Reductions of points on algebraic groups, II

Let $A$ be the product of an abelian variety and a torus over a number field $K$, and let $m$ be a positive integer. If $\alpha \in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that the reduction $(\alpha \bmod \mathfrak p)$ is well defined and has order coprime to $m$. This set admits a natural density, which we are able to express as a finite sum of products of $\ell$-adic integrals, where $\ell$ varies in the set of prime divisors of $m$. We deduce that the density is a rational number, whose denominator is bounded (up to powers of $m$) in a very strong sense. This extends the results of the paper"Reductions of points on algebraic groups"by Davide Lombardo and the second author, where the case $m$ prime is established.


INTRODUCTION
This article is the continuation of the paper Reductions of points on algebraic groups by Davide Lombardo and the second author [4]. We refer to this other work for the history of the problem and further references.
Let A be the product of an abelian variety and a torus over a number field K, and let m be a positive square-free integer. If α ∈ A(K) is a point of infinite order, we consider the set of primes p of K such that the reduction (α mod p) is well defined and has order coprime to m. This set admits a natural density (see Theorem 7), which we denote by Dens m (α).
The main question is whether we can write We are able to express Dens m (α) as an integral over the image of the m-adic representation (see Theorem 17), and also as a finite sum of products of ℓ-adic integrals (see Theorems 20 and 21). The latter decomposition allows us to prove that Dens m (α) is a rational number whose denominator is uniformly bounded in a very strong sense (see Corollary 22).
Finally, we study Serre curves in detail in Section 6. With our results one can explicitly compute Dens m (α) if the m n -Kummer extensions of α (defined in Section 3) have maximal degree for all n or, more generally, if the degrees of these extensions are known and are the same with respect to the base fields K and K(A[m]).

INTEGRATION ON PROFINITE GROUPS
For every compact topological group G, we write µ G for the normalised Haar measure on G. 1 Let G be a profinite group, and let H be a closed subgroup of finite index in G. For every integrable function f : H → C, we define (2) I H,f := H f dµ H .
Let ℓ vary in a finite set of prime numbers, and suppose that we have G = ℓ G ℓ , where each G ℓ is a profinite group containing a pro-ℓ-group G ′ ℓ as a closed subgroup of finite index. Note that we may assume that G ′ ℓ is normal in G ℓ (up to replacing G ′ ℓ by the intersection of its finitely many conjugates in G ℓ ).
The profinite group G ′ := ℓ G ′ ℓ has finite index in G, and the profinite group is a closed subgroup of finite index in each of G, H and G ′ . Because the G ′ ℓ are pro-ℓ-groups for pairwise different ℓ, every closed subgroup of G ′ is similarly a product of pro-ℓ-groups. We can therefore write where the f x,ℓ : H ℓ (x) → C are locally constant functions that are integrable with respect to µ H ℓ (x) . We then have Proof. The assertion follows from (4) because we have a product decomposition for f and because µ H(x) is the product of the µ H ℓ (x) .
Proof. The assertion follows from the product decomposition for f and the fact that µ H is the product of the µ H ℓ .

THE ARBOREAL REPRESENTATION
Let K be a number field, and let K be an algebraic closure of K. Let A be a connected commutative algebraic group over K, and let b A be the first Betti number of A. We fix a square-free positive integer m, and we let ℓ vary in the set of prime divisors of m. We also fix a point α ∈ A(K).
We define T m A as the projective limit of the torsion groups A[m n ] for n 1; we can write We define the torsion fields The Galois action on the m-power torsion points of A gives the m-adic representation of A, which maps Gal(K/K) to the automorphism group of T m A. We can also speak of the mod m n representation, which describes the Galois action on A[m n ]. Choosing a Z ℓ -basis for T ℓ A, we can identify the image of the m-adic representation with a subgroup of ℓ GL b A (Z ℓ ) and the image of the mod m n representation with a subgroup of ℓ GL b A (Z/ℓ n Z). For n 1, let m −n α be the set of points in A(K) whose m n -th multiple equals α. We also write This is the set of sequences β = {β n } n 1 such that mβ 1 = α and mβ n+1 = β n for every n 1; it is a torsor under T m A. We note that m −n 0 = A[m n ] and m −∞ 0 = T m A. We define the fields K m −n α := K(m −n α) for n 1 We call the field extension K m −n α /K m −n the m n -Kummer extension defined by the point α. We view the m-adic representation as a representation of Gal(K m −∞ α /K).
We fix an element β ∈ m −∞ α, and define the arboreal representation where M is the image of σ under the m-adic representation and t = σ(β)−β. The arboreal representation is an injective homomorphism of profinite groups identifying Gal( the image of the m-adic representation and by G(m n ) the image of the mod m n representation. Consider the image of the ℓ-adic representation in GL b A (Z ℓ ): the dimension of its Zariski closure in GL b A ,Q ℓ is independent of ℓ and K, and we denote it by d A . For example, if A is an elliptic curve, we have d A = 2 if A has complex multiplication, and d A = 4 otherwise.
Definition 3. We say that (A/K, m) satisfies eventual maximal growth of the torsion fields if there exists a positive integer n 0 such that for all N n n 0 we have We say that (A/K, m, α) satisfies eventual maximal growth of the Kummer extensions if there exists a positive integer n 0 such that for all N n n 0 we have Remark 4. Condition (9) means that there is eventual maximal growth of the torsion fields, that K m −n α and K m −N are linearly disjoint over K m −n , and that we have If there is eventual maximal growth of the Kummer extensions, the rational number is independent of n for n n 0 . Proof. By [4, Lemma 10], if A is a semiabelian variety and ℓ is a prime divisor of m, then (A/K, ℓ, α) satisfies eventual maximal growth of the torsion fields. We also know that the degree [K ℓ −n : K ℓ −1 ] is a power of ℓ. Therefore the extensions K m −1 K ℓ −n for ℓ | n are linearly disjoint over K m −1 and the first assertion follows. By [4,Remark 7], the second assertion holds for (A/K, ℓ, α), where ℓ is any prime divisor of m. We conclude because these Kummer extensions have order a power of ℓ. Remark 6. This is not a restriction if A is the product of an abelian variety and a torus by Proposition 5. Indeed, consider the number of connected components of the Zariski closure of Zα. If this number is not coprime to m, then the density Dens m (α) is zero by [5,Main Theorem] while if it is coprime to m we may replace α by a multiple to reduce to the case where the Zariski closure of Zα is connected. Finally, we may replace A and reduce to the case where Zα is Zariski dense.
The T m A-torsor m −∞ α from Section 3 defines a Galois cohomology class For any choice of β ∈ m −∞ α, this is the class of the cocycle We also consider the restriction map with respect to the cyclic subgroup generated by some element σ: Proof. Similarly to [4, Remark 19], we may equivalently consider S α as a subset of either Gal(K/K) or Gal(K m −∞ α /K) with their respective normalised Haar measures.
Proof. The generalisation of [4,Proposition 20] to the composite case is straightforward.

4.2.
Counting elements in the image of the arboreal representation.
For every prime divisor ℓ of m and every n 1, we consider the Galois group of the compositum K ℓ −n α K m −1 over K and the inclusion Definition 10. For all x ∈ G(m) and V ∈ G(ℓ n ), we define We denote by π * the projection onto G( * ).
We have Im(M − I) = ℓ Im(π ℓ n M − I), and since the extensions K ℓ −n α K m −1 /K m −1 have pairwise coprime degrees and hence are linearly disjoint, we have t ∈ W m n (M ) if and only if for every ℓ we have t ℓ ∈ W x,ℓ n (π ℓ n M ). This implies the claim.
By Lemma 12, we can define From (15) we deduce that for all M ∈ G(m ∞ ), the value w m n (M ) is also constant for n sufficiently large, so we can analogously define Proof. Taking the limit as n → ∞ in (15) yields (18). For every group F of the form ℓ F ℓ , where F ℓ is a finite abelian ℓ-group with at most b A cyclic components, we define the sets We denote by M F ( * ) and M x,F ( * ), respectively, the images of these sets under the reduction map G(m ∞ ) → G( * ). We also write the union being taken over all x ∈ G(m) and over all groups F = ℓ F ℓ as above, up to isomorphism.
Proposition 14. The following holds: satisfies eventual maximal growth of the torsion fields, then we have Proof. This is proved as in [4, Lemma 21].

THE DENSITY AS AN INTEGRAL
By Theorem 7, computing Dens m (α) comes down to computing the Haar measure of S α in Gal(K m −∞ α /K). The generalisation of [4,Remark 17] to the composite case gives In view of (21), we consider the sets By Proposition 14, the set S α is the disjoint union of the sets S x,F up to a set of measure 0. To see that the Haar measure of S x,F is well defined and to compute it, we define for every n 1 the set Theorem 16. We have where C m is the constant of (10) and w m ∞ is as in (17).
Proof. Choose n large enough so that n > n 0 and n > max ℓ {v ℓ (exp F)} for every ℓ, where n 0 is as in Definition 3. By (11), we can write By (9) the left-hand side is a non-increasing function of n, and therefore it admits a limit for n → ∞, which is µ(S x,F ). The right-hand side is an integral over M x,F (m n ) with respect to the normalized counting measure of G(m n ), and the matrices in M x,F are exactly the matrices in G(m ∞ ) whose reduction modulo m n lies in M x,F (m n ). Taking the limit in n we thus find the formula in the statement.
Theorem 17. We have where the function w m ∞ is as in (17), and the constant C m is as in (10).
Recalling the sets M F from (19), and that by Proposition 14 their union has measure 1 in G(m ∞ ), we can rewrite (25) as where F varies over the products over the primes ℓ | m of finite abelian ℓ-groups with at most b A cyclic components.

Proof.
We have just shown that we may prove (26) instead. Notice that we have M F = ∪ x M x,F . By Theorem 7 we may write Dens m (α) = µ(S α ) = x,F µ(S x,F ) and then it suffices to apply (24).
Corollary 18 ([4, Theorem 1 and Remark 25]). In the special case m = ℓ we have where F varies among the finite abelian ℓ-groups with at most b A cyclic components. We can also write Proof. It suffices to specialize the notation of (25) and (26)  Proof. Let ℓ vary among the prime divisors of m. Note that we have C m = ℓ C ℓ . By assumption, we also know G(m ∞ ) = ℓ G(ℓ ∞ ) and w m ∞ (M ) = ℓ w ℓ ∞ (π ℓ ∞ M ). We may then apply Proposition 2 and find that (25) is the product of the expressions (28).
are translates of closed subgroups of finite index. Suppose that for every ℓ and for every n 1 Proof. We can apply Proposition 2 because there is a product decomposition formula for the function w m ∞ . Namely the assumption on the Kummer extensions implies that for every n 1 we have w m n (M ) = ℓ w ℓ n (π ℓ n M ) and hence we find w m ∞ (M ) = ℓ w ℓ ∞ (π ℓ ∞ (M )).
Proof. Write S x = F S x,F and recall from Proposition 14 that the set of matrices M for which ker(M − I) is infinite has measure zero in G(m ∞ ). By (24) we have By (18) the assertion follows from Theorem 1.
Corollary 22. The density Dens m (α) is a rational number. Moreover, fix g 1. There exists a polynomial p g (t) ∈ Z[t] with the following property: whenever K is a number field and A is the product of an abelian variety and a torus with dim(A) = g, then for all α ∈ A(K) we have where ℓ varies over the prime divisors of m.
Proof. Recall that C m is an integer. In view of Lemma 12, we can consider each ℓ-adic integral in (31) and proceed as in the proof of [4, Theorem 33].
6. SERRE CURVES 6.1. Definition of Serre curves. Let E be an elliptic curve over a number field K. We choose a Weierstrass equation for E of the form ) are the x-coordinates of the points of order 2. The discriminant of the right-hand side of (33 We thus have K( √ ∆) ⊆ K(E [2]), and we define a character Let ψ be the unique non-trivial character GL 2 (Z/2Z) → {±1}; this corresponds to the sign character under any isomorphism of GL 2 (Z/2Z) with S 3 . The character ψ E factors as From now on, we take K = Q. All number fields that we will consider will be subfields of a fixed algebraic closure Q of Q.
Let d be an element of Q × . Let m d be the conductor of Q( √ d); this is the smallest positive integer such that √ d lies in the cyclotomic field Q(ζ m d ). Let d sf be the square-free part of d. We have We define a character We view ε d as a character of GL 2 (Z/m d Z) by composing with the determinant.
Fixing a Z-basis for the projective limit of the torsion groups E[n](Q), we have a torsion representation The image of ρ E is contained in the subgroup . This expresses the fact that √ ∆ is contained in both Q(E [2]) and Q(E[m ∆ ]). An elliptic curve is said to be a Serre curve if the image of ρ E is equal to H ∆ . As proven by N. Jones [1], almost all elliptic curves over Q are Serre curves. 6.2. Counting matrices. Let ℓ be a prime number. For all integers a, b 0, we write M ℓ (a, b) for the set of matrices M ∈ GL 2 (Z ℓ ) such that the kernel of M − I as an endomorphism of (Q ℓ /Z ℓ ) 2 is isomorphic to Z/ℓ a Z × Z/ℓ a+b Z.
If N is a subset of M ℓ (a, b) that is the preimage in M ℓ (a, b) of its reduction modulo ℓ n , then we have by [3,Theorem 27] (where the number of lifts is independent of the matrix). N is a subset of M ℓ (a, b) that is the preimage in M ℓ (a, b) of its reduction modulo ℓ, then we have

Proposition 23. If
Proof. We are working with GL 2 (Z ℓ ), so we can apply [3,Proposition 33] (see also [3,Definition 19]). This gives the assertion for the set M ℓ (a, b); we can conclude because of (34).
We now collect some results in the case ℓ = 2. From [3, Theorem 2] we know We consider the action of  N 2 (a, b, z) for the set of matrices in M 2 (a, b) that fix √ z.
Lemma 24. We have More precisely, we want β to be a square modulo ℓ, and in this case we write β = s 2 . On the other hand we would like to know whether ℓ | det(M − I) = χ(1), which means 1 + α + β = 0 (we are working modulo ℓ). Thus α = −(1 + s 2 ). We then have Proof.
(3) There are ℓ 3 −2ℓ matrices in GL 2 (Z/ℓZ) having 1 as an eigenvalue (see for example [3, Proof of Theorem 2]), and we only need to subtract the identity and the matrices from (1). (4) There are 1 2 # GL 2 (Z/ℓZ) matrices satisfying ε ℓ = −1, and we only need to subtract the matrices from (3). Alternatively, there are # GL 2 (Z/ℓZ) − (ℓ 3 − 2ℓ) matrices that do not have 1 as eigenvalue, and we only need to subtract the matrices from (2). 6.3. Partitioning the image of the m-adic representation. Let E be a Serre curve over Q. Let ∆ be the minimal discriminant of E, and let ∆ sf be its square-free part. We write ∆ sf = zu, where z ∈ {1, −1, 2, −2} and where u is an odd fundamental discriminant. Then |u| is the odd part of m ∆ , and we have ε ∆ = ε z · ε u as characters of (Z/m ∆ Z) × . Now let m be a square-free positive integer. If m = 2, or if m is odd, or if u does not divide m, If m = 2 is even and u divides m, then G(m ∞ ) has index 2 in ℓ G(ℓ ∞ ). The defining condition for the image of the m-adic representation is then ψ = ε ∆ , or equivalently We may then partition G(m ∞ ) ⊆ ℓ|m G(ℓ ∞ ) into two sets that are products, namely The image of the 3-adic representation is the inverse image of its reduction modulo 3, the image of the mod 3 representation is isomorphic to the symmetric group of order 6, and the 3-adic Kummer map is surjective [4,Example 6.4]. The image of the mod 3 representation has a unique subgroup of index 2, so the field Q(E [3]) contains as its only quadratic subextension the cyclotomic field Q( √ −3). The image of the 2-adic representation is GL 2 (Z 2 ); see [8]. By [2, Theorem 5.2] the 2-adic Kummer map is surjective: the assumptions of that result are satisfied because the prime p = 941 splits completely in E [4] but the point (α mod p) is not 2-divisible over F p . Since the image of the mod 2 representation has a unique subgroup of index 2, the field Q(E [2]) contains as its only quadratic subextension the field Q( √ −51) (the square-free part of the discriminant of E is −51). We thus have G(6 ∞ ) = G(2 ∞ ) × G(3 ∞ ), the 2 ∞ Kummer extensions are independent from Q(E [3]), and the 3 ∞ Kummer extensions are independent from Q(E [2]). We are thus in the situation that the fields Q(2 −∞ α) and Q(3 −∞ α) are linearly disjoint over Q. We deduce that the equality Dens 6 (α) = Dens 2 (α) · Dens 3 (α) holds for α and for its multiples. The 2-densities can be evaluated by [4,Theorem 32], for the 3densities see [4,Example 6.4]. By testing the primes up to 10 6 , we have computed an approximation to Dens 6 (α) using [9]. ), as can be seen by investigating the residual degree for the reduction modulo the prime 29327, which splits completely in Q(E [2]). Indeed, the residual degree of the extension Q(2 −1 α) equals 4 while the residual degree of the extension Q(E[2 · 43]) is odd because the prime is congruent to 1 modulo 43, and there are points of order 43 in the reductions (the subgroup of the upper unitriangular matrices in GL 2 (Z/43Z) has order 43). To see this, we consider the intersection L of Q 2 −n α and Q (E[43]). This is a Galois extension of Q, and the group G = Gal(L/Q) is a quotient of both (Z/2 n Z) 2 ⋊ GL 2 (Z/2 n Z) and GL 2 (Z/43Z). Because SL 2 (Z/43Z) has no non-trivial quotient that can be embedded into a quotient of (Z/2 n Z) 2 ⋊ GL 2 (Z/2 n Z), the quotient map GL 2 (Z/43Z) → G factors as This implies that L is a subfield of Q(ζ 43 ). Furthermore, L contains Q( √ −43). Using basic Galois theory, we see that the maximal subfield of Q(ζ 43 ) that can be embedded into Q 2 −n α is Q( √ −43), and we conclude that L equals Q( √ −43). It follows that for m = 2 · 43 we have the maximal degree [Q m −n α : Q m −n ] = m 2n and, more generally, that for every multiple P of α we have [Q m −n P : Q m −n ] = [Q 2 −n P : Q 2 −n ] · [Q 43 −n P : Q 43 −n ]. We may then apply [4, Example 26] and various results in this paper to compute the following exact densities, and we use [9] to numerically verify them for the primes up to Now we work with the 2-adic representation, which is surjective, and restrict to counting the contribution to Dens 2 (α) coming from the matrices satisfying ψ = −1. In view of Lemma 25 and Proposition 23, we find µ GL 2 (Z 2 ) (M 2 (0, b)) = 1/2 · 2 −b for b > 0. By [4,Example 26] the contribution to Dens 2 (α) coming from the matrices in G(2 ∞ ) such that ψ = −1 is therefore (40) Dens 2 (ψ = −1) = b>0 1/2 · 2 −2b = 1/6.