NATHAN JONES. Taking the inverse limit over all n 1 (ordered by divisibility), one may consider the action of G K on. n=1

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1 GL 2 -REPRESENTATIONS WITH MAXIMAL IMAGE NATHAN JONES Abstract. For a matrix group G, consider a Gaois representation ϕ: Ga(Q/Q) G(Ẑ) which extends the cycotomic character. For a broad cass of matrix groups G, we prove a theorem characterizing when such a representation has image which is as arge as possibe inside a fixed open subgroup G G(Ẑ). As appications, we obtain such a characterization for the Gaois representation on the torsion of a simpe principay poarized k-dimensiona abeian variety A defined over Q (where G = GSp 2k ) and aso for the Gaois representation on the torsion of a product of k eiptic curves over Q (where G = {(g 1,..., g k ) GL k 2 : det g 1 = = det g k }). Our resuts are motivated by open image theorems for casses of abeian varieties initiated by Serre in the 1960s. 1. Introduction Let K be a number fied and et E be an eiptic curve defined over K. Consider the action of G K := Ga(K/K) on the n-torsion E[n] of E, which gives rise to a Gaois representation ϕ EK,n : G K Aut(E[n]) GL 2 (Z/nZ). Taking the inverse imit over a n 1 (ordered by divisibiity), one may consider the action of G K on E tors := E[n], obtaining a continuous homomorphism n=1 ϕ EK : G K Aut(E tors ) GL 2 (Ẑ), where GL 2 (Ẑ) = im GL 2(Z/nZ). As discussed in [14], it is of interest to understand the image of ϕ EK. For exampe, if K = Q, the image ϕ E (G Q ) 1 pays a crucia roe in a conjecture of Lang and Trotter which counts primes with fixed Frobenius trace. (This conjecture is sti open, athough average versions (see [5] and [4]) have been obtained.) Serre [18] proved that, when E has no compex mutipication, the image of ϕ EK is open in GL 2 (Ẑ) (i.e. has finite index in GL 2 (Ẑ)). He aso noted that ϕ E K can never be surjective when K = Q, because of the foowing cycotomic obstruction. Let us introduce the notation K(E[n]) := K ker ϕ E K,n = K(the x and y-coordinates of a P E[n]), K(E tors ) := K ker ϕ E K = K(the x and y-coordinates of a P E tors ). Serre noticed that if K = Q, then for some positive integer d = d E 1 we have Q( E ) Q(E[2]) Q(µ d ) Q(E[2]) Q(E[d]), where E denotes the discriminant of any Weierstrass mode of E and Q(µ d ) denotes the d-th cycotomic fied. By Gaois theory, this forces ϕ E (G Q ) GL 2 (Ẑ). More precisey, it impies that } (1) ϕ E (G Q ) GL 2 (Ẑ) χ=ε := {g GL 2 (Ẑ) : χ(g) = ε(g), 1 When K = Q, we wi denote the Gaois representation on Etors (resp. on E[n]) simpy by ϕ E (resp. by ϕ E,n ). 1

2 where the signature (2) ε: GL 2 (Ẑ) GL 2(Z/2Z) GL 2 (Z/2Z)/[GL 2 (Z/2Z), GL 2 (Z/2Z)] {±1} is the unique non-trivia character of GL 2 (Z/2Z) pre-composed with reduction moduo 2 and χ: GL 2 (Ẑ) Ẑ {±1} ( ) ( ) E E is defined by χ(g) =, with the Kronecker symbo, i.e. the unique character which satisfies det g Frob p ( ( ) E E E ) = for each prime p not dividing d. Frob p Serre aso gave exampes of eiptic curves E over Q for which ϕ E has image as arge as possibe moduo this obstruction, i.e. for which (3) ϕ E (G Q ) = GL 2 (Ẑ) χ=ε. Foowing Lang and Trotter, we ca an eiptic curve E over Q a Serre curve if (3) hods. One has (4) E is a Serre curve [GL 2 (Ẑ) : ϕ E(G Q )] = 2. It has been shown (see [10], [3]) that amost a eiptic curves E over Q are Serre curves (see aso [15], which, buiding on [6], gives an asymptotic formua for the number of Serre curves of bounded height). In the present paper, we consider the foowing generaization of the above. Fix once and for a a eve m 1 and a subgroup G(m) GL 2 (Z/mZ), and et (5) G = π 1 (G(m)) GL 2 (Ẑ) be the entire pre-image of G(m) under the canonica projection. Suppose that we have found an eiptic curve E over Q for which the associated Gaois representation maps into G: (6) ϕ E : G Q G. Our goa is to describe the anaogue of (1) in this context, thus making precise the notion that ϕ E has image as arge as possibe, given that it ands in G and aowing us to define the reative concept of a G-Serre curve; this is Definition 2.9 beow. We wi then prove a theorem (see Theorem 2.6) which characterizes G-Serre curves in terms of expicit criteria at finite eve. There is a moduar curve X G(m) whose non-cuspida Q-rationa points correspond to j-invariants of eiptic curves E over Q for which (6) hods (up to GL 2 (Ẑ)-conjugation). When the genus of X G(m) is zero and X G(m) (Q), one may fix a morphism f : A 1 X G(m), and count the rationa points t 0 Q for which f(t 0 ) corresponds to a G-Serre curve. One may use the same techniques as in [3] to prove that amost a eiptic curves in a one-parameter famiy on X G(m) are G-Serre curves (see Theorem 2.11). As another appication, in [11] we use these resuts to prove that, when ordered by height, amost a k-tupes of eiptic curves over Q have division fieds with composita as arge as possibe. Finay, we remark that the techniques used in the current paper are excusivey group-theoretica. Consequenty, our resuts are appicabe in a wider context than we have stated here; see Theorem 2.18 in Section 2.1. For instance, in Coroary 2.20, we appy Theorem 2.18 to the Gaois representation on the torsion of a simpe principay poarized k-dimensiona abeian variety. 2. Statement of resuts To motivate our definitions, we begin by re-examining (1) in more detai. Let E be an eiptic curve over Q without compex mutipication and et E denote the discriminant of any Weierstrass mode of E. Reca that, thanks to the Wei pairing (see [20]), for any d 1 the d-th cycotomic fied Q(µ d ) is contained 2

3 in Q(E[d]). Furthermore, choosing a Z/dZ-basis of E[d] (and thus an imbedding ι: Ga(Q(E[d])/Q) GL 2 (Z/dZ)), the foowing diagram is commutative: (7) res Ga(Q(E[d])/Q) Ga(Q(µ d )/Q) ι GL 2 (Z/dZ) det (Z/dZ), where the unabeed vertica arrow is the usua (canonica) isomorphism. Since we see (assuming ϕ E,2 (G Q ) = GL 2 (Z/2Z)) that [GL 2 (Z/2Z), GL 2 (Z/2Z)] = (ker ε (mod 2)) SL 2 (Z/2Z), Q = Q(µ 2 ) = Q(E[2]) SL2(Z/2Z) Q(E[2]) [GL2(Z/2Z),GL2(Z/2Z)] = Q( E ). Since Q( E ) is an abeian extension of Q, it is contained in some Q(µ d ), by the Kronecker-Weber Theorem. This containment, together with (7), impies (1). This may be re-cast as foows. We wi denote by Q cyc := d 1 Q(µ d ) the cycotomic cosure of Q and by Q ab the maxima abeian extension of Q. By the Kronecker-Weber theorem, one has Q cyc = Q ab. Let us denote the Gaois representation ϕ E simpy by ϕ. The commutator subgroup 2 [ϕ(g Q ), ϕ(g Q )] satisfies (8) [ϕ(g Q ), ϕ(g Q )] ϕ(g Q ) SL 2 (Ẑ). Furthermore, by Gaois theory (and since Q cyc Q(E tors )) we have Q cyc = Q ab = Q(E tors ) [ϕ(g Q),ϕ(G Q )] Q(E tors ) ϕ(g Q) SL 2(Ẑ) = Q cyc, which impies that we must have equaity in (8): This motivates the foowing definitions. [ϕ(g Q ), ϕ(g Q )] = ϕ(g Q ) SL 2 (Ẑ). Definition 2.1. A subgroup H GL 2 (Ẑ) is caed commutator-thick if [H, H] = H SL 2 (Ẑ). Definition 2.2. A subgroup H GL 2 (Ẑ) is caed determinant-surjective if det(h) = Ẑ. Remark 2.3. The above discussion shows that ϕ E (G Q ) GL 2 (Ẑ) is aways a commutator-thick, determinantsurjective subgroup. As in the previous section, assume now that G GL 2 (Ẑ) is a fixed subgroup of finite index, and that ϕ E (G Q ) G. The foowing definition captures the notion that ϕ E (G Q ) is as arge as possibe, given that it s a subset of G. Definition 2.4. We ca a commutator-thick subgroup H G a G-maxima commutator-thick subgroup of G if [H, H] = [G, G]. Remark 2.5. Suppose that H G is determinant-surjective and commutator-thick. sequence (9) 1 H SL 2 (Ẑ) H Ẑ 1, Noting the exact and that the kerne H SL 2 (Ẑ) = [H, H] [G, G], it foows that, if H is a G-maxima commutator-thick subgroup of G in the sense of Definition 2.4, then H is aso maxima (with respect to subset incusion) 2 For a profinite group H, we are defining the commutator subgroup [H, H] to be the cosure of the subgroup generated by its set of commutators {xyx 1 y 1 : x, y H}. 3

4 among the determinant-surjective commutator-thick subgroups of G. Furthermore, it foows from (9) and the corresponding sequence with G repacing H that [ ] H SL 2 (Ẑ) = [G, G] [G : H] = G SL 2 (Ẑ) : [G, G]. Thus, Definition 2.4 is equivaent to H having minima index in G among the determinant-surjective commutator-thick subgroups of G. Our main resut gives the foowing theorem, which expicity characterizes G-maxima commutator-thick subgroups of G. Reca the set-up: m 1 is any positive integer, G(m) GL 2 (Z/mZ) is an arbitrary subgroup, and G := π 1 (G(m)) GL 2 (Ẑ) is the fu pre-image of G(m) under the natura projection. Define the positive integer m 0 by (10) m 0 := cm , prime m 2ord (m)+1 Theorem 2.6. With m 1 and G GL 2 (Ẑ) as in (5), et m 0 be defined by (10). Let H G be any subgroup. Then [H, H] = [G, G] if and ony if (11) [H (mod n), H (mod n)] = [G (mod n), G (mod n)] ( n {m 0 } { prime : m 0 }). In particuar, if H is commutator-thick, then H is a G-maxima commutator-thick subgroup if and ony if (11) hods. We remark that, since for any prime 5, [SL 2 (Z/Z), SL 2 (Z/Z)] = SL 2 (Z/Z) (this foows because PSL 2 (Z/Z) is simpe for 5), Theorem 2.6 is equivaent to the foowing theorem. Theorem 2.7. With m 1 and G GL 2 (Ẑ) as in (5), et m 0 be defined by (10). Let H G be any subgroup. Then [H, H] = [G, G] if and ony if the foowing two conditions hod. (1) For each prime not dividing m 0, one has SL 2 (Z/Z) H (mod ). (2) One has [H (mod m 0 ), H (mod m 0 )] = [G (mod m 0 ), G (mod m 0 )]. In particuar, if H is commutator-thick, then H is a G-maxima commutator-thick subgroup if and ony if conditions (1) and (2) above hod. In appications, it is often usefu to have the eve m 0 given expicity in terms of m, which is part of the motivation for our theorem. Remark 2.8. If each prime which divides m satisfies / ± 1 (mod 5), then Theorems 2.6 and 2.7 hod with (12) m 0 = cm , prime m 2ord (m)+1 Returning to our origina exampe of an eiptic curve, we make the foowing definition. Definition 2.9. Let G GL 2 (Ẑ) be an open subgroup. An eiptic curve E over Q which satisfies ϕ E(G Q ) G is caed a G-Serre curve if ϕ E (G Q ) is a G-maxima commutator-thick subgroup. If the group G is understood, one may refer to a G-Serre curve simpy as a reative Serre curve. As an immediate coroary of Theorem 2.6, we wi give a characterization of G-Serre curves. For any positive integer n 1 denote by G(n) the reduction of G moduo n, viewed as a subgroup of GL 2 (Z/nZ), and define the foowing set of subgroups of G(n): Note that, for any prime not dividing m 0,.. M G (n) := {H G(n) : [H, H] [G(n), G(n)]}. M G () = {H GL 2 (Z/Z) : SL 2 (Z/Z) H}. 4

5 Aso, et X(n) denote the compete moduar curve of eve n, which parametrizes eiptic curves together with chosen Z/nZ-bases of E[n]. Let H be a subgroup of GL 2 (Z/nZ) satisfying I H and for which the determinant map det: H (Z/nZ) is surjective, and consider the quotient curve X H := X(n)/H together with the j-invariant j : X H P 1. Any x P 1 (Q) is in the image of j if and ony if there exists an eiptic curve E over Q with j-invariant x for which ϕ E,n (G Q ) is contained in a subgroup conjugate to H in GL 2 (Z/nZ). We define the foowing set of moduar curves: X G := prime m 0 Now suppose we choose the group {X H : H M G ()} {X H : H M G (m 0 )}. G = π 1 (G(m)) GL 2 (Ẑ) as above so that the corresponding moduar curve X G(m) has genus zero and satisfies X G(m) (Q), and suppose that is an eiptic curve over Q(t) satisfying E: y 2 = x 3 + A(t)x + B(t) (A(t), B(t) Q(t)) (13) ϕ E,Q(t) (G Q(t) ) = G. (Note in particuar that E then defines a morphism P 1 X G(m).) For a rea parameter T 0, define the sets F E (T ) :={t 0 Q : H(t 0 ) T, E t0 /Q is an eiptic curve}, E E,non-G-Serre (T ) :={t 0 F E (T ) : E t0 is not a G-Serre curve}, E E,XH (T ) :={t 0 F E (T ) : j E (t 0 ) Q is the image under j of a point in X H (Q) non-cusp. }, where E t0 denotes the speciaization of E at t 0, H(t 0 ) denotes the height of t 0 (i.e. if we write t 0 = a/b in owest terms, H(a/b) := max{ a, b }), and j : X H P 1 denotes the j-map associated to X H. We have the foowing coroary of Theorem 2.6. Coroary Let m 1 and G GL 2 (Ẑ) be as in (5), and et E/Q(t) be an eiptic curve satisfying (13). Then, for any T 0, E E,non-G-Serre (T ) = E E,XH (T ). X H X G Coroary 2.10 aows us to deduce the foowing generaization of [3, Main Theorem]. Theorem Let ε > 0 be arbitrary. One has E E,non-G-Serre (T ) = O E,ε (T 1+ε ). Since F E (T ) T 2, Theorem 2.11 impies that amost a speciaizations of E are G-Serre curves. The main theorem of [3] gives Theorem 2.11 when G = GL 2 (Ẑ), and aso gives much better bounds in some cases. The proof of Theorem 2.11 proceeds aong the same ines (with Coroary 2.10 repacing [3, Coroary 19]), and aso gives better bounds in the appropriate cases. For the sake of brevity, Theorem 2.11 states ony the weakest form of what one can deduce from Coroary 2.10, and (since its proof is exacty the proof of [3, Main Theorem], mutatis mutandis) we wi not incude it in the present paper. 5

6 2.1. The genera case. The definitions and theorems we have described appy more generay to the foowing situation. Let r 1 be a positive integer and et G GL r be any matrix group, i.e. G is a subgroup scheme of GL r. Assume further that there is a homomorphism δ : G GL 1 for which G(Ẑ) GL1(Ẑ) is surjective, and et S := ker δ denote its kerne, which is aso a matrix group. We have an exact sequence 1 S(Ẑ) G(Ẑ) δ Ẑ 1. Note that (since they are agebraic groups) the Chinese remainder theorem hods for G and for S: (14) G(Ẑ) G(Z ), S(Ẑ) S(Z ). prime prime We wi be considering the reductions of G(Ẑ) and S(Ẑ) moduo positive integers, and so we make the definitions (15) G(m) := G(Ẑ)(mod m), S(m) := S(Ẑ)(mod m). It foows from (14) that, for any positive integer m, one has G(m) G( n ), S(m) n m n m S( n ). The subsets g n M r r (Z ) and s n M r r (Z ) defined by the exact sequences (16) 1 I + n g n G(Z ) G( n ) 1 and (17) 1 I + n s n S(Z ) S( n ) 1 wi pay an important roe for us, and in particuar, we wi consider their reductions moduo. We define g n(), s n() M r r (Z/Z) to be the reductions moduo of g n and s n respectivey, so that there is an exact sequence 1 I + n s n() S( n+1 ) S( n ) 1. It foows from the proof of [17, Lemma 3, IV-23] that we have an increasing sequence of Z/Z-vector spaces s () s 2() s n()... M r r (Z/Z) ( odd) s 4 (2) s 8 (2) s 2 n(2)... M r r (Z/2Z) ( = 2), which must stabiize. We wi assume that this sequence stabiizes immediatey, i.e. we wi assume that for any prime, s () = s 2() = = s n() = M r r (Z/Z). Fix now a positive integer m 1 and an arbitrary subgroup G(m) G(m), and et (18) G := {g G(Ẑ) : g (mod m) G(m)} denote the corresponding finite index subgroup of G(Ẑ). Furthermore, suppose that ϕ: G Q G is any Gaois representation for which δ ϕ: G Q Ẑ agrees with the cycotomic character. In this context, Definition 2.1 may be phrased as foows. Definition A subgroup H G(Ẑ) is caed commutator-thick if [H, H] = H S(Ẑ). Definition 2.4 remains the same, and we repace Definition 2.2 with Definition A subgroup H G(Ẑ) is caed δ-surjective if δ(h) = Ẑ. We remark that (for the same reason as in the GL 2 case) the image ϕ(g Q ) must be commutator-thick and δ-surjective. Foowing [17, IV-25], we make the foowing definition. 6

7 Definition Given a topoogica group G and a finite simpe group Σ we say that Σ occurs in G if and ony if there are cosed subgroups G 1 G and G 2 G 1 with G 2 a norma subgroup of G 1 and such that G 1 /G 2 Σ. Let us now make the foowing assumptions about the groups G and S. Reca that g n() := g n (mod ) and s n() := s n (mod ), with g n, s n M r r (Z ) as in (16) and (17). A0 For any prime and any n 1, one has s n() = s (). A1 There is a finite set L of primes such that, for any prime / L there is a finite simpe non-abeian group PS() and a finite set of surjective homomorphisms satisfying the conditions ϖ i : S() PS() (19) norma subgroup N S(), either N ker ϖ i for some i, or N = S() and (20) prime, = PS() does not occur in S( ). We wi assume for convenience that {2, 3} L. A2 For any prime, one has CD DC : C, D g () = s (). A3 For every prime, s () may be generated as a Z/Z-vector space by a set of matrices {u i } which satisfy i, u 2 i = 0 i, I + u i S(). Remark The discussion in [17, IV-25] shows that assumption A1 impies that in fact, for any prime / L and any positive integer n, (21) ( n, PS() does not occur in S( )) = PS() does not occur in n S(Z ). Remark Taken together with the proof of [17, Lemma 3, IV-23], assumption A0 impies that, for each prime, there is an exponent e 3 such that, whenever K S(Z ) is a cosed subgroup and K (mod e ) = S( e ), one has K = S(Z ). If is an odd prime, one may take e 2. Assumption A3 additionay impies that e 2, e 3 2, and e = 1 for any prime 5. Remark For a but finitey many prime powers n, we have that G(Z/ n Z) (resp. S(Z/ n Z)) consists entirey of smooth points. For such primes, it foows that G(Z/ n Z) = G( n ) (resp. S(Z/ n Z) = S( n )), and aso that assumption A0 hods. In particuar, assumption A0 need ony be verified for a finite set B of bad primes for which G(Z/Z) (resp. S(Z/Z)) contains singuar points. For these primes, it is possibe that S( n ) S(Z/ n Z) for some n 1. In particuar, the sequence 1 S(m) G(m) may fai to be exact when m is divisibe by a prime B. We now define (22) m 0 := cm 3, L δ (Z/mZ) 1 prime m 2ord (m)+1 We wi prove the foowing genera theorem, of which Theorem 2.6 is a specia case. Theorem Let G GL r be any matrix group satisfying assumptions A0, A1, A2, and A3 above, and for any m N, define G(m) by (15). Let G(m) G(m) be any subgroup and G G(Ẑ) the corresponding finite index subgroup defined by (18). Define m 0 by (22). Let H G be any subgroup. Then [H, H] = [G, G] if and ony if the foowing two conditions hod. 7.

8 (1) For each prime not dividing m 0, one has [H (mod ), H (mod )] = [G (mod ), G (mod )]. (2) One has [H (mod m 0 ), H (mod m 0 )] = [G (mod m 0 ), G (mod m 0 )]. In particuar, if H is commutator-thick, then H is a G-maxima commutator-thick subgroup if and ony if conditions (1) and (2) above hod. Remark Remark 2.16 together with assumptions A1 and A3 impy that for any / L and any n 1, one has [S( n ), S( n )] = S( n ) and aso (23) N S( n ) = In particuar, condition (1) in Theorem 2.18 is equivaent to { N (mod ) ker ϖ i for some i, or N = S( n ). For each prime not dividing m 0, one has S() H (mod ). In Section 6, we appy Theorem 2.18 to the Gaois representation ϕ A on the torsion of a k-dimensiona simpe principay poarized abeian variety A over Q, which maps into G(Ẑ) for G = GSp 2k, the group of degree k sympectic simiitudes. Fix any subgroup G(m) GSp 2k (Z/mZ) and et (24) G = π 1 GSp 2k (Ẑ)(G(m)) GSp2k(Ẑ) be the corresponding finite index subgroup of GSp (Ẑ). In this case m 2k 0 is given by (25) m 0 = cm , prime m 2ord (m)+1 Coroary Let A be a simpe principay poarized abeian variety over Q of dimension k 2 and assume that ϕ A (G Q ) G, where G is as in (24). Define m 0 by (25). The image ϕ A (G Q ) G is a G-maxima commutator-thick subgroup if and ony if the foowing conditions hod. (1) For each prime m 0, one has Sp 2k (Z/Z) ϕ A (G Q ) (mod ). (2) One has [ϕ A (G Q ) (mod m 0 ), ϕ A (G Q ) (mod m 0 )] = [G (mod m 0 ), G (mod m 0 )]. Finay, we appy our study to the Gaois representation ϕ (Ei) on the torsion of a k-tupe (E 1,..., E k ) of eiptic curves E i over Q, in which case the appropriate group G is Indeed, one has. G = (GL 2 ) k := {(g 1, g 2,..., g k ) GL k 2 : det g 1 = det g 2 = = det g k }. ϕ (Ei) : G Q (GL 2 ) k (Ẑ). In anaogy with (4), we make the foowing definition. Definition A k-tupe (E 1,..., E k ) of eiptic curves over Q is a Serre k-tupe if [(GL 2 ) k (Ẑ) : ϕ (E i)(g Q )] = 2 k. A Serre k-tupe is a k-tupe for which ϕ (Ei)(G Q ) is as arge as possibe. foowing coroary. In Section 6, we obtain the Coroary The k-tupe (E 1, E 2,..., E k ) is a Serre k-tupe if and ony if the foowing conditions hod. (1) For each prime 5, one has SL 2 (Z/Z) k ( ϕ (Ei)(G Q ) (mod ) ). (2) One has [ϕ (Ei)(G Q ) (mod 36), ϕ (Ei)(G Q ) (mod 36)] = (SL 2 (Z/36Z) ker ε) k, where ε is as in (2). The specia case k = 1 gives the foowing coroary, which improves [10, Lemma 5] to an if and ony-if statement. Coroary An eiptic curve E over Q is a Serre curve if and ony if the foowing two conditions hod. 8

9 (1) For each prime 5, one has SL 2 (Z/Z) ϕ E, (G Q ). (2) One has [ϕ E,36 (G Q ), ϕ E,36 (G Q )] = SL 2 (Z/36Z) (ker ε (mod 36)), where ε is as in (2). In [11], Coroary 2.22 is used to prove that, when ordered by height, amost a k-tupes (E i ) are Serre k-tupes. Remark The fact that ϕ E (G Q ) must be commutator-thick is a consequence of the Kronecker-Weber theorem, and so one cannot draw the same concusion if E is defined over a number fied K Q. Indeed, as shown in [7] (see aso [21]), there are number fieds K and eiptic curves E over K for which ϕ EK (G K ) = GL 2 (Ẑ), and (by (34) beow) GL2(Ẑ) is not commutator-thick. 3. Notation and preiminaries Throughout the paper, we wi use the foowing notation. For positive integers n and m, we wi write n m to mean that, for every prime number p, if p divides n then p divides m. The symbos p and wi aways denote prime numbers. We use the usua notation Ẑ := im Z/mZ for the inverse imit of the projective system {Z/m 1 Z Z/m 2 Z : m 1, m 2 1, m 2 m 1 }. The Chinese remainder theorem gives an isomorphism Ẑ Z, where Z denotes the ring of -adic integers. We wi often make impicit use of this isomorphism. For any fixed positive integer m, we wi denote by Z m (respectivey by Z (m) ) the quotient of Ẑ which corresponds under this isomorphism to Z (respectivey to Z ). Given a subgroup H GL r (Ẑ), we wi denote m m by H m the image of H under the canonica projection GL r (Ẑ) GL r(z m ), by H (m) the image of H under GL r (Ẑ) GL r(z (m) ), and by H(m) the image of H under GL r (Ẑ) GL r(z/mz). We wi overwork the symbo π, using it to denote any of the foowing canonica projections: π : G(Ẑ) G(Z m) π : G(Ẑ) G(m) π : G(m 1 ) G(m 2 ) (m 2 m 1 ). As a consequence of the convention mentioned in Section 2.1, G(m) is aways equa to the image moduo m of G(Ẑ), so that each of these maps is surjective. In some cases we wi use π H to denote the restriction of any of these projections to a subgroup H; hopefuy the meaning wi be cear from context. For a pair of eements h and k in any group, we wi use the standard notation for the commutator: [h, k] := hkh 1 k 1. For any positive integer M and any prime dividing M, consider the kerne of the reduction moduo M/ map: I + M ( ) Mr r (Z/MZ) M r r (Z/MZ). M r r (Z/MZ) In ight of the isomorphism M r r(z/mz) M r r (Z/MZ) M r r(z/z), we wi use the abbreviated notation I + M M r r(z/z) M r r (Z/MZ), and refer to I + M A M r r(z/mz) when A M r r (Z/Z), hoping that this wi not cause too much confusion. In particuar, we wi use g () and s () in the exact sequences 1 I + n g () G( n+1 ) G( n ) 1 9

10 and 1 I + n s () S( n+1 ) S( n ) Proof of Theorem 2.18 In this section, we wi prove Theorem Note that the ony if part is trivia, so we wi focus on the if part. We remark that many of the essentia ideas are aready present in the proof of the Proposition on page IV-19 of [17], wherein a sighty weaker hypothesis than that of Theorem 2.18 is used to deduce that H is an open subgroup of GL 2 (Ẑ). We begin with some preparatory emmas Preparatory emmas. Our first emma wi be used repeatedy throughout the paper. Lemma 4.1. (Goursat s Lemma) Let G 0 and G 1 be groups and G G 0 G 1 a subgroup satisfying π i (G) = G i (i {0, 1}), where π i denotes the canonica projection onto the i-th factor. Let N i := π i (G ker π 1 i ). Then there is an isomorphism of groups ψ : G 0 /N 0 G 1 /N 1 (whose graph is induced by G) for which G = {(g 0, g 1 ) G 0 G 1 : ψ(g 0 N 0 ) = g 1 N 1 }. Proof. See [16, Lemma (5.2.1)], which shows that the image of G in G 0 /N 0 G 1 /N 1 is the graph of an isomorphism ψ. Thus, G {(g 0, g 1 ) G 0 G 1 : ψ(g 0 N 0 ) = g 1 N 1 }. Now note that N 0 N 1 G, from which the equaity foows. Remark 4.2. When appying Lemma 4.1 in this paper, we wi usuay formuate the concusion equivaenty as then there exists a group Q and surjective homomorphisms ψ 0 : G 0 Q, ψ 1 : G 1 Q for which G = {(g 0, g 1 ) G 0 G 1 : ψ 0 (g 0 ) = ψ 1 (g 1 )}. The foowing coroary may be viewed as a fibered version of Lemma 4.1. Suppose now that G 0 and G 1 are groups, together with surjective homomorphisms onto a fixed group R. Let η 0 : G 0 R, denote the fibered product of G 0 and G 1 over R. η 1 : G 1 R G 0 R G 1 := {(g 0, g 1 ) G 0 G 1 : η 0 (g 0 ) = η 1 (g 1 )} Coroary 4.3. Suppose that G G 0 R G 1 is any subgroup satisfying π i (G) = G i for i {0, 1}. Then there is a group Q together with surjective homomorphisms f : Q R, ψ 0 : G 0 Q, and ψ 1 : G 1 Q for which f ψ i = η i (i {0, 1}), and G = {(g 0, g 1 ) G 0 G 1 : ψ 0 (g 0 ) = ψ 1 (g 1 )}. Proof. We appy Lemma 4.1 and note that there is a we-defined surjective group homomorphism The coroary foows Working in G(Z m0 ). f : Q G i /N i R (i {0, 1}) g i N i η i (g i ). Lemma 4.4. Fix a prime number, an exponent n 1, and a pair of matrices C, D M r r (Z ). Then one has [I + n C, I + n D] = I + 2n (CD DC) + 3n E, where E M r r (Z ). Proof. A cacuation, using the series representation (I + n C) 1 = I n C + 2n C 2 3n C verifies the statement of the emma. Reca that an integer m 0 is caed square-fu if, for each prime, one has m 0 = 2 m 0. 10

11 Lemma 4.5. Let be any prime number and et A M r r (Z/Z). Suppose that m 0 is any positive squarefu integer divisibe by 2, and in case = 2 assume further either that 8 divides m 0 or that A 2 0 (mod 2). Let H GL r (Z m0 ) be any subgroup and et M be any mutipe of m 0 satisfying m 0 M m 0. We then have I + m 0 A H(m 0) = I + M A H(M). Proof. The proof is by induction on M, the base case M = m 0 being trivia. If M > m 0, then there is a prime p dividing M for which m 0 M/p. By induction, we have that which impies that (26) I + M/p I + m 0 A H(m 0) = I + M/p A H (M/p), pa H(M/p), since H is a subgroup. Case: p. We regard H(M) as a subgroup of the fibered product { ( ) ( ) } M M (27) H(M) (h 0, h 1 ) H H : π 0 (h 0 ) = π 1 (h 1 ) p ( ) ( ) ( ) M M M via the embedding h (h (mod M/p), h (mod M/)). Here, π 0 : H H and π 1 : H ( ) p p M H are the natura projections. By Coroary 4.3, there must be a group Q, a surjective group homomorphism f : Q H and surjective group homomorphisms p ( ) M p ψ 0 : H ( ) M Q, p for which f ψ i = π i (i {0, 1}) and, under (27), { H(M) = (h 0, h 1 ) H ( M p ) H ψ 1 : H ( ) M Q ( ) } M : ψ 0 (h 0 ) = ψ 1 (h 1 ). Furthermore, since the degrees of π 0 and π 1 are reativey prime, we see that Q must be equa to H ψ i = π i, and (27) is in fact an equaity: H(M) = By (26) we have that (h 0, h 1 ) = { (h 0, h 1 ) H ( I + M/p Case: p =. Now we have I + M 2 A H ( M ( ) M H p ) pa, I ( ) } M : π 0 (h 0 ) = π 1 (h 1 ). ( ) M, p beongs to the right-hand side above, and this corresponds under h (h (mod M/p), h (mod M/)) ) to h = I + (M/)A, which therefore beongs to H(M).. Let I + M 2 A + M B H(M) be any ift. In case is odd, note that divides M/ 2 (since m 0 is square-fu), and so I + M A I (mod ). It foows that 2 ( I + M 2 A + M ) ( B I + M ) 2 A (mod M) I + M 11 A (mod M).

12 In case = 2 and A 2 / 0 (mod 2) note that 8 then divides M/2, so 16 divides (M/4) 2, and so we may use the same reasoning. In case = 2 and A 2 0 (mod 2), one computes (I + M4 A + M2 ) 2 2 B M2 M I + A + MB + 16 A2 + M 2 2 M (AB + BA) B2 (mod M) This concudes the proof of Lemma 4.5. I + M A (mod M). 2 The foowing key emma is a coroary of Lemma 4.5. Lemma 4.6. Assume that A3 hods. Let m 0 be any positive square-fu integer and H(m 0 ) S(m 0 ) any subgroup satisfying (28) m 0, I + m 0 s () H(m 0 ). Suppose that K S(Z m0 ) is any cosed subgroup for which K(m 0 ) = H(m 0 ). Then we must have K = π 1 S(Z m0 ) (H(m 0)). Remark 4.7. In case A3 does not hod (in particuar if s 2 (2) cannot be generated as a Z/2Z-vector space by matrices u i simutaneousy satisfying u 2 i 0 (mod 2) and I + u i S(2)), then the same concusion foows from the additiona hypothesis that 8 divides m 0. Proof. Since K is cosed, it suffices to show that K(M) = π 1 (H(m 0 )) S(M), for any positive integer M satisfying m 0 M m 0. This is proved by induction on M, the base case M = m 0 being trivia. If M > m 0, then there is a prime for which m 0 (M/). By induction, K(M/) = π 1 (H(m 0 )) S(M/). Considering the exact sequence 1 I + (M/)s () S(M) S(M/) 1, we see that K(M) = π 1 (H(m 0 )) if and ony if I + (M/)s () K(M). This ast containment foows from (28) and Lemma 4.5. Proposition 4.8. Suppose G satisfies assumptions A2 and A3. Let m 1 be any positive integer and define m 0 by (22). Let G(m) G(m) be any subgroup and et G m0 = π 1 G(Z m0 )(G(m)) be the corresponding finite index subgroup. Suppose H m0 G m0 is any subgroup which satisfies (29) [H(m 0 ), H(m 0 )] = [G(m 0 ), G(m 0 )]. Then [H m0, H m0 ] = [G m0, G m0 ] = π 1 S(Z m0 ) ([G(m 0), G(m 0 )]). Proof. By Lemma 4.6 (and noting (29)), the proposition wi foow from the containment m 0, I + m 0 s () [G(m 0 ), G(m 0 )]. Define m 0 by m 0 = m 0 ord (m 0), so that m 0, and view G(m 0 ) G(m 0) G( ord (m 0) ) via the Chinese remainder theorem. The above condition then becomes ( ) (30) m 0, {I} I + ord (m 0) 1 s () [G(m 0 ), G(m 0 )] G(m 0) G( ord (m 0) ). To prove this, fix a prime dividing m 0. If divides m, then since G m0 any C g(z ), one has (I, I + ord(m) C) G m0 G m 0 G. = π 1 G(Z m0 )(G(m)), we see that for Likewise, pick (I, I + ord (m) D) G m0 for another arbitrary D g(z ). Appying Lemma 4.4 to the commutator [(I, I + ord (m) C), (I, I + ord (m) D)], and using assumption A2, we see that (30) is satisfied. If does not divide m, then simiary one finds that (I, I + C), (I, I + D) G m0 G m 0 G, for any C, D g(z ), impying (30) again by assumption A2. This competes the proof of the proposition. 12

13 4.3. Working in G(Z (m0)). Lemma 4.9. Assume that A1 and A3 hod. Let m 0 be any integer divisibe by L (where L is as in assumption A1) and suppose that K S(Z (m0)) is any cosed subgroup satisfying Then K = S(Z (m0)). m 0, K() = S(). Proof. Since K is cosed, it suffices to show that, for each integer M with gcd(m 0, M) = 1, K(M) = S(M). We prove this by induction on the number of prime divisors of M. The base case where M = n is a prime power foows immediatey from Remark For the induction step, suppose that M is divisibe by more than one prime and write M = n M, where M and M > 1. By induction, we have that K(M ) = S(M ) and K( n ) = S( n ). By Lemma 4.1, there is a common quotient group Q, together with surjective homomorphisms ψ 0 : K(M ) Q, ψ 1 : K( n ) Q, such that under the isomorphism of the Chinese remainder theorem, we have K(M) = {(h 0, h 1 ) K(M ) K( n ) : ψ 0 (h 0 ) = ψ 1 (h 1 )}. Now we appy the observation (23) (with N = ker ψ 1 ), concuding that either Q has PS() as a quotient, or ese Q = 1. But since M is not divisibe by, (21) impies that Q cannot have PS() as a quotient, and so Q = 1 and K(M) = S(M), finishing the proof of Lemma 4.9. Appying Lemma 4.9 with K a commutator subgroup, and using Remark 2.16, we concude the foowing coroary. Coroary Assume that A1 and A3 hod. Suppose that m 0 is any positive integer divisibe by L and H (m0) G(Z (m0)) is any subgroup satisfying m 0, Then, [H (m0), H (m0)] = [G(Z (m0)), G(Z (m0))] = S(Z (m0)). S() H() Finishing the proof of Theorem We wi now finish the proof of Theorem If H G is any subgroup satisfying n {m 0 } {primes : m 0 }, [H(n), H(n)] = [G(n), G(n)], then by Proposition 4.8, Coroary 4.10 and Lemma 4.1, we see that there is a group Q and surjective homomorphisms ψ 0 : [G m0, G m0 ] Q and ψ 1 : S(Z (m0)) Q for which (regarding H H m0 H (m0)) [H, H] = {(h 0, h 1 ) [G m0, G m0 ] S(Z (m0)) : ψ 0 (h 0 ) = ψ 1 (h 1 )}. We now ony need to show that Q = {1}. Consider the subgroup ker ψ 1 S(Z (m0)), and its projection ker ψ 1 () S(). By assumption A1, either S()/ ker ψ 1 () has PS() as a quotient, or ker ψ 1 () = S(). If S()/ ker ψ 1 () had PS() as a quotient, then so woud Q, and hence so woud [G m0, G m0 ], contradicting (21). Thus we see that, for each prime m 0, ker ψ 1 () = S(). By Lemma 4.9, it foows that ker ψ 1 = S(Z (m0)), and so Q = 1 and [H, H] = [G, G], finishing the proof of Theorem Exampes and remarks Having proved Theorem 2.18, we wi give a few exampes which iustrate some of the subtety of this study. The first exampe highights the fact that, even though there may exist a determinant-surjective, commutator-thick subgroup H of a given finite index subgroup G GL 2 (Ẑ), there may nevertheess be no eiptic curve E over Q for which ϕ E (G Q ) G. 13

14 Exampe 5.1. Let be a prime, G() := {( ) } a b : b Z/Z; a, d (Z/Z), 0 d and G = π 1 (G()) GL 2 (Ẑ). Even though there do exist commutator-thick, determinant-surjective subgroups H G, Mazur has shown (see [13]) that, for > 163, there is no eiptic curve E over Q for which ϕ E (G Q ) G. More generay, when the index of G in GL 2 (Ẑ) is arge enough, one expects that no subgroup of G arises as ϕ E (G Q ) for E over Q (see [18, 4.3] and [1]). The next exampe shows that there do exist finite index subgroups G GL 2 (Ẑ) which have no commutatorthick, determinant-surjective subgroups. Exampe ( 5.2. Let be an odd prime and fix any eement ε (Z/Z) which is not a square, i.e. for which ε ) = 1. Let C ns () denote the non-spit Cartan subgroup {( )} x εy C ns () := GL y x 2 (Z/Z). The group G := π 1 (C ns ()) GL 2 (Ẑ) has no subgroup H which is simutaneousy commutator-thick and determinant-surjective. Proof. If there were a commutator-thick, determinant-surjective subgroup H π 1 (C ns ()) then there woud necessariy be a group homomorphism ψ : Ẑ C ns () for which the diagram (31) Ẑ Ẑ red ψ C ns () det (Z/Z) commutes. We wi show that such a homomorphism ψ cannot exist. Suppose for the sake of contradiction that such a ψ does exist. The group C ns () is a cycic group of order 2 1, from which it foows that ψ factors as Ẑ red (Z/m Z) ψ ψ((z/m Z) ) C ns (), where does not divide m. Let g be a generator of C ns () and consider the image under ψ of (Z/Z) {1} (Z/Z) (Z/m Z). By order considerations, we must have ψ((z/z) {1}) g +1, from which it foows by (31) (since g +1 is a scaar matrix) that the canonica projection (Z/m Z) (Z/Z) maps into [(Z/Z) ] 2, a contradiction. Thus, there is no ψ making (31) commute, proving the assertion. Remark 5.3. By Remark 2.3, the previous exampe gives another proof (See aso [19, Lemme 17, p. 197]) that E/Q for which ϕ E, (G Q ) C ns (). Our third exampe iustrates, among other things, that the Z/Z-vector space V M 2 2 (Z/Z) defined by the exact sequence 1 I + mv H(m) H(m) 1 may shrink when we repace m by a mutipe of m (thus, care must be taken in Lemma 4.6). spit-cartan subgroup C s (Z/ n Z) and the Bore subgroup B(Z/ n Z) be defined as usua by {( )} {( )} C s (Z/ n x 0 x y Z) := =: B(Z/ n Z) GL 0 z 0 z 2 (Z/ n Z). Let χ (1), χ (2) n : B(Z/ n Z) (Z/ n Z) be the characters defined by n (( )) (( )) χ (1) x y := x, χ (2) x y n 0 z := z, n 0 z 14 Let the

15 and et us use the same symbos to denote their corresponding restrictions to C s (Z/ n Z). Note that, if π 1 (C s (Z/Z)) GL 2 (Z/ 2 Z) denotes the fu pre-image of C s (Z/Z), then there is a surjective group homomorphism µ: π 1 (C s (Z/Z)) C s (Z/ 2 Z) (( )) ( ) x + 3a 3b x + 3a 0 µ :=. 3c z + 3d 0 x + 3d Finay, we use L : (Z/Z) (Z/3Z) to denote the (unique) surjective homomorphism which sends a generator of (Z/Z) to 1. Exampe 5.4. Suppose that G GL 2 (Ẑ) is a subgroup with { } G(3 7 13) = (g 3, g 7, g 13 ) C s (Z/3Z) B(Z/7 13Z) : det(g 3 ) = L 7 (χ (1) 7 (g 7)), χ (1) 3 (g 3) = L 13 (χ (1) 13 (g 13)). It is possibe that G(9 7 13) GL 2 (Z/9 7 13) is smaer than the fu pre-image π 1 (G(3 7 13)) GL 2 (Z/9 7 13Z). For exampe, one coud have { } G(9 7 13) = (g 9, g 7, g 13 ) π 1 (C s (Z/3Z)) B(Z/7 13Z)) : det(g 9 ) = θ 7 (χ (1) 7 (g 7)), χ (1) 9 (µ(g 9)) = θ 13 (χ (1) 13 (g 13)), where θ : (Z/Z) (Z/9Z) denotes any surjective homomorphism. Suppose that G = π 1 (G(9 7 13)) GL 2 (Ẑ), where G(9 7 13) is as above. To see that π(g(9 7 13)) = G(3 7 13), note that θ (χ (1) (g )) (mod 3) = L (χ (1) (g )) and χ (1) 9 (µ(g 9)) (mod 3) = χ (1) 3 (g 9 (mod 3)). Aso note that the exact sequence has a arger kerne than {( ) a b 1 I c a whose kerne is sti arger than 1 I I + 3M 2 2 (Z/3Z) G(9) G(3) 1 } (mod 3) G(9 7) G(3 7) 1, {( ) } 0 b (mod 3) G(9 7 13) G(3 7 13) 1. c 0 Remark 5.5. The torsion conductor m E of an eiptic curve E over Q is defined in [9] to be the smaest positive integer m 1 for which ϕ E (G Q ) = π 1 (ϕ E,m (G Q )). Exampe 5.4 indicates that the determination of m E can be quite deicate. 6. The specia cases G = GSp 2k and G = (GL 2 ) k For any positive integer k 1, our study may be appied to the group G = GSp 2k of degree k sympectic simiitudes. This group arises when one considers the action of Gaois on the torsion of a simpe principay poarized abeian variety of dimension k. For any commutative ring R, the group GSp 2k (R) of R-vaued points is GSp 2k (R) = {g GL 2k (R) : c = c g R with g t Ωg = cω}, where Ω is the 2k 2k matrix given in bock form by Ω := ( 0 Ik I k 0 Note that GSp 2 = GL 2, and the k = 1 case coincides with that of an eiptic curve; this case wi be treated beow when considering G = (GL 2 ) k. Therefore, for most of our consideration of G = GSp 2k, we wi assume that k 2. It is we-known that the function ). δ : GSp 2k (R) R, g c g 15

16 is a group homomorphism, whose kerne defines the sympectic group Sp 2k, which is aso a subgroup of SL 2k : Furthermore, if Sp 2k (R) = {g SL 2k (R) : g t Ωg = Ω}. ϕ A : G Q GSp 2k (Ẑ) is the Gaois representation defined by etting G Q operate on the torsion of A, then (by virtue of the Wei pairing) δ ϕ A : G Q Ẑ agrees with the cycotomic character. We wi now verify assumptions A0, A1, A2 and A3 for G = GSp 2k. Lemma 6.1. If k 2, then assumption A1 hods for G = GSp 2k, with PS() = PSp 2k (Z/Z) := Sp 2k (Z/Z)/{±I}, L = {2, 3} and a singe map given by the natura projection. ϖ : Sp 2k (Z/Z) PSp 2k (Z/Z) Proof. In [8, Hauptsatz 9.22] one may find the proof that PSp 2k (Z/Z) is simpe in the stated cases. Fix any norma subgroup N Sp 2k (Z/Z). Considering the exact sequence 1 {±I} Sp 2k (Z/Z) PSp 2k (Z/Z) 1, one sees that if N ker ω, it foows that N = Sp 2k (Z/Z) since the above sequence doesn t spit. Finay, it foows by considering Syow subgroups that PSp 2k (Z/Z) does not occur in Sp 2k (Z/ Z) for 5 and, competing the proof of Lemma 6.1. Lemma 6.2. Assumptions A0, A2 and A3 hod for the group G = GSp 2k. Proof. Note that, for any n 1, one has s n() = {A M 2k 2k (Z/Z) : A t Ω + ΩA = 0}. In particuar, assumption A0 hods. For indices i and j with 1 i, j k, et E i,j denote the k k bock matrix with a 1 in the i-th row and j-th coumn and with a other entries equa to zero, and set { E i,j + E j,i if i j S i,j := otherwise. One verifies that U := {( Ei,j 0 0 E j,i )}1 i,j k i j E i,i {( )} 0 Si,j i,j k i j {( )} {( )} 0 0 Ei,i E i,i S i,j 0 1 i,j k E i,i E i,i 1 i k i j is a basis of s () as a Z/Z-vector space. Furthermore, one verifies that for each u i U, u 2 i 0 (mod ) and aso that I + u i S(). Thus, assumption A3 hods for G = GSp 2k. Now note that where s () = {A M 2k 2k (Z/Z) : A t Ω + ΩA = 0} = Ω Sym 2k (Z/Z), Sym 2k (Z/Z) := {A M 2k 2k (Z/Z) : A t = A} is the set of symmetric matrices. Consider the Lie bracket ΩSym 2k (Z/Z) ΩSym 2k (Z/Z) ΩSym 2k (Z/Z) [ΩX, ΩY ] = ΩX ΩY ΩY ΩX. Since Ω 1 Sym 2k (Z/Z)Ω = Sym 2k (Z/Z), we may repace X with ΩXΩ, so that the Lie bracket becomes Writing X andy Y in bock form as ( ) A B X = B t D [Ω ΩXΩ, ΩY ] = XY + ΩY XΩ. ( ) E F and Y = F t, J 16

17 one computes that XY + ΩY XΩ = ( ) AE JD + BF t F t B BJ + JB t + AF + F t A B t E + EB + DF t + F D EA + DJ + B t F F B t. Taking A = D = E = J = 0 and F = E i,i, B = E i,j (with i j), we concude that ( ) Ei,j 0 CD DC : C, D g 0 E (). j,i Simiary, taking A = E i,i, E = I and B = D = F = J = 0 gives us that ( ) Ei,i 0 CD DC : C, D g 0 E (). i,i Taking A = E i,i, F = E i,j (i j) and B = D = E = J = 0 and simiary B = E i,j, E = E i,i and A = D = F = J = 0 gives ( ) ( ) 0 Ei,j + E j,i 0 0, CD DC : C, D g 0 0 E i,j + E j,i 0 (), whie taking A = E i,i, F = E i,i and B = D = E = J = 0 and aso B = E i,i, E = E i,i and A = D = F = J = 0 gives ( ) ( ) 0 2Ei,i 0 0, CD DC : C, D g 0 0 2E i,i 0 (). This verifies A3 when is an odd prime. For = 2, one uses the fact that so that in particuar, Thus, g () = {A M 2k 2k (Z/Z) : A t Ω + ΩA Z/Z Ω}, ( 0 Ei,i 0 0 and a simiar cacuation verifies that ( 0 Ei,i 0 0 finishing the proof of Lemma 6.2. ) ( ) I ), ( ) 0 0 g 0 I (). ( I ) ( 0 Ei,i 0 0 ) = ( ) 0 Ei,i, 0 0 ( ) 0 0 CD DC : C, D g E i,i 0 (), As mentioned in Section 2, we may appy Theorem 2.18 to the Gaois representation ϕ A on the torsion of a k-dimensiona simpe abeian variety A over Q. Fix any subgroup G(m) GSp 2k (Z/mZ) and et G = π 1 GSp 2k (Ẑ)(G(m)) GSp2k(Ẑ) be the corresponding finite index subgroup of GSp (Ẑ). In this case m 2k 0 is given by m 0 = cm , 2ord (m)+1. m The foowing coroary restates Coroary 2.20 from Section 2, which we may now deduce from Theorem Coroary 6.3. Let A be a simpe principay poarized abeian variety over Q of dimension k 2, and assume that ϕ A (G Q ) G. The image ϕ A (G Q ) G is a G-maxima commutator-thick subgroup if and ony if the foowing conditions hod. (1) For each prime m 0, one has Sp 2k (Z/Z) ϕ A (G Q ) (mod ). (2) One has [ϕ A (G Q ) (mod m 0 ), ϕ A (G Q ) (mod m 0 )] = [G(m 0 ), G(m 0 )]. 17

18 If a k-dimensiona abeian variety is a product of eiptic curves, the representation ϕ A maps into a different group, which we now consider. Define the group (GL 2 ) k by specifying its R-vaued points, for any commutative ring R as (GL 2 ) k (R) := {(g 1, g 2,..., g k ) GL 2 (R) k : det(g 1 ) = det(g 2 ) = = det(g k )}. Note that the diagona imbedding g g (32) (g 1, g 2,..., g k ) g k reaizes (GL 2 ) k as an agebraic subgroup of GL 2k. Taking δ((g 1, g 2,..., g k )) := det g i to be the common determinant of any of the entries, one has S = (SL 2 ) k. We wi presenty verify the assumptions A0 through A3. Lemma 6.4. Assumption A1 hods for G = (GL 2 ) k, with L = {2, 3, 5}, PS() = PSL 2(Z/Z), and ϖ i : SL 2 (Z/Z) k PSL 2 (Z/Z) (i = 1, 2,..., k) given by projection onto the i-th factor foowed by the projection SL 2 (Z/Z) PSL 2 (Z/Z). Proof. The we-known fact that PSL 2 (Z/Z) is simpe for 5 may be found in [8, Hauptsatz 6.13]. To prove that any norma subgroup N SL 2 (Z/Z) k must satisfy either N ker ϖ i for some i or N = SL 2 (Z/Z) k, we proceed by induction on k 1. The k = 1 case foows immediatey from the fact that the exact sequence 1 {±I} SL 2 (Z/Z) PSL 2 (Z/Z) 1 does not spit. By induction we assume that the statement of the emma hods for some fixed k and now et N SL 2 (Z/Z) k+1 be any norma subgroup satisfying N ker ϖ i for each i. Let π 1 : SL k+1 2 SL k 2 be the projection onto the first k factors and π 2 : SL k+1 2 SL 2 the projection onto the ast factor. By induction, we have that π 1 (N) = SL 2 (Z/Z) k and π 2 (N) = SL 2 (Z/Z). By Lemma 4.1, we have that N = SL 2 (Z/Z) k Q SL 2 (Z/Z) for some group Q. Either Q has PSL 2 (Z/Z) as a quotient, or Q = {1}. Conjugation by {I} k SL 2 (Z/Z), one sees that the first possibiity contradicts N SL 2 (Z/Z) k being a norma subgroup. Thus Q = {1}, and the induction step is compete. Finay, we verify that PSL 2 (Z/Z) does not occur in SL 2 (Z/ Z) k for 7 and. Define the notation Then for any exact sequence of groups one has Occ(G) := { finite simpe non-abeian groups Σ occurring in G }. 1 G 1 G G 2 1, Occ(G) = Occ(G 1 ) Occ(G 2 ). This observation reduces to the case k = 1, which in turn foows from the fact that Occ(SL 2 (Z/Z)) = Occ(PSL 2 (Z/Z)) {PSL 2 (Z/Z), A 5 }, where A 5 is the aternating group on 5 eements (see [8, Hauptsatz 8.27]). This finishes the proof of Lemma 6.4. Lemma 6.5. Assumptions A0, A2 and A3 hod for the group G = (GL 2 ) k. 18

19 Proof. To verify assumption A2, first note that the imbedding (32) reaizes g () as A A g () = ; A i M 2 2 (Z/Z), tr A 1 = tr A 2 = = tr A k A k If is odd, we observe that A xi g (), xi where 2x = tr A 1, and simiary with the other main diagona entries. If = 2 then we observe that A B I , 0 B g 2(Z/2Z) I B (where tr A = 0 and tr B = 1), and simiary for the other main diagona entries. These observations reduce us to the case k = 1, which is a specia case of Lemma 6.2. This verifies assumption A2. For assumptions A0 and A3, one takes the specia case k = 1 of the basis U occurring in the proof of Lemma 6.2, appied to each main diagona entry. This verifies assumptions A0 and A3, finishing the proof of Lemma 6.5. As mentioned above, the group G = (GL 2 ) k just considered arises when one considers the Gaois representation ϕ (Ei) : G Q (GL 2 ) k (Ẑ) attached to a k-tupe (E 1, E 2,..., E k ) of eiptic curves E i over Q. Theorem 2.18 gives a criterion for detecting Serre k-tupes (see Definition 2.21) that invoves the eve m 0 = = 27, 000. With a bit more work, using particuar information about GL 2, one can improve this to m 0 = = 36, as foows. First, we observe that Remark 2.8 hods. Indeed, choosing L minimay so that A1 is satisfied, we may decompose L as a disjoint union L = L 0 L 1, where finite simpe non-abeian group PS() and set {ϖ i } L 0 := L : of surjective homomorphisms ϖ i : S() PS(), {2, 3} N S(), with i N ker ϖ i and N S() and L 1 := L L 0. Thus, L 0 is the subset of primes in L for which condition (19) fais (together with the primes 2 and 3), whie L 1 is the subset of primes in L for which condition (20) fais. For the group G = (GL 2 ) k, one has L 0 = {2, 3} and L 1 = {5}. We now show that, repacing m 0 with (12), the proof of Theorem 2.18 sti hods in this case. Indeed, the proof of Proposition 4.8 remains vaid for the new vaue of m 0. Furthermore, repacing L = {2, 3, 5} with {2, 3} in Lemma 4.9, and choosing in the decomposition M = n M occurring in its proof so that 5, one sees that Lemma 4.9 aso remains vaid. Finay, for any prime 5 with / ± 1 (mod 5), since 5 doesn t divide SL 2 (Z/Z), the group PSL 2 (Z/5Z) A 5 does not occur in SL 2 (Z/Z). Thus, under the hypothesis that no prime dividing m satisfies ±1 (mod 5), it foows from (21) that PSL 2 (Z/5Z) does not occur in [G m0, G m0 ], and so the proof occurring in Section 4.4 aso remains vaid. Thus, Theorem 2.18 hods with m 0 given by (12) when G = (GL 2 ) k. In particuar, this justifies Remark 2.8. In case m = 1, we may further reduce m 0 from = 216 to = 36. For this, we wi need the foowing technica emma. Suppose G 1 and G 2 are groups together with surjective group homomorphisms ϖ 1 : G 1 A ϖ 2 : G 2 A onto a common abeian group A. Let G = G 1 A G 2 denote the fibered product G = {(g 1, g 2 ) G 1 G 2 : ϖ 1 (g 1 ) = ϖ 2 (g 2 )}. 19

20 Lemma 6.6. With notation as above, suppose that there exists a subset B 2 G 2 satisfying ϖ 2 (B 2 ) = A and a of whose eements commute with one another. Then one has Proof. See [12, Lemma 1, p. 174]. [G, G] = [G 1, G 1 ] [G 2, G 2 ]. We wi now use Lemmas 6.6 and 4.6 to deduce Coroary 2.22 from Section 2: Coroary 6.7. The k-tupe (E 1, E 2,..., E k ) is a Serre k-tupe if and ony if the foowing conditions hod. (1) For each prime 5, one has SL 2 (Z/Z) k ( ϕ (Ei)(G Q ) (mod ) ). (2) One has [ϕ (Ei)(G Q ) (mod 36), ϕ (Ei)(G Q ) (mod 36)] = (SL 2 (Z/36Z) ker ε) k, where ε is as in (2). Proof. In this case one has G = (GL 2 ) k and m = 1. Put m 0 = 36. As in the argument of Section 4.4, Lemma 4.6 and Coroary 4.10 reduce the proof to showing that [ (33) (GL2 (Z/36Z)) k, (GL 2 (Z/36Z)) k ] = (SL2 (Z/36Z) ker ε) k. We observe that for any eve n, the image of the embedding (Z/nZ) GL 2 (Z/nZ), x ( ) x defines a subgroup B 2 GL 2 (Z/nZ) which reaizes the conditions of Lemma 6.6 for G 1 = (GL 2 (Z/nZ)) k 1 and G 2 = GL 2 (Z/nZ). By induction, we are reduced to verifying (33) in the case k = 1. This case foows from [12, Theorem 2.1, p. 166] and the discussion on pages of [12]. (In fact, these observations impy that (34) [GL 2 (Ẑ), GL2(Ẑ)] = SL2(Ẑ) ker ε, an index 2 subgroup of SL 2 (Ẑ).) This finishes the proof of Coroary Acknowedgments Some of the research eading to this paper was done whie visiting the Hausdorff Research Institute for Mathematics and subsequenty the Max Panck Institute for Mathematics (both in Bonn, Germany). I woud ike to thank each of these institutes for providing stimuating work environments. I woud aso ike to thank D. Zywina for comments on a previous version, as we as A.C. Cojocaru, D. Grant and G. Banaszak for usefu conversations. Finay, I woud ike to thank the anonymous referee for carefuy reading the manuscript and offering severa hepfu comments. References [1] K. Arai, On uniform ower bound of the Gaois images associated to eiptic curves, Journa de théorie des nombres de Bordeaux, 20 no. 1 (2008), [2] Y. Biu and P. E. Parent, Serre s uniformity probem in the spit Cartan case, Ann. of Math. 173, no. 1 (2011), [3] A.C. Cojocaru, D. Grant and N. Jones, One-parameter famiies of eiptic curves over Q with maxima Gaois representations, Proc. Lond. Math. Soc. 103, no. 3 (2011), [4] C. David and F. Papaardi, Average Frobenius distributions of eiptic curves, Internationa Math. Research Notices, 4 (1999), [5] É. Fouvry and M.R. Murty, On the distribution of supersinguar primes, Can. J. Math. 48 (1996), [6] D. Grant, A Formua for the Number of Eiptic Curves with Exceptiona Primes, Compos. Math. 122 (2000), [7] A. Greicius, Eiptic curves with surjective goba Gaois representation, Experiment. Math. 19, no. 4 (2010), [8] B. Huppert, Endiche Gruppen, Grundehren Math. Wiss. 134, Springer-Verag, Berin (1967). [9] N. Jones, A bound for the torsion conductor of a non-cm eiptic curve, Proc. Amer. Math. Soc. 137 (2009), [10] N. Jones, Amost a eiptic curves are Serre curves, Trans. Amer. Math. Soc. 362 (2010), [11] N. Jones, Pairs of eiptic curves with maxima Gaois representations, J. Number Theory 133 (2013), [12] S. Lang and H. Trotter, Frobenius distribution in GL 2 extensions, Lecture Notes in Math. 504, Springer (1976). [13] B. Mazur, Rationa isogenies of prime degree, Invent. Math., 44, no. 2 (1978), [14] B. Mazur, Rationa points on moduar curves, in Lecture Notes in Math. 601, Springer, NY 1977, [15] V. Radhakrishnan, An asymptotic formua for the number of non-serre curves in a two-parameter famiy of eiptic curves, Ph.D. dissertation, University of Coorado at Bouder (2008). [16] K. Ribet, Gaois action on division points of Abeian varieties with rea mutipications, Amer. J. Math. 98, no. 3 (1976),