On the effective version of Serre's open image theorem

Let E/Q$E/\mathbb {Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod ℓ$\ell$ Galois representation ρ¯E,ℓ$\overline{\rho }_{E, \ell }$ of E$E$ is surjective for each prime number ℓ$\ell$ that is sufficiently large. Under the generalized Riemann hypothesis, we give an explicit upper bound on the largest prime ℓ$\ell$ , linear in the logarithm of the conductor of E$E$ , such that ρ¯E,ℓ$\overline{\rho }_{E, \ell }$ is nonsurjective.


Introduction
Let E be an elliptic curve defined over Q.For a prime number ℓ, let E[ℓ] denote the ℓ-torsion subgroup of E(Q) and let T ℓ (E) denote the ℓ-adic Tate module of E. Recall that E[ℓ] and T ℓ (E) are free modules of rank two over F ℓ and Z ℓ , respectively, where F ℓ denotes the finite field with ℓ elements and Z ℓ denotes the ring of ℓ-adic integers.Fixing bases, we obtain the module isomorphisms The absolute Galois group Gal(Q/Q) acts coordinate-wise on elements of E[ℓ] and on T ℓ (E).These actions respect the above isomorphisms, and give rise to the mod ℓ Galois representation and ℓ-adic Galois representation of E, which are denoted respectively by ρE,ℓ : The above Galois representations carry a considerable amount of information about E. For instance, consider the reduction E p of the curve E at a prime number p that is distinct from ℓ.The well-known Néron-Ogg-Shafarevich criterion gives that E has good reduction at p if and only if ρ E,ℓ is unramified at p. Further, if E has good reduction at p, then tr ρ E,ℓ (Frob p ) = a p (E) and det ρ E,ℓ (Frob p ) = p, where Frob p ∈ Gal(Q/Q) denotes a Frobenius automorphism associated with p and a p (E) is defined by the equation #E p (F p ) = p + 1 − a p (E).
Suppose from now on that E is without complex multiplication, i.e., assume that the geometric endomorphism ring of E is trivial.A celebrated theorem of Serre [40], known as the open image theorem, establishes that if ℓ is sufficiently large, then ρE,ℓ is surjective.Let c(E) denote the least positive integer such that if ℓ > c(E), then ρE,ℓ is surjective. 1 Serre asked if c(E) ≤ 37 holds for each elliptic curve E/Q without complex multiplication.This is known as Serre's uniformity question and, following theoretical advances and vast numerical evidence, is now articulated in precise conjectures of Sutherland [47,Conjecture 1.1] and Zywina [53,Conjecture 1.12].Mazur's landmark work on modular curves implies that c(E) ≤ 11 if E is semistable [34,Theorem 4].Further progress toward resolving Serre's question has since been made by studying modular curves, which we discuss in §2.In an adjacent direction, there has been progress in bounding c(E) in terms of invariants of E, such as the Faltings height h E or the conductor N E of E, which we address in this paper.
Serre's original proof of the open image theorem is ineffective and does not give a bound on c(E).The first unconditional bound on c(E) is due to Masser and Wüstholz [33] who proved in 1993 that there exist absolute constants C 1 and γ such that c(E) ≤ C 1 h γ E .This bound has subsequently been improved and made explicit in [28,32,39].Under the now superfluous assumption that E is modular [8], in 1995 Kraus [23] where rad n := p|n p denotes the radical of an integer n.In 2005, Cojocaru [12] proved independently using a similar approach that ( Serre mentioned without proof in [44,Note 632.6] that the "log log" term in (1) may be removed by employing an ℓ-adic technique of Faltings [21, §6, pp. 362-363].The technique and its extension is called the Faltings-Serre method (see [9,Section 2]) and is widely recognized for its important role in establishing various modularity results: such as the modularity of elliptic curves over (infinitely) many imaginary quadratic fields [11,14,18,22,48], the paramodularity of certain abelian surfaces with a trivial endomorphism ring [7,9], and the modularity of some Calabi-Yau threefolds [15,19].
In 2014, Larson and Vaintrob [27, Theorem 1] proved, without applying the Falting-Serre method, that under GRH, there exists an absolute constant C 3 such that Not only is their bound linear in log N E , but it also holds over an arbitrary number field K (with C 3 depending only on K).However, even over Q, the bound (2) is ineffective in the sense that no method is presently known for computing C 3 .In order to compute C 3 via the proof in [27], one would need to understand the rational points on the modular curve X + ns (ℓ) for some prime ℓ ≥ 53.The main result of this paper is an explicit conditional bound on c(E) of the same asymptotic quality as (2) for elliptic curves over Q.Specifically, we shall prove the following.
where rad n := p|n p denotes the radical of an integer n.
Our proof of Theorem 1 follows the strategy set forth by Serre in [41, Théorème 22] and [44,Note 632.6].The key improvement comes from the following sharpening of his conditional bound in the effective version of Faltings's isogeny theorem for elliptic curves [41,Théorème 21].
Theorem 2. Assume GRH.Let E 1 /Q and E 2 /Q be elliptic curves without complex multiplication.Suppose that E 1 and E 2 are not Q-isogenous.Then there exists a prime number p of good reduction for E 1 and E 2 such that a p (E 1 ) = a p (E 2 ) satisfying the inequality The structure of our paper is as follows.In §3, we give a variant of the effective Chebotarev density theorem due to Bach and Sorenson.We use this tool, together with a refinement of a technique of Faltings, to prove Theorem 2 in §4.With this result in hand, in §5, we follow in the footsteps of Serre's elegant proof of (1) to complete our proof of Theorem 1. Afterward, we illustrate our result with a numerical example in §5.3.
We conclude the introduction with some remarks on extensions of the effective Serre's open image theorem.
We recall from the modularity theorem [49,52] that Galois representations of elliptic curves over Q arise from Galois representations of weight 2 cuspidal eigenforms.Therefore, Theorem 1 can also be interpreted as an effective open image theorem for modular forms of weight 2. We refer the reader to [5,38] for other effective results for higher weight modular forms.
Let K be a number field and E/K be a non-CM elliptic curve.Serre's open image theorem also applies over a number field, so we can define c(E) in a similar way as before.It is known that there is a uniform bound of c(E) for certain families of Q-curves E over a quadratic field K (see [28,30]).
The main result of this paper is to make explicit the conditional bound of Larson and Vaintrob [27, Theorem 1] for elliptic curves over Q.However, as their bound holds for elliptic curves over number fields, it is natural to ask if we could extend our result to E/K.The approach that we follow relies on Mazur's cyclic isogeny theorem.Generalizing Mazur's result to elliptic curves over arbitrary number fields appears to be challenging.Nonetheless, assuming GRH, if K is among a certain finite set of quadratic fields K, Banwait [3] and Banwait, Najman, and Padurariu [4], building on the earlier work of David [16], Larson and Vaintrob [26], and Momose [35], proved an analog of Mazur's cyclic isogeny theorem for E/K.Thus it is promising that one may be able to extend our work to give an explicit open image theorem for elliptic curves defined over these quadratic fields.

Progress toward Serre's uniformity question
The most significant progress toward a resolution of Serre's uniformity question comes from studying rational points on certain modular curves.By doing so, one limits the possibilities for G E (ℓ) := im ρE,ℓ , i.e., the image of ρE,ℓ .As G E (ℓ) is a subgroup of GL 2 (F ℓ ), in order to describe what is known in this direction, we first state a well-known classification of subgroups of GL 2 (F ℓ ) (which dates back to Dickson [17]) and give some necessary terminology.
Let ℓ be an odd prime number and fix a nonsquare element ε ∈ F × ℓ .The split Cartan subgroup and non-split Cartan subgroup of GL 2 (F ℓ ) are, respectively, and C ns (ℓ), respectively.One may show that Let B(ℓ) denote the Borel subgroup of GL 2 (F ℓ ), i.e., the subgroup of upper triangular matrices, Let A n and S n denote the alternating and symmetric groups, respectively, on n elements.Finally, for a subgroup G of GL 2 (F ℓ ), let G denote the image of G in the projective linear group PGL 2 (F ℓ ).
With notation set, we now state the classification.For further details, we refer the reader to [40, §2].
Returning to the world of elliptic curves, recall that by the Weil pairing on E, the composition Consequently, in order to prove that ρE,ℓ is surjective for a particular prime number ℓ, it suffices to rule out possibilities (2) through ( 6 Theorem 4. Let E/Q be an elliptic curve without complex multiplication.If ℓ is a prime number such that ℓ > 37, then either ρE,ℓ is surjective or G E (ℓ) is conjugate to a subgroup of C + ns (ℓ) but is not conjugate to any subgroups of C ns (ℓ).

The effective Chebotarev density theorem
In this section, we offer a modest extension of the version of the effective Chebotarev density theorem given in [1].This extension will serve as a crucial tool in our proof of Theorem 1. First, let us fix some relevant notation.
Let K be a number field with absolute discriminant d K and ring of integers O K .For a prime ideal p ⊆ O K , we write p | p to indicate that p lies above a prime number p. Let v p : K → Z denote the normalized p-adic valuation on K.We write N (p) for the ideal norm of p, which extends multiplicatively to arbitrary ideals of O K .One has that v p (p) = e p and N (p) = p fp , where e p and f p are the ramification index and inertia degree of p, respectively.
Assume that K/Q is Galois.For a prime number p that is unramified in K/Q, write ( p K/Q ) to denote the Artin symbol of K/Q at p. Let C be a subset of the Galois group Gal(K/Q) that is closed under conjugation.Associated with C, consider the counting function The Chebotarev density theorem states that where π(x) is the prime counting function and f ∼ g means that lim x→∞ f (x) g(x) = 1.Assuming GRH, Lagarias and Odlyzko [24] gave an effective version of the Chebotarev density theorem that provides an error term in (4).Moreover, they showed that there exists an absolute constant k such that the least prime number p with satisfies the inequality p ≤ k(log d K ) 2 .Oesterlé [37] stated that the absolute constant k may be taken to be 70.Subsequently, Bach and Sorenson offered the following improvement.
Theorem 5. Assume GRH.Let K be a Galois number field and let C ⊆ Gal(K/Q) be a nonempty subset that is closed under conjugation.Then there exists a prime number p that is unramified in K/Q for which (5) holds that satisfies the inequality where a, b, and c are absolute constants that may be taken to be 4, 2.5, and 5, respectively, or may be taken to be the improved values given in [1, Table 3] associated with K.
For our application, we need an extension of Theorem 5 that allows for the avoidance of a prescribed set of primes.Corollary 6. Assume GRH.Let K be a Galois number field, let m be a squarefree positive integer, and set K := K( √ m).Let C ⊆ Gal(K/Q) be a nonempty subset that is closed under conjugation.Then there exists a prime number p not dividing m that is unramified in K/Q for which (5) holds that satisfies the inequality p ≤ (ã log where ã, b, c are absolute constants that may be taken to be 4, 2.5, and 5, respectively, or may be taken to be the improved values given in [1, Table 3] associated with K.
Proof.Notice that K is Galois over then each prime number dividing m is ramified in K/Q, so Theorem 5 provides the desired result.Thus, we assume that K = K.
Let res : Gal( K/Q) −→ Gal(K/Q) denote the restriction map and consider the subset C := res −1 (C) of Gal( K/Q).Since C is closed under conjugation in Gal(K/Q), C is closed under conjugation in Gal( K/Q).By applying Theorem 5 to K and C, we obtain a prime number p that is unramified in K for which (5) holds and satisfies the inequality (6).Note that K is ramified at the ramified primes of K and at the prime divisors of m (and possibly at 2).Thus p is unramified in K and does not divide m.Finally, because p K/Q ⊆ C and res In the corollary, we see that p is bounded above in terms of [ K : Q] and log d K .In our application, the degree [ K : Q] will be absolutely bounded.Thus, it will remain to bound log d K .For this purpose, we employ the following lemma, which can be found in [41,Proposition 6].
Proof.The left-hand inequality follows from the Minkowski bound for the discriminant [41, p. 139].For the right-hand inequality, let D K ⊆ O K denote the different ideal of K and note that By taking logarithms, we obtain For each prime ideal p ⊆ O K lying above p, we have that v p (D K ) = e p − 1 + s p for some integer s p satisfying 0 ≤ s p ≤ v p (e p ) (see, e.g., [36,Theorem 2.6,p. 199]).Thus Since Observe that Applying ( 8), ( 9) and ( 10) to (7), we obtain The claimed inequality now follows by noting that p|d K log p = log rad(d K ).

An effective isogeny theorem for elliptic curves
The objective of this section is to provide an improved conditional bound on the effective version of Faltings's isogeny theorem for elliptic curves.We begin in §4.1 with some preliminaries.4.1.Ramified primes.Let A/Q be an abelian variety.For a positive integer m, let Q(A[m]) denote the m-division field of A, i.e., the field obtained by adjoining to Q the coordinates of all points of A(Q) of order dividing m.For a prime number ℓ, we write We now recall Serre and Tate's extension of the criterion of Néron-Ogg-Shafarevich to abelian varieties in order to specify which primes ramify in the infinite degree algebraic extension Q(A[ℓ ∞ ])/Q.Theorem 8. Let A/Q be an abelian variety.For a prime number p, the following are equivalent: (1) A has good reduction at p, Corollary 9. Let B A be the product of ℓ and the primes of bad reduction for A. The extension Proof.First, we note that ℓ is ramified in We continue the proof by showing that the extension Q(A[ℓ ∞ ])/Q is unramified at exactly the primes not dividing B A .Let p be a prime number such that p ∤ B A .Then A has good reduction at p, so it follows from the equivalence of ( 1) and ( 2) in Theorem 8 that in the chain of subfields Provided that E 1 and E 2 are without complex multiplication and not Q-isogenous, Serre proved in [41,Théorème 21] that under GRH, the least prime number p ∤ N E 1 N E 2 for which equality in (11) fails to hold satisfies the inequality for some absolute, computable constant C 4 .In this section, we improve upon the bound in ( 12) by removing the "log log" factor.We begin by offering a refinement of a proposition due to Serre.Serre mentioned his result, without proof, in [44,Note 632.6].Much later, he communicated an elegant proof that appears in [10,Theorem 4.7].Our proof builds on the one appearing in [10] by considering a quotient by scalar matrices as in the proof of [41,Théorème 21'].Our bound on the order of G (as below) coincides with Serre's when ℓ = 2, which is the prime for which we will apply the result in the proof of Theorem 2. However, it does offer an improvement in the case when ℓ = 2, so we find it worthwhile to include it nonetheless.
Proposition 12. Let ℓ be a prime number and r be a positive integer.Let Γ be a group and ρ 1 , ρ 2 : Γ −→ GL r (Z ℓ ) be group homomorphisms.Suppose there is an element γ ∈ Γ such that tr ρ 1 (γ) = tr ρ 2 (γ).Then there exists a quotient G of Γ and a subset C ⊆ G for which (1) the order of G is at most ℓ 2r 2 −1 ℓ−1 , (2) the set C is nonempty and closed under conjugation in G, and (3) if the image in G of an element γ ∈ Γ belongs to C, then tr ρ 1 (γ) = tr ρ 2 (γ).
Proof.Let M := Mat r×r (Z ℓ ) be the Z ℓ -algebra of the r × r matrices with coefficients in Z ℓ .Let A denote the Z ℓ -algebra generated by the image of Γ under the product map Let G ′ be the image of Γ under ρ 1 × ρ 2 in A/ℓA.Because of the existence of the identity element in Γ, the algebra A contains the set of scalar matrices We write H to denote the image of Λ 2r \ℓΛ 2r in A/ℓA.Then we have the group isomorphism H ∼ = Z/ℓZ × , as the image of (µI r , µI r ) in A/ℓA is determined by the image of µ in Z/ℓZ.Clearly, H ∩ G ′ is a normal subgroup of G ′ .Consider the group G := G ′ /(H ∩ G ′ ).From the second isomorphism theorem in group theory, we have G ∼ = G ′ H/H.Since the rank of A as a free Z ℓmodule is at most 2r 2 and both G ′ H and H are groups in A/ℓA, we obtain the bound Let m be the largest nonnegative integer such that for each γ ∈ Γ, one has that tr ρ 1 (γ) ≡ tr ρ 2 (γ) (mod ℓ m ).
Let C be the set of elements in G whose preimages in G ′ all take nonzero values under λ.From the definition of m and the linearity of the trace map, there exists a γ 0 ∈ Γ such that and in particular tr ρ 1 (γ) = tr ρ 2 (γ).
We now employ the above proposition, together with the effective Chebotarev density theorem in the form of Corollary 6 to give an improvement on the bound in (12), as recorded in Theorem 2.

Proof of Theorem 2. Let
Let G and C be as in the conclusion of Proposition 12. Let K be a subfield of Q(A[2 ∞ ]) for which Gal(K/Q) = G, which exists by the fundamental theorem of infinite Galois theory.From Proposition 12, the size of G is bounded above by 255.Since One can check that the largest divisor of (6 Applying Corollary 6 with K and C as above and m := rad(2N E 1 N E 2 ), we obtain a prime number p not dividing m such that ( p K/Q ) ⊆ C which satisfies inequality (6).As Frob Consequently, a p (E 1 ) = a p (E 2 ).It remains to show that p satisfies the claimed bound.
As in the statement of Corollary 6, let K := K( √ m).We have that Thus by Corollary 6, where ã, b, c are absolute constants that may be taken to be 4, 2.5, and 5, respectively, or may be taken to be the improved values given in [1, We have that by Remark 11, K/Q is unramified outside of the prime divisors of m = rad(2N E 1 N E 2 ).As K is the compositum of K and Q( √ m), the primes that ramify in K are precisely those that Hence by Lemma Considering ( 14), (15), and ( 16), we conclude that p ≤ (1.257(383 log rad(2N Partially expanding the right-hand side above gives the claimed bound for p. Remark 13.In the proof above, for fields K with log d K ≤ 100, we use the general bound of [1, Theorem 5.1] rather than the improved bounds appearing in Table 3 of [1].We do so because Table 3 does not give constants in boxes where some combination of degree and discriminant would violate Minkowski's theorem.This only affects certain entries in the table for which the logarithm of the absolute value of the discriminant is less than 100.For example, the maximal totally real subfield of Q(ζ 7 ) has degree 3 and discriminant 49, yet the table gives no constants for number fields with degree 3-4 for which the logarithm of the absolute value of the discriminant is less than 5.

A bound on the largest non-surjective prime
We begin in §5.1 with some necessary background on quadratic Galois characters and quadratic twists of elliptic curves.In §5.2, we put together the pieces to complete the proof of Theorem 1. Finally, we present a numerical example in §5.3 that illustrates the theorem.Since ker χ is an index two subgroup of Gal(Q/Q), there exists a nonzero squarefree integer D such that ker where res D denotes the restriction homomorphism.Thus, for each σ ∈ Gal(Q/Q), We write χ D to denote the quadratic Galois character described by (17).Note that each quadratic Galois character may be written as χ D for a unique squarefree integer D.
For a prime number p, let I p and I p (D) denote the inertia subgroups of Gal(Q/Q) and Gal(Q( √ D)/Q), respectively.One says that χ D is unramified at p if I p ⊆ ker χ D .From the description of χ D given in (17) and upon noting that res D (I p ) = I p (D), we see that Now consider an elliptic curve E/Q given by a Weierstrass equation The quadratic twist of E by χ D (or, equivalently, by D) is the elliptic curve E D /Q given by See [13, §4.3] or [46, X.2] for background on quadratic twists.By taking (20) to be a minimal model for E, upon computing and comparing the discriminants of ( 20) and ( 21), we find that rad(N E D ) divides rad(2DN E ).
Lemma 14.If D = 1 is squarefree and E/Q is without complex multiplication, then E and E D are not Q-isogenous.
Proof.We know from the Chebotarev density theorem that the natural density of primes p for which χ D (Frob p ) = −1 is 1 2 .For such a p, if p ∤ 2DN E , then a p (E) = −a p (E D ) by (23).Thus either a p (E) = a p (E D ) or a p (E) = 0.The density of primes p for which p ∤ 2DN E and a p (E) = 0 is 0 by [41, p. 123], [20, p. 131].Thus there exists a prime p ∤ 2DN E such that a p (E) = a p (E D ).As such, E and E D are not Q-isogenous.

5.2.
Completing the proof.We now turn to the problem of bounding c(E).Suppose that ℓ is an odd prime such that G E (ℓ) satisfies (5) of Proposition 3.With an appropriate choice of F ℓ -basis of E[ℓ] in defining ρE,ℓ , we may assume that G E (ℓ) ⊆ C + ns (ℓ) and G E (ℓ) ⊆ C ns (ℓ).Following Serre, we consider the quadratic Galois character given by the composition We list some basic properties of ǫ ℓ , which are previously noted in [40, p. 317] and [12, p. 18].
Lemma 15.With the above notation and assumptions, ǫ ℓ satisfies the following properties.
(1) For each prime p ∤ N E , ǫ ℓ is unramified at p.
(2) One has that ǫ ℓ = χ D for some integer D | N E .
(1) If p ∤ ℓN E , then by Remark 10, ρ E,ℓ is unramified at p.In particular, ǫ ℓ is unramified at p.When ℓ ∤ N E and p = ℓ, the claimed property follows by a more delicate analysis; see [40, p. 295 We are now in a position to prove the main theorem.
Proof of Theorem 1.Let ℓ be a prime number such that ℓ > 37 and ρE,ℓ is non-surjective.By Theorem 4, we know that up to conjugation, G E (ℓ) ⊆ C + ns (ℓ) yet G E (ℓ) ⊆ C ns (ℓ).Consider the quadratic Galois character ǫ ℓ defined in (24).Let D be the squarefree integer such that ǫ ℓ = χ D and consider the quadratic twist E D of E. By Lemma 15(2) and ( 22), we have that rad(N E D ) divides rad(2N E ). ( For each prime number p ∤ 2N E , by (23), we have that In about one second total, SageMath's [50] built-in is surjective command confirms that ρE,ℓ is surjective for each prime number ℓ ≤ 10915.Thus, conditional on GRH, ρE,ℓ is surjective for all primes ℓ.Calling Zywina's ExceptionalSet script [54] on E confirms this unconditionally.
Q(A[m])/Q is unramified at p for each positive integer m, not divisible by p, and (3) Q(A[m])/Q is unramified at p for infinitely many positive integers m, not divisible by p. Proof.See [45, Theorem 1].

.
Recently, Zywina[54, Proposition 1.8 and Theorem 1.10] improved the bounds of Kraus and Cojocaru.Considerably better bounds for c(E) are known under the assumption of the generalized Riemann hypothesis (GRH) for Dedekind zeta functions.Assuming GRH, Serre gave an elegant proof in 1981 [41, Théorème 22] that there exists an absolute, computable constant C 2 such that (3)the equivalence of (1) and(3)in Theorem 8, p is a prime of good reduction of A, so p ∤ B A .Remark 10.Equivalently, ρ A,ℓ is ramified at exactly the prime divisors of B A .Remark 11.Let A = E 1 × E 2 be the product of two elliptic curves E 1 and E 2 , with conductors N E 1 and N E 2 , respectively.It follows from Theorem 8 that A has good reduction at a prime ℓ if and only if E 1 and E 2 both have good reduction at ℓ. Thus rad(B A ) = rad(ℓN E 1 N E 2 ).4.2.Effective isogeny theorem.In 1968, Serre proved the isogeny theorem for elliptic curves with non-integral j-invariant [43, p. IV-14].This was generalized by Faltings for arbitrary abelian varieties [21, Korollar 2, p. 361].For elliptic curves E 1 /Q and E 2 /Q, the isogeny theorem gives that E 1 and E 2 are Q-isogenous if and only if for each prime p ∤ N E 1 N E 2 , one has that