1}} ⊕ ⋯ ⊕ (ℤ/p^{ℓ}ℤ)^{m_{ℓ}} .

If λ = 1^{m_{1}} ⋯ ℓ^{m_{ℓ}} is an integer partition, we denote by λ′ := (λ′_{1} ⩾ λ′_{2} ⩾ ⋯) its conjugate defined by λ′_{k} = ∑_{j=k}^{ℓ} 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 finite abelian p-group of type λ, which can be expressed by

(1)  C_{λ, μ}(p) = p^{∑_{i⩾1} μ′_{i+1}(λ′_{i}−μ′_{i})} ∏_{i⩾1} \left[{λ′_{i} − μ′_{i+1} \atop λ′_{i} − μ′_{i}}\right]_{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, we obtained the following conjecture ([DJ12, Conjecture 1]).

Conjecture 1. For any positive integer ℓ, let λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} be an integer partition, and assume^{1} that p ⩾ 3. As d is varying over the set of fundamental negative discriminants, the average of | Cℓ(K_{d})[p]|^{m_{1}}| Cℓ(K_{d})[p^{2}]|^{m_{2}} ⋯ | Cℓ(K_{d})[p^{ℓ}]|^{m_{ℓ}} is equal to

∑_{μ⊆λ} C_{λ,μ}(p),

1In order to simplify the notations, we will exclude p = 2 for class groups. However, one can obtain directly the case p = 2 by replacing Cℓ(K_{d}) by Cℓ(K_{d})^{2} all along the article.

where the sum is over all integer partitions μ ⊆ λ. Similarly, as d is varying over the set of fundamental positive discriminants, the average of | Cℓ(K_{d})[p]|^{m_{1}}| Cℓ(K_{d})[p^{2}]|^{m_{2}} ⋯ | Cℓ(K_{d})[p^{ℓ}]|^{m_{ℓ}} is equal to

∑_{μ⊆λ} C_{λ,μ}(p)p^{−|μ|}.

Concerning the probability laws of p^{j}-ranks rk_{p^{j}}(Cℓ(d)), the following conjecture comes naturally from [Del11, Corollary 11].

Conjecture 2. Let ℓ be a positive integer and μ := μ_{1} ⩾ μ_{2} ⩾ ⋯ ⩾ μ_{ℓ} ⩾ 0 a partition of length ℓ(μ) ⩽ ℓ (i.e., μ_{ℓ+1} = 0). Assume that p ⩾ 3. Then, as d is varying over the set of fundamental negative discriminants, the probability that rk_{p^{j}}(Cℓ(K_{d})) = μ_{j} for all 1 ⩽ j ⩽ ℓ is equal to

\frac{(1/p^{μ_{ℓ}+1}; 1/p)_{∞}}{p^{μ_{1}^{2}+···+μ_{ℓ}^{2}} ∏_{j=1}^{ℓ}(1/p; 1/p)_{μ_{j}−μ_{j+1}}} .

Moreover, as d is varying over the set of fundamental positive discriminants, the probability that rk_{p^{j}}(Cℓ(K_{d})) = μ_{j} for all 1 ⩽ j ⩽ ℓ is equal to

\frac{(1/p^{μ_{ℓ}+2}; 1/p)_{∞}}{p^{μ_{1}^{2}+···+μ_{ℓ}^{2}+(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(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^{1}. In [FK07], the authors proved Conjecture 1 for p = 2 (replacing Cℓ(K_{d}) by Cℓ(K_{d})^{2}) and any λ = 1^{m_{1}}. 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_{1}} and Conjecture 2 for ℓ = 1. More precisely, if λ = 1^{m}, Conjecture 1 says that the average of | Cℓ(K_{d})[p]|^{m} for class groups of imaginary (resp. real) quadratic fields is equal to (with the notations of [DJ12])

M_{0}(x^{1^{n}}) = ∑_{k=0}^{n} \left[{n \atop k}\right]_{p} (resp. M_{1}(x^{1^{n}}) = \frac{1}{p} ∑_{k=0}^{n} \left[{n \atop k}\right]_{p})

Those correspond to 𝒩(n, p) (resp. ℳ_{n}(p)) used in [FK07, FK06]. Fouvry and Klüners proved in [FK06] that if Conjecture 1 is true with λ = 1^{m} for all m, then Conjecture 2 is true with ℓ = 1 for all μ_{1} ⩾ 0. We will adapt their proof and use a result in [DJ12] to simplify one of its step in order to obtain the following generalization.

Theorem 1. Assume that p ⩾ 3. Let ℓ be a positive integer and assume that for any λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, as d is varying over the set of fundamental negative discriminants, the average of | Cℓ(K_{d})[p]|^{m_{1}}| Cℓ(K_{d})[p^{2}]|^{m_{2}} ⋯ | Cℓ(K_{d})[p^{ℓ}]|^{m_{ℓ}} is equal to ∑_{μ⊆λ} C_{λ,μ}(p). Then for any μ_{1} ⩾ μ_{2} ⩾ ⋯ ⩾ μ_{ℓ} ⩾ 0, as d is varying over the set of fundamental negative discriminants, the probability that rk_{p^{j}}(Cℓ(K_{d})) = μ_{j} for all 1 ⩽ j ⩽ ℓ is equal to

\frac{(1/p^{μ_{ℓ}+1}; 1/p)_{∞}}{p^{μ_{1}^{2}+···+μ_{ℓ}^{2}} ∏_{j=1}^{ℓ}(1/p; 1/p)_{μ_{j}−μ_{j+1}}} .

Furthermore, assume that for any positive integer ℓ and any λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, as d is varying over the set of fundamental positive discriminants, the average of | Cℓ(K_{d})[p]|^{m_{1}}| Cℓ(K_{d})[p^{2}]|^{m_{2}} ⋯ | Cℓ(K_{d})[p^{ℓ}]|^{m_{ℓ}} is equal to ∑_{μ⊆λ} C_{λ,μ}(p)p^{−|μ|}. Then for any μ_{1} ⩾ μ_{2} ⩾ ⋯ ⩾ μ_{ℓ}, as d is varying over the set of fundamental positive discriminants, the probability that rk_{p^{j}}(Cℓ(K_{d})) = μ_{j} for all 1 ⩽ j ⩽ ℓ is equal to

\frac{(1/p^{μ_{ℓ}+2}; 1/p)_{∞}}{p^{μ_{1}^{2}+···+μ_{ℓ}^{2}+(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p; 1/p)_{μ_{j}−μ_{j+1}}} .

One can also adapt the Cohen-Lenstra heuristics for Tate-Shafarevich groups of elliptic curves. If E is an elliptic curve defined over ℚ, we denote by � (E) its Tate-Shafarevich group. In this context, we assume in this article that � (E)[p^{∞}] is finite for all elliptic curves E/ℚ (which is a classical conjecture). In that case, � (E)[p^{∞}] is a group of type^{2} S (i.e. it is endowed with a bilinear, alternating, non degenerate pairing β: � (E)[p^{∞}] × � (E)[p^{∞}] → ℚ/ℤ). Let ℱ_{u} be the family of elliptic curves E defined over^{3} ℚ with rank u, ordered by their conductor (denoted by N(E)). If f is a real valued function defined over isomorphism classes of groups of type S (see [Del01, Del07]), then we say that f(� (E)) has average value c ∈ ℝ for E varying over ℱ_{u} if

∑_{\begin{matrix}E∈ℱ_{u} \\ N(E)<X \end{matrix}} f(� (E)) = (c + o(1)) ∑_{\begin{matrix}E∈ℱ_{u} \\ N(E)<X \end{matrix}} 1, as X → ∞.

If f is the characteristic function of some property, we say that c is the probability of this property (or the density of the set of Tate-Shafarevich groups satisfying it) for E varying over ℱ_{u}. We raised the following conjecture in [DJ12].

Conjecture 3. Let ℓ be a positive integer, let λ = 1^{m_{1}}2^{m_{2}} ⋯ 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 |� (E)[p]|^{m_{1}}|� (E)[p^{2}]|^{m_{2}} ⋯ |� (E)[p^{ℓ}]|^{m_{ℓ}} is equal to

∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)}.

Concerning the probability laws of p^{j}-ranks rk_{p^{j}}(� (E)), we have the following ([Del11]).

Conjecture 4. Let ℓ be a positive integer, let μ = μ_{1} ⩾ μ_{2} ⩾ ⋯ ⩾ μ_{ℓ} ⩾ 0 be an integer 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}}(� (E)) = 2μ_{j} for all 1 ⩽ j ⩽ ℓ is equal to

\frac{(1/p^{2u+2μ_{ℓ}+1}; 1/p^{2})_{∞}}{p^{2(μ_{1}^{2}+···+μ_{ℓ}^{2})+(2u−1)(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p^{2}; 1/p^{2})_{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 elliptic curves E/ℚ, where S(E)_{p} is the p-Selmer group of E/ℚ. Their results, together with a strong form of the rank conjecture (asserting that the rank of E is 0 or 1 with probability 1/2 each and that elliptic curves with rank ⩾ 2 do not contribute in the averages), imply Conjecture 3 for λ = 1^{1} and p = 2, 3. Heath-Brown [HB93, HB94], then Swinnerton-Dyer [SD08] and Kane [Kan11] also obtained results about S(E)_{2} when E is varying over some families of quadratic twists. Their results, together with a strong rank conjecture, imply Conjecture 3 for λ = 1^{1} and p = 2 for some families of quadratic twists. Furthermore, Conjecture 3 is compatible with the conjecture of Poonen and Rains ([PR12]).

2This notion is different from the previously mentioned groups of type λ, where λ is a partition.

3One can replace in our discussion ℚ by any other number field K.

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. Assume that for any λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, as E/ℚ, ordered by conductors, is varying over ℱ_{u}, the average of |� (E)[p]|^{m_{1}}|� (E)[p^{2}]|^{m_{2}} ⋯ |� (E)[p^{ℓ}]|^{m_{ℓ}} is equal to ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)}. Then for any μ_{1} ⩾ μ_{2} ⩾ ⋯ ⩾ μ_{ℓ}, as E/ℚ is varying over ℱ_{u}, the probability that rk_{p^{j}}(� (E)) = 2μ_{j} for all 1 ⩽ j ⩽ ℓ is equal to

\frac{(1/p^{2u+2μ_{ℓ}+1}; 1/p^{2})_{∞}}{p^{2(μ_{1}^{2}+···+μ_{ℓ}^{2})+(2u−1)(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p^{2}; 1/p^{2})_{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 in some families. In particular, we can obtain similar results for Selmer groups of elliptic curves (see section 5). 2- From the laws of the p^{j}-ranks for all j = 1, …, ℓ as in theorems 1 and 2, one can also deduce the probability that the p^{ℓ}-rank is equal to some fixed value for a 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 probability laws of the p^{j}-ranks for all j = 1, …, ℓ. For this, it seems that we need to know an error term for the probability laws and this error term is not given by the heuristic philosophy. However, the theoretical results of [FK07, HB93, SD08, Kan11] concern the p^{j}-ranks of the groups studied (with an explicit error term) from which the moments are deduced.

2. An auxiliary analytic tool

In this section, we prove a generalization of [FK06, Lemma 6] which will be useful later.

Lemma 3. Let a ∈ ℂ with |a| > 1 and g(z) = ∑_{r⩾0} w_{r}z^{r} be an entire function satisfying the following properties:

• there exists an absolute constant C > 0 and α ∈ ℝ such that for all r ∈ ℕ, |w_{r}| ⩽ Ca^{−r^{2}/2+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−1/2, we have g ≡ 0 (i.e. ω = ∞).

Proof. Let k be a nonnegative integer. Taking |z| = |a|^{k}, a direct computation shows that |g(z)| ⩽ C′|a|^{(k+α)^{2}/2}, where C′ = C ∑_{r∈ℤ} |a|^{−(r−α)^{2}/2}. Assume that g ≢ 0. Then [FK06, Lemma 6] gives

sup_{|z|=|a|^{k}} |g(z)| ≫ |a|^{k(k+1)/2+kω}.

Hence, we must have (k + α)^{2} ⩾ k(k + 1) + 2kω for all k ∈ ℕ, which implies ω ⩽ α − 1/2. □

Corollary 4. Let ℓ ∈ ℕ^{∗}, let a ∈ ℂ with |a| > 1 and let g(\underline{z}) = ∑_{r} w_{r}z_{1}^{r_{1}}z_{2}^{r_{2}} ⋯ z_{ℓ}^{r_{ℓ}} with \underline{z} = (z_{1}, z_{2}, ⋯, z_{ℓ}) ∈ ℂ^{ℓ} and where the sum is over all integer partitions r = r_{1} ⩾ r_{2} ⩾ ⋯ ⩾ r_{ℓ} ⩾ 0. We assume that:

• |w_{r}| ⩽ C a^{−(r|r)/2+α|r|} for some absolute constant C and α < 3/2;

• g(a^{m_{1}}, a^{m_{2}}, …, a^{m_{ℓ}}) = 0 for all nonnegative integers m_{1}, m_{2}, …, m_{ℓ}.

If α < 1/2, then g ≡ 0 and w_{r} = 0 for all r. If α ∈ [1/2, 3/2[ and w_{0,0,...,0} = 0, then g ≡ 0 (therefore w_{r} = 0 for all r).

Proof. If ℓ = 1, this is proved by the above lemma, since we have ω > α − 1/2 in both cases under consideration. If ℓ ⩾ 2, we fix (m_{2}, …, m_{ℓ}) ∈ ℕ^{ℓ−1} and set

f(z) = ∑_{r_{1}} z^{r_{1}} ( ∑_{r_{1}⩾r_{2}⩾···⩾r_{ℓ}} w_{r_{1},r_{2},...,r_{ℓ}}a^{r_{2}m_{2}} ⋯ a^{r_{ℓ}m_{ℓ}} ) .

Then f(z) satisfies the condition of the previous lemma since

| ∑_{r_{2}⩾...⩾r_{ℓ}⩾0} w_{r_{1},r_{2},...,r_{ℓ}}a^{r_{2}m_{2}} ⋯ a^{r_{ℓ}m_{ℓ}} |  ≪_{m_{2},...,m_{ℓ}} a^{−(r_{1}|r_{1})/2+αr_{1}}.

With the conditions of the corollary, we deduce that f(z) = 0, therefore for any fixed r_{1} ⩾ 0, we must have

∑_{r_{1}⩾r_{2}⩾···⩾r_{ℓ}⩾0} w_{r_{1},r_{2},...,r_{ℓ}}a^{r_{2}m_{2}} ⋯ a^{r_{ℓ}m_{ℓ}} = 0,

for all m_{2}, …, m_{ℓ}. We use the fact that when r_{1} is fixed,

∑_{r_{1}⩾r_{2}⩾···⩾r_{ℓ}⩾0} w_{r_{1},r_{2},...,r_{ℓ}}z_{2}^{r_{2}} ⋯ z_{r}^{r_{ℓ}}

is a polynomial to conclude. □

3. Class groups of number fields

We will actually prove a more general result displayed in Theorem 5 (which clearly implies Theorem 1). If K is a number field, we denote by Cℓ(K) its class group. Let 𝒦 be a fixed set of number fields ordered by the absolute value of their discriminant disc(K). If f is a real valued function defined over isomorphism classes of finite abelian groups, then, as before, we say that f(Cℓ(K)) has average value c ∈ ℝ for K varying over 𝒦 if

∑_{\begin{matrix}K∈𝒦 \\ | disc(K)|<X \end{matrix}} f(Cℓ(K)) = (c + o(1)) ∑_{\begin{matrix}K∈𝒦 \\ | disc(K)|<X \end{matrix}} 1, as X → ∞.

As before, if f is the characteristic function of some property, we say that c is the probability of this property (or the density of the set of the class groups satisfying it) for K varying in 𝒦.

Theorem 5. Let u be a nonnegative integer and let ℓ be a positive integer. Assume that for every integer partition λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, as K is varying over 𝒦, the average of | Cℓ(K)[p]|^{m_{1}} ⋯ | Cℓ(K)[p^{ℓ}]|^{m_{ℓ}} is equal to ∑_{μ⊆λ} C_{λ,μ}(p)p^{−u|μ|}. Then for any μ_{1} ⩾ μ_{2} ⩾ ⋯ ⩾ μ_{ℓ}, as K is varying over 𝒦, the probability that rk_{p^{j}}(Cℓ(K)) = μ_{j} for all 1 ⩽ j ⩽ ℓ, is equal to

\frac{(1/p^{u+μ_{ℓ}+1}; 1/p)_{∞}}{p^{μ_{1}^{2}+···+μ_{ℓ}^{2}+u(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p; 1/p)_{μ_{j}−μ_{j+1}}} .

We follow and generalize the proof of [FK06]. First, we will need the following proposition.

Proposition 6. Let u ∈ ℕ and ℓ ∈ ℕ. For all λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} we have

(2)  ∑_{μ⊆λ} C_{λ,μ}(p)p^{−u|μ|} = O_{p,ℓ}(p^{(λ′|λ′)/2}).

Proof. Set C_{λ} := ∑_{μ⊆λ} C_{λ,μ}(p)p^{−u|μ|}. The equality

\left[{n \atop k}\right]_{p} = \frac{(p; p)_{n}}{(p; p)_{k}(p; p)_{n−k}} = p^{k(n−k)} \frac{(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 coefficients C_{λ,μ}(p), imply that

C_{λ} ⩽ C ∑_{μ⊆λ} p^{∑_{i} μ′_{i}(λ′_{i}−μ′_{i})} ⩽ C p^{∑_{i} λ_{i}^{′2}/4} ∑_{μ⊆λ} 1,

for some constant C depending only on p and ℓ (we used μ′_{i}(λ′_{i} − μ′_{i}) ⩽ λ_{i}^{′2}/4 for all 0 ⩽ μ′_{i} ⩽ λ′_{i}, noting that μ ⊆ λ if and only if μ′ ⊆ λ′). Now, since ℓ is fixed, the number of sub-partitions μ ⊆ λ is certainly bounded by the product (λ_{1} + 1)(λ_{2} + 1) ⋯ (λ_{ℓ} + 1). By the arithmetico-geometric mean inequality, we obtain

∑_{μ⊆λ} 1 ⩽ (1 + \frac{|λ|}{ℓ})^{ℓ} = O_{ℓ}(|λ|^{ℓ}).

Finally, we have C_{λ} = O_{p,ℓ}(p^{(λ′|λ′)/4}|λ|^{ℓ}) = O_{p,ℓ}(p^{(λ′|λ′)/2}). □

Remark. As it can be seen in the proof of the above proposition, we have the more precise upper bound C_{λ} = O_{p,ℓ}(p^{(λ′|λ′)/4}|λ|^{ℓ}). Nevertheless, the upper bound given in the proposition will be sufficient for our application.

Proof of Theorem 5. For X ⩾ 1 and r := r_{1} ⩾ r_{2} ⩾ ⋯ ⩾ r_{ℓ} ⩾ 0, set

N(X) := |{K ∈ 𝒦: | disc(K)| ⩽ X}|, and
N(X, r) := |{K ∈ 𝒦: | disc(K)| ⩽ X and rk_{p^{i}}(Cℓ(K)) = r_{i} for all i = 1, …, ℓ}|.

For every λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, the assertion of Theorem 5 implies that

(3) ∑_{r} \frac{N(X, r)}{N(X)}p^{m_{1}r_{1}+m_{2}(r_{1}+r_{2})+···+m_{ℓ}(r_{1}+r_{2}+···+r_{ℓ})} = ∑_{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_{1} ⩾ r_{2} ⩾ ⋯ ⩾ r_{ℓ} ⩾ 0. Note that m_{1}r_{1} + m_{2}(r_{1} + r_{2}) + ⋯ + m_{ℓ}(r_{1} + r_{2} + ⋯ + r_{ℓ}) = (λ′|r). Hence, equation (3) can be written as

(4)  ∑_{r} \frac{N(X, r)}{N(X)}p^{(λ′|r)} = ∑_{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 in the compact set [0, 1]. We deduce that there exists a real number d_{r} ∈ [0, 1] and an infinite subset ℳ of ℕ such that

N(m, r)/N(m) → d_{r} (m ∈ ℳ, m → ∞).

Replacing m_{i} by 2m_{i} + 1, we see from (3) and Proposition 6 that

(5)  \frac{N(X, r)}{N(X)} ≪_{λ} p^{−(2m_{1}+1)r_{1}−(2m_{2}+1)(r_{1}+r_{2})−···−(2m_{ℓ}+1)(r_{1}+···+r_{ℓ})},

uniformly in X and r, from which we deduce that

∑_{r} \frac{N(m, r)}{N(m)}p^{(λ′|r)} = O_{λ}(1).

Hence by Lebesgue’s Dominated Convergence Theorem we have

∑_{r} d_{r}p^{(λ′|r)} = ∑_{μ⊆λ} C_{λ,μ}(p)p^{−|μ|u}.

If we consider the infinite multi-dimensional system

(𝒮)  ∑_{r} x_{r}p^{(λ′|r)} = ∑_{μ⊆λ} C_{λ,μ}(p)p^{−|μ|u} for all λ = 1^{m_{1}} ⋯ ℓ^{m_{ℓ}},

where the unknowns are x_{r} ⩾ 0 then d_{r} is a solution of (𝒮). We already know that

x_{r} = \frac{(1/p^{μ_{ℓ}+u+1}; 1/p)_{∞}}{p^{μ_{1}^{2}+···+μ_{ℓ}^{2}+u(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(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 equivalent to

∑_{r} x_{r}p^{(λ|r)} = C_{λ′} for all λ,

where C_{λ′}(p) = ∑_{μ⊆λ′} C_{λ′,μ}(p)p^{−|μ|u} = O(p^{(λ|λ)/2}) by Proposition 6. From x_{r} ⩾ 0, we deduce x_{r} = O(p^{−(λ|r)+(λ|λ)/2}), so when λ = r, we get

0 ⩽ x_{r} ⩽ c_{0}p^{−(r|r)/2},

for some absolute constant c_{0}. Now, if (x′_{r})_{r} is another solution of (𝒮), then setting w_{r} = x_{r} − x′_{r}, we have a function

(6)  f(\underline{z}) = f(z_{1}, z_{2}, …, z_{ℓ}) = ∑_{r} w_{r}z_{1}^{r_{1}}z_{2}^{r_{2}} ⋯ z_{ℓ}^{r_{ℓ}}

satisfying f(\underline{z}) = 0 if z_{1} = p^{m_{1}}, z_{2} = p^{m_{2}}, …, z_{ℓ} = p^{m_{ℓ}} for all m_{1}, m_{2}, …, m_{ℓ} ∈ ℕ, and

|w_{r}| ⩽ 2c_{0}p^{−(r|r)/2}.

Thus we can apply Corollary 4 to conclude that x_{r} = x′_{r}. So, as X → ∞, the sequence N(X, r)/N(X) has only one limit point, which is d_{r} = x_{r}. □

Corollary 7. Let u be a nonnegative integer and let ℓ be a positive integer. Assume that for every integer partition λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, as K is varying over 𝒦, the average of | Cℓ(K)[p]|^{m_{1}} ⋯ | Cℓ(K)[p^{ℓ}]|^{m_{ℓ}} is equal to ∑_{μ⊆λ} C_{λ,μ}(p)p^{−u|μ|}. Then for k ∈ ℕ, as K is varying over 𝒦, the probability that rk_{p^{ℓ}}(Cℓ(K)) = k is equal to

\frac{(1/p^{u+k+1}; 1/p)_{∞}}{(1/p; 1/p)_{k}p^{ℓk(u+k)}} Q_{1/p,ℓ,1} (\frac{1}{p^{2k+u−1}} ) ,

where Q_{q,ℓ,1}(x) := ∑_{n⩾0} \frac{(−1)^{n}x^{ℓn}q^{n(n+1)(2ℓ+1)/2−n}(1 − xq^{2n+1})}{(q; q)_{n}(xq^{n+1}; q)_{∞}}.

The series Q_{q,ℓ,k}(x) was defined by Andrews (see [And74]). The formula of the above corollary is the u-probability that the p^{ℓ}-rank of a finite abelian p-group is equal to k, as obtained in [Coh85] (note that we use the definition of u-averages and 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}}(Cℓ(K)) = k}|.

We have

(7)  \frac{N(X, ℓ, k)}{N(X)} = ∑_{μ_{1}⩾μ_{2}⩾···⩾μ_{ℓ−1}⩾k} \frac{N(X, μ)}{N(X)},

where the sum is over integer partitions μ = μ_{1} ⩾ μ_{2} ⩾ ⋯ ⩾ μ_{ℓ−1} ⩾ μ_{ℓ} = k. By the assumptions and Theorem 5, we have

lim_{X→∞} \frac{N(X, μ)}{N(X)} = \frac{(1/p^{u+μ_{ℓ}+1}; 1/p)_{∞}}{p^{μ_{1}^{2}+···+μ_{ℓ}^{2}+u(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p; 1/p)_{μ_{j}−μ_{j+1}}} .

In equation (7) we take the limit as X → ∞ and use Lebesgue’s Dominated Convergence Theorem (with equation (5)) to obtain that the probability that rk_{p^{ℓ}}(Cℓ(K)) = k is equal to

∑_{μ_{1}⩾···⩾μ_{ℓ−1}⩾μ_{ℓ}=k} \frac{(1/p^{u+μ_{ℓ}+1}; 1/p)_{∞}}{p^{μ_{1}^{2}+···+μ_{ℓ}^{2}+u(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p; 1/p)_{μ_{j}−μ_{j+1}}}

= \frac{(1/p^{u+k+1}; 1/p)_{∞}}{p^{ℓk(u+k)}(1/p; 1/p)_{k}} ∑_{μ_{1}⩾···⩾μ_{ℓ−1}⩾0} \frac{(1/p)^{μ_{1}^{2}+···+μ_{ℓ−1}^{2}+(u+2k)(μ_{1}+···+μ_{ℓ−1})}}{∏_{j=1}^{ℓ−1}(1/p; 1/p)_{μ_{j}−μ_{j+1}}} ,

the equality being derived by shifting all indices by k. Now, from [Del11, Proposition 13] (or [And74, Eq. (2.5)]), the last sum is exactly

Q_{1/p,ℓ,1} (\frac{1}{p^{2k+u−1}} ) ,

as expected (note that Q_{q,1,1}(x) = 1). □

4. Tate-Shafarevich groups of elliptic curves

In this section, we prove Theorem 2 and we follow the previous one. We just need an upper bound for the coefficients ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)}, which is given in the following result.

Proposition 8. Let u ∈ ℕ and ℓ ∈ ℕ^{∗}. For all λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, we have

∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)} = O_{p,ℓ}(p^{(λ′|λ′)}).

Proof. We have

∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)} = ∑_{μ⊆λ} p^{2(∑_{i} μ′_{i+1}(λ′_{i}−μ′_{i}))} ∏_{i} \left[{λ′_{i} − μ′_{i+1} \atop λ′_{i} − μ′_{i}}\right]_{p^{2}}p^{−|μ|(2u−1)}

⩽ p^{|λ|} ∑_{μ⊆λ} p^{2(∑_{i} μ′_{i+1}(λ′_{i}−μ′_{i}))} ∏_{i} \left[{λ′_{i} − μ′_{i+1} \atop λ′_{i} − μ′_{i}}\right]_{p^{2}}.

Using the same method as in the proof of Proposition 6, we obtain

(8)  ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)} = O_{p,ℓ}(p^{(λ′|λ′)/2+|λ|}|λ|^{ℓ}),

which implies the upper bound given in the proposition. □

Proof of Theorem 2. For r := r_{1} ⩾ r_{2} ⩾ ⋯ ⩾ r_{ℓ} ⩾ 0, we denote by 2r the integer partition 2r := 2r_{1} ⩾ 2r_{2} ⩾ ⋯ ⩾ 2r_{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}}(� (E)) = r_{i} for all i = 1, …, ℓ}|.

Note that since � (E) is a group of type S then rk_{p^{j}}(� (E)) must be even. Hence if one of the r_{j}’s is odd, then N(X, r) = 0 for all X. So, for any λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, the assertion of Theorem 2 implies that

(9)  ∑_{r} \frac{N(X, 2r)}{N(X)}p^{(λ′|2r)} = ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)} + o_{λ}(1),

where the sum is over all integer partition r = r_{1} ⩾ ⋯ ⩾ r_{ℓ}. As before, we will prove that the sequence N(X, 2r)/N(X) has only one limit point as X → ∞. We are led to consider the system

(𝒯)  ∑_{r} x_{2r}p^{(λ′|2r)} = ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)} for all λ = 1^{m_{1}} ⋯ ℓ^{m_{ℓ}},

where the unknowns are x_{2r} ⩾ 0. If e_{2r} is a limit point of N(X, 2r)/N(X), then e_{2r} is a solution of (𝒯). We already know that

x_{2r} = \frac{(1/p^{2u+2μ_{ℓ}+1}; 1/p^{2})_{∞}}{p^{2(μ_{1}^{2}+···+μ_{ℓ}^{2})+(2u−1)(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p^{2}; 1/p^{2})_{μ_{j}−μ_{j+1}}}

is a solution (see [DJ12], remark after Theorem 14). If x_{2r} is a solution of (𝒯), then we have

∑_{r} x_{2r}p^{2(λ|r)} = ∑_{μ⊆λ} C_{λ′,μ}(p^{2})p^{−|μ|(2u−1)} for all λ,

where ∑_{μ⊆λ} C_{λ′,μ}(p^{2})p^{−|μ|(2u−1)} = O(p^{(λ|λ)}), from which we deduce

x_{2r} ≪ p^{(r|r)−2(r|r)} ≪ p^{−(r|r)} = (p^{2})^{−(r|r)/2}.

Therefore Corollary 4 with a = p^{2} gives the unicity. □

Corollary 9. Let u be a nonnegative integer, and let ℓ be a positive integer. Assume that for any λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, as E/ℚ, ordered by conductors, is varying over ℱ_{u}, the average of |� (E)[p]|^{m_{1}}|� (E)[p^{2}]|^{m_{2}} ⋯ |� (E)[p^{ℓ}]|^{m_{ℓ}} is equal to ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|(2u−1)}. Then for k ∈ ℕ, as E/ℚ is varying over ℱ_{u}, the probability that rk_{p^{ℓ}}(� (E)) = 2k is equal to

\frac{(1/p^{2u+2k+1}; 1/p^{2})_{∞}}{(1/p^{2}; 1p^{2})_{k}p^{ℓk(2u+2k−1)}} Q_{1/p^{2},ℓ,1}(1/p^{4k+2u−3}).

Proof. We proceed as for the proof of Corollary 7. □

The formula in the above corollary is the u-probability that the p^{ℓ}-rank of a finite abelian p-group of type S is equal to 2k, as obtained in [Del11].

5. Selmer groups of elliptic curves

In the proof of Theorem 2, it is essential to use the fact that � (E) is a group of type S and that rk_{p^{j}}(� (E)) is even, since on the left hand side of (9) the sum involves partitions with only even parts μ_{j}. Nevertheless, one can ask what should be the p^{j}-rank probability laws for other families of groups, if we assume that their moments are given as in Conjecture 3. This question can be naturally asked in particular for Selmer groups of elliptic curves (or more precisely for the p-primary parts of the Selmer groups). If E is an elliptic curve defined over ℚ, then we denote by 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 sequence

0 → E(ℚ) ⊗ ℚ_{p}/ℤ_{p} → S(E) → � (E)[p^{∞}] → 0,

which can be seen as the limit of

0 → E(ℚ)/p^{n}E(ℚ) → S(E)_{p^{n}} → � (E)[p^{n}] → 0.

We assume for simplicity that E(ℚ)_{tors} is trivial: it is not a restriction since we are considering averaging over elliptic curves and on average elliptic curves have trivial rational torsion. The Selmer group S(E) can be an infinite group, nevertheless its subgroup 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 have rk_{p^{j}}(S(E)) = rk_{p^{j}}(S(E)[p^{k}]) for all k ⩾ j. Since � [p^{∞}] is finite by assumption, we have S(E) ≃ (ℚ_{p}/ℤ_{p})^{r(E)}, where r(E) is the rank of the Mordell-Weil group E(ℚ) and

rk_{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_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} is an integer partition, then

|S(E)[p]|^{m_{1}}|S(E)[p^{2}]|^{m_{2}} ⋯ |S(E)[p^{ℓ}]|^{m_{ℓ}}

is meaningful and we can consider the average value of this function as E is varying over a family of elliptic curves. The works of [PR12] suggests that the p-Selmer groups should behave in a “global” way independently of the rank of E (except for the 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_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} be an integer partition. As E/ℚ, ordered by conductors, is varying over all elliptic curves, the average of |S(E)[p]|^{m_{1}}|S(E)[p^{2}]|^{m_{2}} ⋯ |S(E)[p^{ℓ}]|^{m_{ℓ}} is equal to

∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{|μ|}.

If ℓ = 1, this conjecture is originally due to Poonen and Rains in [PR12], where they use a completely different model for Selmer group.

Proposition 10. Let ℓ be a positive integer and let δ ∈ {0, 1}. Assume that for any partition λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}}, the average of |S(E)[p]|^{m_{1}}|S(E)[p^{2}]|^{m_{2}} ⋯ |S(E)[p^{ℓ}]|^{m_{ℓ}} is equal to ∑_{μ⊆λ} C_{λ,μ}(p^{2})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. 1 − α). Then, for all μ_{1} ⩾ μ_{2} ⋯ ⩾ μ_{ℓ}, as E/ℚ is varying over ℱ, the probability that rk_{p^{j}}(S(E)) = 2μ_{j} + δ for all 1 ⩽ j ⩽ ℓ is equal to

(δ(1 − α) + α(1 − δ)) \frac{(1/p^{2δ+2μ_{ℓ}+1}; 1/p^{2})_{∞}}{p^{2(μ_{1}^{2}+···+μ_{ℓ}^{2})+(2δ−1)(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p^{2}; 1/p^{2})_{μ_{j}−μ_{j+1}}} .

Proof. For X ⩾ 1 and r = r_{1} ⩾ r_{2} ⩾ ⋯ ⩾ 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_{1}} ⋯ ℓ^{m_{ℓ}} be an integer partition. Since the rk_{p^{j}}(S(E))’s have all the same parity for j ∈ ℕ, and by the assumptions of the theorem, we have

∑_{r} \frac{N(X, 2r)}{N(X)}p^{(λ′|2r)} = α ∑_{μ⊆λ} C_{λ,μ}p^{|μ|} + o_{λ}(1),

and

∑_{r} \frac{N(X, 2r + 1)}{N(X)}p^{(λ′|2r+1)} = (1 − α) ∑_{μ⊆λ} C_{λ,μ}p^{|μ|} + o_{λ}(1).

For δ ∈ {0, 1}, set

e_{2r+δ} = \frac{(1/p^{2δ+2μ_{ℓ}+1}; 1/p^{2})_{∞}}{p^{2(μ_{1}^{2}+···+μ_{ℓ}^{2})+(2δ−1)(μ_{1}+···+μ_{ℓ})} ∏_{j=1}^{ℓ}(1/p^{2}; 1/p^{2})_{μ_{j}−μ_{j+1}}} .

Thus for δ = 0, we recover e_{2r} which was defined in the previous section, where we already saw that

∑_{r} αe_{2r}p^{(λ′|2r)} = α ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{|μ|},

that αe_{2r} is the only solution of the above system, and that N(X, 2r)/N(X) → e_{2r} as X → ∞. Now, by the same arguments as before, we obtain that there exists an unique solution to the system

(𝒯’) ∑_{r} (x_{2r+1}p^{(λ′|2r+1)}) = (1 − α) ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{|μ|}, for all λ = 1^{m_{1}} ⋯ ℓ^{m_{ℓ}},

where the unknown are x_{2r+1}. Furthermore x_{2r+1} = (1 − α)e_{2r+1} is a solution of (𝒯’). Indeed, by [DJ12, Remark after Theorem 14]

∑_{r} e_{2r+1}p^{(λ′|2r+1)} = p^{|λ′|} ∑_{r} e_{2r+1}p^{(λ′|2r)}
= p^{|λ|} ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|},

and moreover by [DJ12, Theorem 1], we have

p^{|λ|} ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{−|μ|} = ∑_{μ⊆λ} C_{λ,μ}(p^{2})p^{|μ|}.

Finally N(X, 2r + 1)/N(X) → e_{2r+1} as X → ∞. □

Now, adapting the proof of Corollary 7, we have the following result.

Corollary 11. Let ℓ be a positive integer and set δ ∈ {0, 1}. Assume that for every λ = 1^{m_{1}}2^{m_{2}} ⋯ ℓ^{m_{ℓ}} the average of |S(E)[p]|^{m_{1}}|S(E)[p^{2}]|^{m_{2}} ⋯ |S(E)[p^{ℓ}]|^{m_{ℓ}} is equal to ∑_{μ⊆λ} C_{λ,μ}(p^{2})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. 1 − α). Then, for k ∈ ℕ, the probability that rk_{p^{ℓ}}(S(E)) = 2k + δ is equal to

(δ(1 − α) + α(1 − δ)) \frac{(1/p^{2k+2δ+1}; 1/p^{2})_{∞}}{(1/p^{2}; 1/p^{2})_{k}p^{ℓk(2k+2δ−1)}} Q_{1/p^{2},ℓ,1}(1/p^{4k+2δ−3}).

The value of α can be of course ≠ 1/2. Furthermore, even in the case of a family of quadratic twists of an elliptic curve E defined over a number field K, it is possible to have α ≠ 1/2 (see [KMR13a], in that case we have K ≠ ℚ).

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 ∈ {0, 1}. Then, as E/ℚ, ordered by conductor, is varying over all elliptic curves, the probability that rk_{p^{ℓ}}(S(E)) = 2k + δ is equal to

f(p, ℓ, 2k + δ) := \frac{1}{2} \frac{(1/p^{2k+2δ+1}; 1/p^{2})_{∞}}{(1/p^{2}; 1/p^{2})_{k}p^{ℓk(2k+2δ−1)}} Q_{1/p^{2},ℓ,1}(1/p^{4k+2δ−3}).

For ℓ = 1, we recover the conjectural distribution X_{Sel_{p}} of [PR12] and the proved distribution of Sel_{2} in [HB93, HB94, Kan11, SD08, KMR13b] for some families of quadratic twists of an elliptic curve.

The conjectural distribution on the p^{j}-rank of the Selmer groups given above is of course compatible with the rank conjecture. Indeed, note that we have Q_{q,∞,1}(x) = 1/(xq^{2}; q)_{∞} from which we easily deduce that

lim_{ℓ→∞} f(p, ℓ, 2k + δ) = \{ \begin{matrix}0 & if k ⩾ 1, \\ 1/2 & if k = 0. \end{matrix}

Since for ℓ large enough, rk_{p^{ℓ}}(S(E)) = r(E), we recover the fact that on average half 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, 1, 2k + δ) = \{ \begin{matrix}0 & if k ⩾ 1, \\ 1/2 & if k = 0. \end{matrix}

We give some numerical approximations for the function f(p, ℓ, 2k+δ) for p = 2, 3, 5 and for small values of ℓ and of 2k + δ in the following tables.

\begin{matrix}\begin{matrix}2k + δ\ℓ & 1 & 2 & 3 \\ 0 & 0.2097 & 0.3541 & 0.4271 \\ 1 & 0.4194 & 0.4899 & 0.4987 \\ 2 & 0.2796 & 0.1456 & 0.0729 \\ 3 & 0.0798 & 0.1009 & 0.0012 \end{matrix} & \begin{matrix}2k + δ\ℓ & 1 & 2 & 3 \\ 0 & 0.3195 & 0.4398 & 0.4799 \\ 1 & 0.4792 & 0.4992 & 0.4999 \\ 2 & 0.1797 & 0.0601 & 0.0201 \\ 3 & 0.0207 & 0.0007 & 2 · 10^{−5} \end{matrix} \\ p = 2 & p = 3 \end{matrix}

\begin{matrix}2k + δ\ℓ & 1 & 2 & 3 \\ 0 & 0.3966 & 0.4793 & 0.4959 \\ 1 & 0.4958 & 0.4999 & 0.4999 \\ 2 & 0.1033 & 0.0207 & 0.0041 \\ 3 & 0.0042 & 3 · 10^{−5} & 2 · 10^{−7} \end{matrix} ; p = 5

6. Remark on the uniqueness of the solution

In our study related to Tate-Shafarevich group, we were led to consider and to discuss the unicity of the solution of the following infinite multi-dimensional system

(𝒰)  ∑_{r} x_{r}p^{(λ|r)} = ∑_{μ⊆λ′} C_{λ′,μ}(p^{2})p^{−|μ|(2u−1)} for all λ = 1^{m_{1}} ⋯ ℓ^{m_{ℓ}},

where the unknowns are x_{r} ⩾ 0. We only considered solution (x_{r})_{r} such that x_{r} = 0 if in r = r_{1} ⩾ r_{2} ⩾ ⋯ ⩾ r_{ℓ} at least one of the r_{j}’s has not the same parity as r_{1}. In that case, the term p^{(λ|r)} involved in the sum is of the form p^{(λ|2r+δ)}, and the factor 2 allowed to have an asymptotic 0 ⩽ x_{2r+δ} ≪ p^{(−r|r)/2} which implied the unicity 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 same parity.

Set μ ∈ ℝ, and for a partition r we define

y_{r}(μ) = \{ \begin{matrix}μe_{r} & if r is even, \\ (1 − μ)e_{r} & if r is odd, \\ 0 & otherwise. \end{matrix}

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 have

0 ⩽ x_{r} ≪ p^{−(r|r)/2+α|r|}

for all r. Now, if we let w_{r} = x_{r} − y_{r}(x_{0,0,...,0}) then it is easy to see that we also have

|w_{r}| ≪ p^{−(r|r)/2+α|r|}

(w_{r} is not necessarily nonnegative), and the function g(\underline{z}) = ∑_{r} w_{r}z_{1}^{r_{1}} ⋯ z_{ℓ}^{r_{ℓ}} satisfies the hypothesis of Corollary 4 with α ∈]1, 3/2[ and with w_{0,0,...,0} = 0. Hence w_{r} = 0 for all r and x_{r} = y_{r}(x_{0,0,...,0}). We deduce the following proposition.

Proposition 12. If 0 ⩽ x_{r} is a solution of equation (𝒰) then x_{r} = y_{r}(μ) for some μ. In particular, x_{r} = 0 if r is not an even nor an odd partition.

It would be interesting to study the (unicity of the) solutions of equation (𝒰) if we do not assume that x_{r} ⩾ 0.

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Université de Franche-Comté, Laboratoire de Mathématiques de Besançon, CNRS UMR UMR 6623, Facultés des Sciences et Techniques, 16 route de Gray, 25030 Besançon, France.

   E-mail address: christophe.delaunay@univ-fcomte.fr

Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France.

   E-mail address: jouhet@math.univ-lyon1.fr