On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields

Transcription

1 On the Surjectivity of Gaois Representations Associated to Eiptic Curves over Number Fieds Eric Larson and Dmitry Vaintrob Abstract Given an eiptic curve E over a number fied K, the -torsion points E[] of E define a Gaois representation Ga(K/K) GL 2 (F ). A famous theorem of Serre [11] states that as ong as E has no Compex Mutipication (CM), the map Ga(K/K) GL 2 (F ) is surjective for a but finitey many. We say that a prime number is exceptiona (reative to the pair (E, K)) if this map is not surjective. Here we give a new bound on the argest exceptiona prime, as we as on the product of a exceptiona primes of E. We show in particuar that conditionay on the Generaized Riemann Hypothesis (GRH), the argest exceptiona prime of an eiptic curve E without CM is no arger than a constant (depending on K) times og N E, where N E is the absoute vaue of the norm of the conductor. This answers affirmativey a question of Serre in [12]. MSC cass: 11G05. 1 Introduction Let E be an eiptic curve over a number fied K, and for each prime number, et E[] be the group of -torsion points of E over K. This group is isomorphic to (Z/Z) 2 and has action by the absoute Gaois group G K := Ga(K/K), which we denote ρ E, : G K GL(E[]) GL 2 (F ). The coection of representations ρ E, encode many important properties of E, such as its primes of bad reduction and its number of points over finite fieds. As ong as E has no compex mutipication (CM), these representations are surjective for a but finitey many, which we ca exceptiona primes for E. This resut was proven in Serre s 1968 paper [11], and concuded the proof of the ong-conjectured Open Image Theorem the statement that the inverse imit of the images im ρ E,m (G K ) im GL m 2 (Z/mZ) m Z Harvard University Department of Mathematics earson3@gmai.com Massachusetts Institute of Technoogy Department of Mathematics mitkav@math.mit.edu 1

2 has finite index in im m GL 2 (Z/mZ) = GL 2 (Ẑ). Serre s origina proof was ineffective, even over the ground fied Q. But in the ater paper [12], he gave in the case of K = Q an expicit upper bound on the argest exceptiona prime of an eiptic curve E over the rationa numbers without CM, conditionay on the Generaized Riemann Hypothesis (GRH). Namey he showed that the argest exceptiona prime E is bounded by the foowing expression in the conductor N E of the eiptic curve: E C 1 og N E (og og N E ) 3, (1) for C 1 an absoute (and effectivey computabe) constant. In the same paper, he conjectured that, conditionay on GRH, a simiar bound shoud hod for eiptic curves defined over arbitrary number fieds K. An effective bound over arbitrary number fieds K was ater given, unconditionay, by the paper of Masser and Wüsthoz [10], with bound C 2 max(h E, n K ) γ for absoute constants C 2 and γ, where h E is the ogarithmic height of the j-invariant of E and n K is the degree of K. By Theorem 4.2 of [8], we can take γ = 2 if we et C 2 depend on K; our resuts impy that conditionay on GRH, we can take γ = 1 + ɛ if we et C 2 depend on K. Over Q, Kraus and Cojocaru ([5] and [2]) gave another unconditiona bound in terms of the conductor using the moduarity of eiptic curves over Q, namey E C 3 N E (og og N E ) 1/2. Moreover, in [14], Zywina shows that the product A E := exceptiona for E can be bounded by the b E th power of each of the above bounds on E, where b E is the number of primes of bad reduction for E. The gradua improvements in the bound on exceptiona primes have paid off. A recent paper of Biu and Parent [1] which made a breakthrough in the search for a uniform bound on exceptiona primes over Q (showing that X spit ()(Q) consists ony of CM points and cusps for sufficienty arge) reied cruciay the vaue of γ appearing in the Masser- Wüsthoz bound. This paper continues this tradition. We bound, conditionay on GRH, both the argest exceptiona prime E and the product of a exceptiona primes A E. Our proof is in the spirit of Serre s origina bound in [12], but we aow E to be defined over an arbitrary number fied K, which entais a more deicate anaysis. The bound on the argest exceptiona prime we get is, as conjectured in [12], the same as what Serre obtained when K = Q (equation (1), with the constant C 1 repaced by a constant C(K) depending on the number fied K). In fact we show that an asymptoticay better bound hods. Namey, conditionay on GRH, the argest exceptiona prime E satisfies E C (K) og N E, 2

3 where N E is the absoute vaue of the norm of the conductor of E and C (K) is a constant depending on K. We make the constant C(K) in our first bound effective, but have at the moment no effective way of determining the constant C (K) in the second, asymptoticay better bound (even over K = Q). We aso give a conditiona bound on the product of a exceptiona primes, A E. We show, in particuar, that for fixed K and fixed ɛ > 0, we have A E < NE ɛ for a but finitey many curves E. The bound one woud get by mutipying together a primes up to our upper bound for E as we as bounds on A E given in earier papers give vaues which are asymptotic to a positive power of N E. Our proof can be roughy outined as foows. First we compare an exceptiona prime and an unexceptiona prime p, and show that the two Gaois representations ρ E, and ρ E,p impose conditions on traces of Frobenius of E which are incompatibe if is sufficienty arge compared to p and N E. This part reies on the effective Chebotarev Theorem of Lagarias and Odyzko together with a resut of our paper [7]. Next, we give an upper bound for the smaest unexceptiona prime p. The anaysis here bifurcates. Ineffectivey, it can be easiy shown that the smaest such p is bounded above by a constant depending ony on K. The effective bound is trickier, and uses Serre s method in [12], which depends on GRH in an essentia way. Combining the bound on the unexceptiona p with the bound on the exceptiona in terms of p competes the proof. We then show that the bound on can be tweaked to give an upper bound on the product A E of a exceptiona primes. Throughout the paper, we treat separatey two different kinds of exceptiona primes: those for which ρ E, is absoutey irreducibe, and those for which it is not. Whie the anaysis in the two cases is remarkaby parae, our bound on the product of exceptiona primes of the second kind (such that ρ E, is reducibe over F ) turns out to be significanty better, poynomia in og N E (see Lemma 17). Notationa Conventions. Fix a number fied K, and write n K, r K, R K, h K, and K for the degree, rank of the unit group, reguator, cass number, and discriminant of K respectivey. Let us choose for every prime idea v of K, a corresponding Frobenius eement π v G K := Ga(K/K). We et E be an eiptic curve without compex mutipication (CM). Uness otherwised specified, this wi be taken to mean without CM over K. Write N E and a E for the absoute vaue of the norm of the conductor of E, and the number of primes of additive reduction of E, respectivey. We say that X K Y if there are effectivey computabe constants A and B depending ony on K for which X AY + B. Moreover, we say that X K Y if X AY + B for constants A and B that are not assumed to be effectivey computabe. If the constants A and B are absoute, we drop the K subscript on the and. With this notation, our resuts are as foows. 3

4 Theorem 1 (Theorem 23). Assume GRH. Let E be an eiptic curve over a number fied K without CM. Then any exceptiona prime satisfies K og N E. Moreover, the product of a exceptiona primes satisfies K 4 ae (og N E ) 14. Theorem 2 (Theorem 25). Assume GRH. Let E be an eiptic curve over a number fied K without CM. Then any exceptiona prime satisfies K og N E (og og N E ) 3. Moreover, the product of a exceptiona primes satisfies K 4 ae (og N E ) 14 (a E + og og N E ) 6 (og og N E ) 36 K 4 ae (og N E ) 21. Acknowedgements We woud ike to thank David Zureick-Brown, Bryden Cais and Ken Ono for suggesting us the probem that ed to this paper and many hepfu discussions. Thanks aso to Barry Mazur for vauabe comments and suggestions. 2 Possibe Images of the Representation ρ E, In this section, we anayze the possibe images of ρ E,. The proofs of a of the resuts of this section are in the papers [11] and [12] by Serre. We begin by singing out some subgroups of GL 2 (F ). Definition 3. A Cartan subgroup is a subgroup of GL 2 (F ) GL 2 (F ) that consists of a eements which preserve two one-dimensiona subspaces W 1 and W 2 of F 2, where W 1 and W 2 are either both defined over F, or are both defined over F 2 and are Gaois conjugate. In some basis of F 2, such a subgroup contains a eements of GL 2 (F ) that ook ike ( ) 0. 0 A Cartan subgroup is index two in its normaizer. The normaizer consists of matrices of the form ( ) ( ) 0 0 or 0 0 (i.e. matrices which either fix or permute the two subspaces fixed by the Cartan subgroup). 4

5 Lemma 4. Let G be any subgroup of GL 2 (F ). Then, one of the foowing hods: 1. (Reducibe Case) G acts reduciby on F (Normaizer Case) G is contained in the normaizer of a Cartan subgroup, but not in the Cartan subgroup itsef. 3. (Specia Linear Case) G contains SL 2 (F ). 4. (Irreguar Case) The image of G under the projection GL 2 (F ) PGL 2 (F ), is contained in a subgroup which is isomorphic to A 4, S 4, or A 5. Remark 5. We use the term irreguar subgroup to avoid a cash of notation; usuay they are caed exceptiona subgroups. Proof. See Section 2 of [11]. Definition 6. Having fixed the fied K, we ca a prime number p acceptabe if p is unramified in K/Q and p 53. (So amost a primes are acceptabe.) Lemma 7. If p is acceptabe, then inertia at p provides an eement in the image of Pρ E,p of order at east 13. Proof. This foows from Lemma 18 of [12] (which is stated for K = Q, but the same proof works as ong as p is unramified in K). Lemma 8. Let be an acceptabe exceptiona prime. Then the image of ρ E, fas into either the reducibe case or the normaizer case of Lemma 4. Proof. Since K, it foows that det ρ E, is surjective, so the image of ρ E, cannot fa into case 3 because is exceptiona. By Lemma 7, the image of ρ E, cannot fa into case 4. The two remaining cases wi require separate anaysis, and throughout the paper we wi separate them as the reducibe case and the normaizer case. 3 The Effective Chebotarev Theorem We have the foowing effective version of the Chebotarev Density Theorem, due to Lagarias and Odyzko. Theorem 9 (Effective Chebotarev). Assume GRH. Let L/K be a Gaois extension of number fieds with L Q. Then every conjugacy cass of Ga(L/K) is represented by the Frobenius eement of a prime idea v such that Nm K Q (v) (og L) 2. 5

6 Proof. See [6], remark at end of paper regarding the improvement to Coroary 1.2. Coroary 10 (Effective Chebotarev with avoidance). Assume GRH. Let L/K be a Gaois extension of number fieds with L Q and Σ Σ K a finite set of primes which incudes the primes at which L/K is ramified. Let N be the norm of the product of the primes in Σ, and write d = [L : K]. Then every conjugacy cass of Ga(L/K) is represented by the Frobenius eement of a prime idea v Σ K Σ such that Nm K Q (v) d2 ( og N + og K + n K og d ) 2 K d 2 ( og N + og d ) 2. Proof. Let H be the Hibert cass fied of K, of degree h K over K. Then H = h K K, so any eement of the cass group is represented by a prime idea v Σ K of norm (h K og K ) 2. It foows from a resut of Lenstra (Theorem 6.5 in [9]) that h K 3/2 K, so we can take Nm(v) 4 K. Now, we et I = v Σ v, and appy this resut to the image in the cass group of the idea I 1. We get a prime idea v 0 with Nm(v 0 ) 4 K such that v 0I is principa, generated by x K. Define L = L[ 3 x, ω], for a primitive cube root of unity ω. The set Σ Σ K of primes ramified in L /K consists of a eements of Σ, pus some primes dividing 6v 0. Now, we appy effective Chebotarev again, to Ga(L /K), to concude that every conjugacy cass of Ga(L /K) is represented by a Frobenius eement of a prime idea v Σ K which is unramified in L, and thus not in Σ, with Nm K Q (v) (og L )2. We now turn to bounding og L. For a prime v of K, write e v and f v for the ramification and inertia degrees of v respectivey. We have og L = [L : K] og K + og Nm L K d L K 6d og K + v Σ ((6d 1)f v og p v + 6df v e v va pv (d) og p v ) 6d (og K + v Σ f v og p v + v Σ f v e v va pv (d) og p v ) 6d (og K + og(n 6 4 K) + n K og d ) d (og N + og K + n K og d). Throughout the paper, we wi frequenty appy the above coroary to Gaois representations buit out of the representations ρ E,. For this purpose, reca the we-known Néron-Ogg-Shafarevich criterion: Theorem 11 (Néron-Ogg-Shafarevich). Let E be an eiptic curve over K. Then ρ E, is ramified ony at primes dividing and the conductor of E. Proof. This is we known; see e.g. Proposition 4.1 of [13]. 6

7 4 Bounds In Terms of The Smaest Unexceptiona Prime Reca that we have fixed an eiptic curve E over a number fied K, and is an exceptiona prime for (E, K). In this section we give bounds on both the argest exceptiona prime and the product of a exceptiona primes, in terms of the smaest unexceptiona prime. 4.1 The Reducibe Case Suppose that E[] is reducibe over F, and write ( ρ E, F F = ψ (1) 0 ψ (2) Theorem 12. Assume GRH. There exists a finite set S K of primes numbers depending ony on K such that if / S K, then there exists a CM eiptic curve E, which is defined over K and whose CM-fied is contained in K, such that for some character ɛ : Ga(K/K) µ 12, { ψ (1) = ϕ (1) ɛ ψ (2) = ϕ (2) ɛ 1 ) where ρ E, F F =. ( ϕ (1) 0 0 ϕ (2) ). (2) Moreover the eiptic curve E depends ony on E (i.e. is independent of ), and ɛ is ramified ony at primes dividing the conductor of E or the discriminant of K. Proof. See Theorem 1 of [7] and Remark 1.1 foowing the theorem. This ets us reate the Frobenius poynomias of E and E at sma primes of K. We make the foowing definitions. Definition 13. Fix E and E as above. We define R E to be the product of a reducibe primes satisfying equation (2) for some character ɛ : Ga(K/K) µ 12. The fact that E depends ony on E (for K 1) impies that K R E. ρ E, reducibe (Moreover, this is sharp, as R E divides the product on the eft.) Definition 14. For a monic P Z[x], define its 12th Adams operation Ψ 12 P Z[x] to be the (monic) poynomia whose roots (in C) are the twefth powers of the roots of P. Using this notation and writing P E (v) = x 2 + Tr E (π v )x + Nm(v) for the Frobenius poynomia of π v G K, we have the foowing resut (where E is the CM eiptic curve from above). 7

8 Lemma 15. Let v be a prime of K at which E has good reduction. If 4(Nm v) 6 < R E, then Ψ 12 P E (v) = Ψ 12 P E (v); moreover, if R E is such that 4 Nm v < and ɛ (π v ) = 1 (where ɛ : G K µ 12 is as in Theorem 12), then P E (v) = P E (v). Proof. Suppose R E, i.e. satisfies equation (2) for some ɛ. Hence, (ψ (1) ) 12 = (ϕ (1) ) 12 and (ψ (2) ) 12 = (ϕ (2) ) 12, i.e. Ψ 12 P E Ψ 12 P E mod. Since this hods for a R E, by pugging in v we obtain Ψ 12 P E (v) Ψ 12 P E (v) mod R E. If moreover ɛ (π v ) = 1, then ψ (1) (π v ) = ϕ (1) (π v ) and ψ (2) (π v ) = ϕ (2) (π v ). Equivaenty, P E (v) P E (v) mod. From the Wei bounds, P E0 (v) has nonpositive discriminant and constant term Nm v for any eiptic curve E 0 and prime v of good reduction for E 0. In other words, P E0 (v) = x 2 ax + Nm v and Ψ 12 P E0 (v) = x 2 bx + Nm v 12, with a 2 Nm v and b 2(Nm v) 6. It foows that P E (v) P E (v) = Ax for some A 4 Nm v and Ψ 12 P E Ψ 12 P E = Bx for some B 4(Nm v) 6. On the other hand, we have seen above that A and R E B. The emma foows, using that A < and A impy A = 0 (and simiary for B). Now we are in a position to bound any prime with reducibe ρ E, (or the product of a such) in terms of a sma irreducibe prime p. Lemma 16. Suppose that p is an acceptabe prime of irreducibe type. Let E be as in Theorem 12, and et H GL 2 (F p ) GL 2 (F p ) be the image of ρ E,p ρ E,p. Then there exists a surjective homomorphism f : H G to some group G with G p 3, and a g G such that for any (X, Y ) H with f(x, Y ) = g, we have Tr(X 12 ) Tr(Y 12 ). Proof. First suppose that p is unexceptiona. By the theory of compex mutipication, the image of ρ E,p is contained in a Cartan subgroup. Hence, the image of the projectivization Pρ E,p is contained in a cycic group of order p ± 1. Since p was assumed acceptabe, p ± It foows that there is an M PGL 2 (F p ) whose 12th power is not conjugate to anything in the image of Pρ E,p. Taking G = PGL 2 (F p ) and f : GL 2 (F p ) GL 2 (F p ) G to be projection onto the first factor foowed by the projection GL 2 (F p ) PGL 2 (F p ) competes the proof in this case. 8

9 Hence we can assume that our second prime p is of normaizer type. Since both Cartan subgroups and their normaizers in GL 2 (F p ) have p 2 eements, and det ρ E,p = det ρ E,p is surjective onto F p, the image of ρ E,p ρ E,p has order p 3. By Lemma 7, the projective image of Pρ E,p contains an eement of order at east 13. In particuar, in some basis of F 2, it must contain something of the form A = ( a 0 0 b with a 12 b 12. Let B be an eement in the image of ρ E,p that is not in the Cartan group. Since the image of ρ E,p is abeian, it foows that the image of ρ E,p ρ E,p contains (M, 1), where M = ABA 1 B 1. By expicit computation, M = ), ( ab ba 1 Since a 12 b 12, we have (ab 1 ) 12 + (ba 1 ) Taking G = H and g = (M, 1) thus competes the proof. Lemma 17. Assume GRH. Let p be the smaest acceptabe prime of irreducibe type. Then, R E K p 36 (og N E + og p) 12. Moreover, any prime R E satisfies ). K p 3 (og N E + og p). Proof. Let f : H G and g G be as in Lemma 16. First, we bound R E. By Coroary 10 appied to g G, Néron-Ogg-Shafarevich, and Lemma 16, there is a prime v of good reduction for E such that Tr ρ E,p (πv 12 ) Tr ρ E,p(πv 12 ), which moreover satisfies Nm v K p 6 (og N E + og p) 2. (3) In particuar, Ψ 12 P E (v) Ψ 12 P E (v), so by Lemma 15 we have R E 4(Nm v) 6 K p 36 (og N E + og p) 12. To bound the argest exceptiona prime, we consider the direct sum of ɛ and G. Since ɛ has order 12, its image contains (1, g 12 ). Appying Coroary 10 to g 12 G, Néron- Ogg-Shafarevich, and Lemma 16, we can find a prime v of good reduction for E such that Tr ρ E,p (π v ) Tr ρ E,p(π v ) and ɛ (π v ) = 1, which satisfies the bound (3). In particuar, P E (v) P E (v), so Lemma 15 gives 4 Nm v K p 6 (og N E + og p), as desired. 9

10 4.2 The Normaizer Case Let be a prime such that the image of ρ E, fas into the normaizer case of Lemma 4. Write C for our Cartan subgroup and N for its normaizer. Then we have a quadratic character χ on Ga(Q/K) given by χ: Ga(Q/K) N N/C {±1}. Lemma 18. The character χ is ramified ony at paces of bad additive reduction. Proof. See Lemma 2 in Section 4.2 of [11]. In this case, we say that is χ-exceptiona (of normaizer type). More generay, if V Hom(G K, Z/2) is a finite-dimensiona F 2 -vector space of Gaois characters, we say that is V -exceptiona if is χ-exceptiona for some χ V. Note that the space V of characters induces a Gaois extension of K with Gaois group the dua F 2 -vector space V, via the foowing construction. Definition 19. For V Hom(G K, Z/2Z), we write ρ V : G K V for the map induced by the pairing V G ab K F 2. This gives (functoriay) a one-to-one correspondence between finite F 2 -vector spaces of Gaois characters and finite abeian fied extensions with Gaois group annihiated by 2. Lemma 20. The vector space V of a quadratic Gaois characters ramified ony at paces of bad additive reduction satisfies V 2 a E+2nK h K, where (as per our notationa conventions on page 3), a E is the number of primes of additive reduction for E. (In fact, the argument beow shows V 2 a E+2n K 2 r 2(C(K)), where r 2 (C(K)) is the 2-rank of the cass group.) Proof. Note that V = V. Write U K I K for the subgroup of K -mutipes of idèes whose non-archimedean components are integra units at a paces. By cass fied theory, ρ V induces a surjection I K V. Since [I K : U K ] = h K, it suffices to show that ρ V (U K ) V has order at most 2 a E+2n K. However, the restriction ρ V UK factors through the projection U K Ov/(O v) 2. v of additive reduction Now by a standard appication of Hense s emma, if p v 2 then O v/(o v) 2 = F 2, and if p v = 2 then O v/(o v) 2 is a vector space over F 2 of dimension at most 2e v f v. Since v 2 2e vf v = 2n K, this gives the desired bound. 10

11 Lemma 21. Assume GRH. Let V be a d-dimensiona vector space of quadratic Gaois characters ramified ony at paces of bad additive reduction, and et p be the smaest acceptabe prime of irreducibe type that is not V -exceptiona. Then the product of a V -exceptiona primes satisfies ( ) 2 2 K 2 d p 3 1 d (og N E + og p). In particuar, each exceptiona prime of normaizer type satisfies K 2 d p 3 (og N E + og p) (as it is V -exceptiona for V = χ of dimension d = 1). Proof. We start by showing that for any α V, there is some X α PGL 2 (F p ) of nonzero trace such that (α, X α ) is contained in the image of ρ V Pρ E,p. If p is unexceptiona, then Pρ E,p surjects onto PGL 2 (F p ). Hence, the abeianization of Pρ E,p is the quadratic character defined by PGL 2 (F p )/ PSL 2 (F p ). Since V is an abeian group, the image of ρ V Pρ E,p contains everything of the form (α, X) either for every X PSL 2 (F p ), or for every X / PSL 2 (F p ). Either way, the image contains something of the form (α, X α ) where X α has nonzero trace. If p is exceptiona, then by assumption p must be of normaizer type; write C for the Cartan subgroup. Pick some Y α so that (α, Y α ) = (ρ V ρ E, )(g α ) is in the image of ρ V Pρ E,p. Since p is not V -exceptiona, we can choose Y α so that Y α C. If Tr(Y α ) 0, we are done, so suppose Tr(Y α ) = 0. From Lemma 7, there is an eement Z α = Pρ E,p (h α ) of order greater than four in the image of ρ E,p (which must ie C). Now we can take X α = Y α Zα 2 which satisfies (ρ V ρ E,p )(g α h 2 α) = (α, Y α Zα) 2 and has nonzero trace, as desired. Now, for each α V, et X α be the eement constructed above. Appying Coroary 10 and Néron-Ogg-Shafarevich, we can find a prime idea v α with (ρ V Pρ E,p )(π vα ) = (α, X α ), which moreover satisfies Nm v α K 4 d p 6 (og N E + og p + d) 2 K 4 d p 6 (og N E + og p) 2. (The ast inequaity foows from Lemma 20, using a E K og N E.) This gives by the Wei bound that for any α V we can choose v α so that exceptiona of normaizer type 0 Tr E (π vα ) K 2 d p 3 (og N E + og p). (4) Now, Tr E (π vα ) must be divisibe by a V -exceptiona primes whose corresponding character χ satisfies χ (π vα ) = 1. But for any χ, haf of the α V satisfy χ (π vα ) = 1. Putting this together, 2 d 1 ( 2 Tr E (π vα ) c K 2 d p 3 (og N E + og p)) d 1, α 0 V 11

12 where c K is the effective constant impicit in equation (4). Taking the 2 d 1 st root of both sides yieds the desired concusion. 5 The Ineffective Bound Lemma 22. If p is the smaest acceptabe unexceptiona prime for an eiptic curve E without CM, then p K 1. Proof. By Serre s Open Image Theorem [11], it suffices to verify the statement for a but finitey many eiptic curves E over K. In order to do this, et p be some acceptabe prime. In particuar, p 23, so the genera of the moduar curves X 0 (p), X spit (p), and X nonspit (p) are a at east 2. By Fating s theorem [3], there are finitey many points on each of these moduar curves. Because repacing E by a quadratic twist does not change whether or not p is exceptiona, this competes the proof. Theorem 23. Assume GRH. Let E be an eiptic curve over a number fied K without CM. Then any exceptiona prime satisfies K og N E. Moreover, the product of a exceptiona primes satisfies K 4 ae (og N E ) 14. Proof. This is an immediate consequence of Lemmas 17, 21, and The Effective Bound The bound on the smaest unexceptiona prime p in the previous section reies on Fating s theorem, which at the moment is ineffective. Here we give an effective bound on p (which depends on the curve E, but quite genty), using the resuts of Section 4. Lemma 24. Let S be a finite set of primes, p be the smaest acceptabe prime number not in S, and b be a constant depending ony on K. Then for any A, K A p b p K og A. S Proof. Since the product of a unacceptabe primes depends ony on K, it suffices to prove this emma in the case that S contains a of the unacceptabe primes. Using the prime number theorem (which is effective), p <p ( ) og og S ( K og A p b) K og A + og p. 12

13 (The effectiveness of the prime number theorem is ony needed to insure the effectiveness of the K above; any effective error term wi suffice. For exampe, the origina proof of the prime number theorem [4] gives π(x) Li(x) xe c og x for c effective.) The desired resut foows immediatey. Theorem 25. Assume GRH. Let E be an eiptic curve over a number fied K without CM. Then any exceptiona prime satisfies K og N E (og og N E ) 3. Moreover, the product of a exceptiona primes satisfies K 4 ae (og N E ) 14 (a E + og og N E ) 6 (og og N E ) 36. Proof. From the bound on the product in Lemma 17, together with Lemma 24, we concude that the smaest acceptabe prime p of irreducibe type satisfies p K og og N E. Lemma 17 thus gives that the product of a primes of reducibe type is bounded by (og N E ) 12 (og og N E ) 36, up to a constant depending on K. Combining this with the bound on the product in Lemma 21, together with Lemma 24, the smaest acceptabe prime p of irreducibe type that is not V -exceptiona satisfies p K og og N E + dim V. Thus, Lemmas 17 and 21 impy the desired resut. 7 Expicit Constants In this section, we estimate the dependence on K in Theorem 2. Everything used to prove Theorem 2 bois down to the effective Chebotarev theorem (for which the K-dependence is expicit), and Theorem 12. To make Theorem 12 effective, we can use the foowing resut: Theorem 26. Assume GRH. In Theorem 12, the product of a S K is bounded by: ( exp c n K 0 (R K n r K K + h 2 K (og K ) 2 ) ), where c 0 is an effectivey computabe absoute constant. (In particuar, every S K is bounded by this expression.) Proof. See Theorem 7.9 of [7]. Theorem 27. Assume GRH. Let E be an eiptic curve over a number fied K without CM. Then any exceptiona prime satisfies og N E (og og N E ) 3 + exp ( c nk (R K n r K K + h 2 K (og K ) 2 ) ). 13

14 Moreover, the product of a exceptiona primes satisfies 4 ae (og N E ) 14 (a E + og og N E ) 6 (og og N E ) 36 exp ( c nk (R K n r K K + h 2 K (og K ) 2 ) ). Here, the constant c and the constants impied by the symbo are a absoute and effectivey computabe. Proof. This foows from carefuy keeping track of the contributions depending on K in the proof of Theorem 2. It is easy to see that the contributions from the bounds given on the set S K dominate a other contributions coming from the fied K. References [1] Yuri Biu and Pierre Parent. Serre s uniformity probem in the spit Cartan case. Ann. of Math. (2), 173(1): , [2] Aina Carmen Cojocaru. On the surjectivity of the Gaois representations associated to non-cm eiptic curves. Canad. Math. Bu., 48(1):16 31, With an appendix by Ernst Kani. [3] G. Fatings. Endichkeitssätze für abesche Varietäten über Zahkörpern. Invent. Math., 73(3): , [4] J. Hadamard. Sur a distribution des zéros de a fonction ζ(s) et ses conséquences arithmétiques. Bu. Soc. Math. France, 24: , [5] Aain Kraus. Une remarque sur es points de torsion des courbes eiptiques. C. R. Acad. Sci. Paris Sér. I Math., 321(9): , [6] J. C. Lagarias and A. M. Odyzko. Effective versions of the Chebotarev density theorem. In Agebraic number fieds: L-functions and Gaois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pages Academic Press, London, [7] Eric Larson and Dmitry Vaintrob. Determinants of subquotients of gaois representations associated to abeian varieties. Avaiabe at [8] Samue Le Fourn. Surjectivity of gaois representations associated with quadratic q-curves. Avaiabe at [9] H. W. Lenstra, Jr. Agorithms in agebraic number theory. Bu. Amer. Math. Soc. (N.S.), 26(2): , [10] D. W. Masser and G. Wüsthoz. Gaois properties of division fieds of eiptic curves. Bu. London Math. Soc., 25(3): ,

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