Quantitative upper bounds related to an isogeny criterion for elliptic curves

For $E_1$ and $E_2$ elliptic curves defined over a number field $K$, without complex multiplication, we consider the function ${\mathcal{F}}_{E_1, E_2}(x)$ counting non-zero prime ideals $\mathfrak{p}$ of the ring of integers of $K$, of good reduction for $E_1$ and $E_2$, of norm at most $x$, and for which the Frobenius fields $\mathbb{Q}(\pi_{\mathfrak{p}}(E_1))$ and $\mathbb{Q}(\pi_{\mathfrak{p}}(E_2))$ are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that $E_1$ and $E_2$ are not potentially isogenous if and only if ${\mathcal{F}}_{E_1, E_2}(x) = \operatorname{o} \left(\frac{x}{\log x}\right)$, we investigate the growth in $x$ of ${\mathcal{F}}_{E_1, E_2}(x)$. We prove that if $E_1$ and $E_2$ are not potentially isogenous, then there exist positive constants $\kappa(E_1, E_2, K)$, $\kappa'(E_1, E_2, K)$, and $\kappa''(E_1, E_2, K)$ such that the following bounds hold: (i) ${\mathcal{F}}_{E_1, E_2}(x)<\kappa(E_1, E_2, K) \frac{ x (\log\log x)^{\frac{1}{9}}}{ (\log x)^{\frac{19}{18}}}$; (ii) ${\mathcal{F}}_{E_1, E_2}(x)<\kappa'(E_1, E_2, K) \frac{ x^{\frac{6}{7}}}{ (\log x)^{\frac{5}{7}}}$ under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) ${\mathcal{F}}_{E_1, E_2}(x)<\kappa''(E_1, E_2, K) x^{\frac{2}{3}} (\log x)^{\frac{1}{3}}$ under GRH, Artin's Holomorphy Conjecture for the Artin $L$-functions of number field extensions, and a Pair Correlation Conjecture for the zeros of the Artin $L$-functions of number field extensions.


Introduction
Let K be a number field, with O K denoting its ring of integers and K denoting a fixed algebraic closure.
In what follows, we use the letter p to denote a non-zero prime ideal of O K and refer to it as a prime of K, N K (p) to denote the norm of p, and F p to denote the finite field O K /p.
Let E 1 and E 2 be elliptic curves over K.We denote by N 1 and N 2 the norms of the conductors of E 1 and E 2 , respectively.For a prime p of K that is of good reduction for both E 1 and E 2 and for each index 1 ≤ j ≤ 2, we consider the polynomial P Ej ,p (X) := X 2 −a p (E j )X +N K (p) ∈ Z[X], where N K (p)+1−a p (E j ) is the number of F p -rational points of the reduction of E j modulo p.We recall that, for any rational prime ℓ distinct from the field characteristic of F p , P Ej ,p (X) is the characteristic polynomial of the image ρ Ej,ℓ (Frob p ) of a Frobenius element Frob p ∈ Gal K/K under the ℓ-adic Galois representation ρ Ej ,ℓ of E j defined by the action of Gal K/K on the ℓ-division points of E j K .Viewing P Ej ,p (X) in C[X] and denoting its roots by π p (E j ) and π p (E j ), we recall that |π p (E j )| = N K (p), which implies that |a p (E j )| ≤ 2 N K (p) and hence that Q(π p (E j )) is either Q or an imaginary quadratic field.In what follows, we refer to a p (E j ) as the Frobenius trace and to Q(π p (E j )) as the Frobenius field associated to E j and p.
From now on, we assume that E 1 and E 2 are without complex multiplication.Given a field extension L of K, we say that E 1 and E 2 are L-isogenous if there exists an isogeny from E 1 to E 2 , defined over L. We say that E 1 and E 2 are potentially isogenous if there exists a finite extension L of K such that E 1 and E 2 are L-isogenous.It is known that the following statements are equivalent: E 1 and E 2 are potentially isogenous; E 1 and E 2 are K-isogenous; E 1 and E 2 are L-isogenous for some quadratic field extension L of K; either E 1 and E 2 are K-isogenous, or there exists a quadratic character χ such that E 1 and the quadratic twist E χ 2 are K-isogenous (e.g., see [LeFNa20, Lemma 3.1, p. 214; proof of Claim 3, p. 215]).Our goal in this paper is to investigate questions arising from a criterion regarding whether E 1 and E 2 are potentially isogenous, as we explain below.
In [KuPaRa16, Theorem 3, p. 90], Kulkarni, Patankar, and Rajan show that E 1 and E 2 are potentially isogenous if and only if the set of primes p of K, of good reduction for E 1 and E 2 , such that Q(π p (E 1 )) = Q(π p (E 2 )), has a positive upper density within the set of primes of K, that is, the counting function Thus, E 1 and E 2 are not potentially isogenous if and only if F E1,E2 (x) = o x log x .In relation to the above result, in [KuPaRa16, Conjecture 1, p. 91], Kulkarni, Patankar, and Rajan mention the following conjecture: E 1 and E 2 are not potentially isogenous if and only if there exists a positive constant c(E 1 , E 2 , K), which may depend on E 1 , E 2 , and K, such that, for any sufficiently large While the "if" implication follows from the aforementioned result of Kulkarni, Patankar, and Rajan, the "only if" implication remains open and motivates the investigation of the growth of the function F E1,E2 (x).
In [CoFoMu05, p. 1174], the authors record the following remark of Serre, highlighting only the main idea of proof: if E 1 and E 2 are not potentially isogenous, then, under the Generalized Riemann Hypothesis for Dedekind zeta functions, there exists a positive constant c ′ (E 1 , E 2 , K), which depends on E 1 , E 2 , and K, such that, for any sufficiently large x, In [BaPa18, Theorem 2, p. 43], Baier and Patankar address the growth of F E1,E2 (x) in the case K = Q and prove that, under the Generalized Riemann Hypothesis for Dedekind zeta functions, there exists a positive constant c ′′ (E 1 , E 2 ), which depends on E 1 and E 2 , such that, for any sufficiently large x, In [BaPa18, Theorem 3, p. 43], Baier and Patankar also prove the following unconditional bound for F E1,E2 (x), resulting from an unconditional variation of the proof of their conditional result: there exists a positive constant c ′′′ (E 1 , E 2 ), which depends on E 1 and E 2 , such that, for any sufficiently large x, of number field extensions.We shall refer to these latter hypotheses as GRH, AHC, and PCC, and state them explicitly in the notation part of Section 1.
Theorem 1.Let E 1 and E 2 be elliptic curves over a number field K, without complex multiplication, and not potentially isogenous.Denote by N 1 and N 2 the norms of the conductors of E 1 and E 2 , respectively.
(i) There exists a positive constant κ(E 1 , E 2 , K), which depends on E 1 , E 2 , and K, such that, for any sufficiently large x, .
(ii) If GRH holds, then there exists a positive constant κ ′ (E 1 , E 2 , K), which depends on E 1 , E 2 , and K, such that, for any sufficiently large x, .
(iii) If GRH, AHC, and PCC hold, then there exists a positive constant κ ′′ (E 1 , E 2 , K), which depends on E 1 , E 2 , and K, such that, for any sufficiently large x, Remark 2. Theorem 1 may be viewed under the general theme of strong multiplicity one results, such as those proven in [JaSh76], [MuPu17], [Ra94], [Ra00], [Wa14], and [Wo22].In particular, the methods developed in [MuPu17] and [Wo22] are applicable to bounding F E1,E2 (x) from above in the case K = Q and under hypotheses different from ours.Specifically, letting E 1 and E 2 be elliptic curves over Q, without complex multiplication, not potentially isogenous, and assuming the Generalized Riemann Hypothesis for the Rankin-Selberg L-functions associated to the symmetric power L-functions of E 1 and E 2 , the methods of (see [Wo22, Remark (ii), p. 567]), while the methods of (log x) The proof of Theorem 1 relies on upper bounds related to the Lang-Trotter Conjecture for Frobenius fields of one elliptic curve.We formulate the relevant results here for the convenience of the reader.Let E/Q be an elliptic curve without complex multiplication and let F be an imaginary quadratic field.Lang and Trotter [LaTr76] conjectured the asymptotic where C(E, F ) is an explicit constant depending on E and F .Zywina [Zy15, Theorem 1.3, p. 236] proved that unconditionally, and that under GRH, .
The proof of Theorem 1 also relies on the following result which relates to the generalization of the Lang-Trotter Conjecture on Frobenius traces formulated by Chen, Jones, and Serban in [ChJoSe22].
Theorem 3. Let E 1 and E 2 be elliptic curves over a number field K, without complex multiplication, and not potentially isogenous.Denote by N 1 and N 2 the norms of the conductors of E 1 and E 2 , respectively.
(iii) If GRH, AHC, and PCC hold, then there exists a positive constant κ ′′ 0 (E 1 , E 2 , K, α 1 , α 2 ), which depends on E 1 , E 2 , K, α 1 , and α 2 , such that, for any sufficiently large x, (ii') If GRH holds, then there exists an absolute, effectively computable, positive constant κ ′ 1 such that, for any sufficiently large x, (iii') If GRH, AHC, and PCC hold, then there exists an absolute, effectively computable, positive constant κ ′′ 1 such that, for any sufficiently large x, T α1,α2 E1,E2 (x Remark 5. Theorem 3 may be viewed under the general Lang-Trotter theme of results about the number of primes for which the Frobenius trace of an abelian variety is fixed, such as those proven in and [Zy15].The connection between Theorem 3, and thus Theorem 1, with the Lang-Trotter Conjectures on Frobenius traces formulated in [LaTr76,p. 33] and [ChJoSe22,p. 382] prompts the question of predicting, conjecturally, the asymptotic behavior of F E1,E2 (x) and T α1,α2 E1,E2 (x) for E 1 , E 2 , α 1 , and α 2 as in the setting of Theorem 3. We relegate such investigations to a future project.

Notation
• Given a number field K, we denote by O K its ring of integers, by K the set of non-zero prime ideals of O K , by n K the degree of K over Q, by d K ∈ Z\{0} the discriminant of an integral basis of O K , and by disc(K/Q) = Zd K ✂ Z the discriminant ideal of K/Q.For a prime ideal p ∈ K , we denote by N K (p) its norm in K/Q.We say that K satisfies the Generalized Riemann Hypothesis (GRH) if the Dedekind zeta function ζ K of K has the property that, for any ρ ∈ C with 0 ≤ Re ρ ≤ 1 and ζ K (ρ) = 0, we have Re(ρ) = 1 2 .When K = Q, the Dedekind zeta function is the Riemann zeta function, in which case we refer to GRH as the Riemann Hypothesis (RH).
• Given a finite Galois extension L/K of number fields and a subset C ⊆ Gal(L/K), stable under conjugation, we denote by π C (x, L/K) the number of non-zero prime ideals of the ring of integers of K, unramified in L, of norm at most x, for which the Frobenius element is contained in C. We set with the dash on the product indicating that each of the primes p therein lies over a non-zero prime ideal ℘ of O L , with ℘ ramified in L.
• Given a finite Galois extension L/K of number fields and an irreducible character χ of the Galois group of of χ, and by A χ (T ) the function of a positive real variable T > 3 defined by the relation • Given a finite Galois extension L/K of number fields, we say that it satisfies Artin's Holomorphy Conjecture (AHC) if, for any irreducible character χ of the Galois group of L/K, the Artin L-function L(s, χ, L/K) extends to a function that is analytic on C, except at s = 1 when χ = 1.We recall that, if we assume GRH for L and AHC for L/K, then, given any irreducible character χ of the Galois group of L/K, and given any non-trivial zero ρ of L(s, χ, L/K), the real part of ρ satisfies Re ρ = 1 2 .In this case, we write ρ = 1 2 + iγ, where γ denotes the imaginary part of ρ.
• Given a finite Galois extension L/K of number fields, let us assume GRH for L and AHC for L/K.For an irreducible character χ of the Galois group of L/K and an arbitrary T > 0, we define the pair correlation function of L(s, χ, L/K) by where γ 1 and γ 2 range over all the imaginary parts of the non-trivial zeroes ρ = 1 2 +iγ of L(s, χ, L/K), counted with multiplicity, and where, for an arbitrary real number u, e(u) := exp(2πiu) and w(u) := 4 4+u 2 .We say that the extension L/K satisfies the Pair Correlation Conjecture (PCC) if, for any irreducible character χ of the Galois group of L/K and for any A > 0 and T > 3, provided 0 ≤ Y ≤ Aχ(1)n K log T , we have

From shared Frobenius fields to shared absolute values of Frobenius traces
We keep the general setting and notation from Section 1.To prove Theorem 1, we reduce the study of the primes p for which the Frobenius fields of E 1 and E 2 coincide to a study of the primes p for which the absolute values of the Frobenius traces of E 1 and E 2 coincide, as follows.Lemma 6.Let E 1 and E 2 be elliptic curves over a number field K, non-isogenous over K. Denote by N 1 and N 2 the norms of the conductors of E 1 and E 2 , respectively.Let p be a degree one prime of K such that the rational prime p : Proof.The "if" implication is clear, since, for each 1 To justify the "only if" implication, we distinguish between p supersingular and ordinary for E 1 and E 2 .If p is supersingular for both E 1 and E 2 , then a p (E 1 ) = a p (E 2 ) = 0.When p is ordinary for both E 1 and E 2 , or ordinary for one of E 1 or E 2 , and supersingular for the other, we look at the prime ideal factorization of p in the ring of integers O F of the imaginary quadratic field . By the lemma's hypothesis, the group of units of , where Tr F/Q (α) denotes the trace of the algebraic number α ∈ F .Since, for each 1 ≤ j ≤ 2, a p (E j ) = Tr Q(πp(Ej ))/Q (π p (E j )), we obtain that say, for E 1 , and supersingular for the other, say, for E 2 , then p splits completely in Q(π p (E 1 )) and ramifies in ).Thus, this case does not occur.

Elliptic curves with shared absolute values of Frobenius traces
We keep the general setting and notation from Section 1.In light of Lemma 6, in order to prove Theorem 1, we focus on the primes p for which |a p (E 1 )| = |a p (E 2 )|.We view this condition as a combination of two linear relations between the traces of E 1 and E 2 , namely a p (E 1 ) + a p (E 2 ) = 0 and a p (E 1 ) − a p (E 2 ) = 0, which are particular cases of Theorem 3,.Our goal in this section is to prove Theorem 3.
3.1.Preliminaries.We follow the methods developed in [CoWa23] and [Wa23].These methods already give rise to the stated conditional estimates for T 1,1 E1,E2 (x), but need to be adjusted for the general conditional and unconditional bounds, as we explain below.

Consider the abelian surface
For an arbitrary rational prime ℓ, consider the residual modulo ℓ Galois representations ρ A,ℓ , ρ E1,ℓ , and ρ E2,ℓ of A, E 1 , and E 2 , respectively, defined by the action of Gal K/K on the ℓ-division groups A[ℓ], E 1 [ℓ], and tr(ρ Ej ,ℓ (Frob p )) ≡ a p (E j )(mod ℓ) for any p with gcd(N K (p), ℓN j ) = 1and for any 1 ≤ j ≤ 2. Setting we recall from [Lo16, Lemma 7.1, p. 409] that, thanks to our assumptions that E 1 and E 2 are without complex multiplication and not potentially isogenous, there exists a positive integer c(A, K), which depends on A and K, such that if ℓ > c(A, K), then Im ρ A,ℓ = G(ℓ), that is, For an arbitrary pair of matrices (M 1 , M 2 ) ∈ G(ℓ) and for each 1 ≤ j ≤ 2, we denote by λ 1 (M j ), λ 2 (M j ) ∈ F ℓ the eigenvalues of M j .Associated to G(ℓ), we consider the groups and the sets G(ℓ) # := the set of conjugacy classes of G(ℓ), P (ℓ) # := the set of conjugacy classes of P (ℓ), With the above notation, our strategy for proving parts (i) and (iii) of Theorem 3 is to relate ) Λ(ℓ) /K , and our strategy for proving part (ii) of Theorem 3 is to relate ℓ) .After establishing these relations, we apply different variations of the effective Chebotarev density theorem to obtain upper bounds for the number of primes p whose Frobenius element satisfies the desired Chebotarev conditions.In the end, we minimize the bounds by choosing ℓ suitably as a function of x.
Before executing this strategy, we record a few properties of the groups and sets introduced above.
Lemma 7.For ℓ an arbitrary rational prime, the following statements hold.Lemma 8.For ℓ an arbitrary rational prime, the following statements hold.
For part (ii), the case In particular, every conjugacy class in C(ℓ) α1,α2 contains an element of B(ℓ).
For part (iii), we provide a short proof.
Lemma 10.For ℓ an odd rational prime such that ℓ does not divide at least one of α 1 , α 2 , the following statements hold.
Proof.For parts (i), the case where α −1 2 (mod ℓ) is the inverse of α 2 (mod ℓ).Note that, since α 1 and α 2 are not both divisible by ℓ, either this inverse exists, or, if it does not, the inverse of α 1 (mod ℓ) exists, in which case a similar argument works using α −1 1 (mod ℓ).This completes the proof of (i).For part (ii), the case α 1 = α 2 = 1 is [CoWa23, Lemma 17, (iv), p.701]In the general case, we first consider the number of matrices in the image of C Borel (ℓ) α1,α2 in B(ℓ)/U (ℓ) ≃ T (ℓ).They are clearly determined by the diagonal entries and can be counted as follows: where α −1 2 (mod ℓ) is the inverse of α 2 (mod ℓ).As before, if the inverse does not exist, a similar argument works using α −1 1 (mod ℓ).Next, we observe that the inverse image of C Borel (ℓ) α1,α2 under the projection Finally, from part (iii) of Lemma 9 and part (i) of the current lemma, we deduce that ≤ 2ℓ 5 .This completes the proof of (iii).
We are now ready to prove Theorem 3. We fix a rational prime ℓ such that ℓ > c(A, K) and such that ℓ does not divide at least one of α 1 , α 2 .
From (1) and (2), we deduce that (4) In what follows, we bound from above the function on the right hand side of the inequality.
Once more relying on [Se81, Proposition 5, p. 129], Lemma 9, and the Néron-Ogg-Shafarevich criterion for abelian varieties, we obtain that From [Lo16, Lemma 7.1, p. 409], we know that there exists an effectively computable, positive constant a(h A , n K ), which depends on the Faltings height h A of A and on n K , such that, if ℓ > a(h A , n K ), then (3) holds.Hence condition (5) on ℓ is ensured by the restrictions for some positive constants a 1 (h A , n K ) and a 2 (h A , n K , d K , N 1 , N 2 ), which depend on h A , n K , d K , N 1 , and and N 2 , we may choose the prime ℓ such that (log log x) for some positive constant and α 2 .
First, we proceed identically to part (i) and deduce that Next, we apply [MuMuWo18, Theorem 1.2, p. 402] (which requires GRH, AHC, and PCC) to estimate the counting function π CProj (ℓ) α 1 ,α 2 x, K(A[ℓ]) Λ(ℓ) /K .By putting all estimates together, we deduce that Then, using Lemmas 9 -10, we infer that Reasoning as in part (i), we may choose the prime ℓ such that for some positive constant a 4 = a 4 (h and α 2 .Finally, recalling (4), we obtain that for some positive constant κ ′′ 0 (E 1 , E 2 , K, α 1 , α 2 ), which depends on E 1 , E 2 , K, α 1 , and α 2 .This completes the proof of part (iii) of Theorem 3. Lemma 11.Let S be a non-empty set of prime ideals of K, let (K p ) p∈S be a family of finite Galois extensions of Q, and let (C p ) p∈S be a family of non-empty sets such that each C p is a union of conjugacy classes of Gal(K p /Q). Assume that there exist an absolute constant c 1 > 0 and a function f : R → (0, ∞) such that For each x > 2, let y = y(x) > 2, u = u(x) > 2 be such that Assume GRH for Dedekind zeta functions.Then, for any ε > 0, there exists a constant c(ε) > 0 such that, for any sufficiently large x, We apply this lemma to the set S α1,α2 := {p : gcd , to the conjugacy classes C p := {id Kp }, and to the function f (v) := 2 log(4v).
Note that, for S 1,1 , this application is precisely the case g = 2 of [CoWa23, Lemma 18, pp.704-705].We obtain that, under the Riemann Hypothesis for the Riemann zeta function and GRH for the Dedekind zeta functions of the number fields K p , the following holds.
For a fixed arbitrary x > 2, let y := y(x) and u := u(x) be real numbers such that 2 < u(x) < y(x).

Elliptic curves with shared Frobenius fields
Let E 1 and E 2 be elliptic curves over a number field K, without complex multiplication, and not potentially isogenous.We keep the associated notation from the previous sections and prove Theorem 1.
Corollary 12. Let E 1 and E 2 be two elliptic curves over a number field K. Then E 1 and E 2 are potentially isogenous if and only if F E1,E2 (x) has a positive upper density within the set of primes of K.
Proof.For the"only if" implication, we assume that E 1 and E 2 are potentially isogenous.This implies that E 1 is isogenous over K to a quadratic twist of E 2 .Therefore, |a p (E 1 )| = |a p (E 2 )| for all but finitely many primes p of K.As in the "if" implication of Lemma 6, we have that Q(π p (E 1 )) = Q(π p (E 2 )) for all but finitely many primes p of K.So F E1,E2 (x) has density one in the set of primes of K.
For the "if" implication, we prove the contrapositive.Assume that E 1 and E 2 are not potentially isogenous.Then, from part (i) of Theorem 1, we deduce that F E1,E2 (x) is bounded from above by a set of density zero in the set of primes of K.

21 (log x) 43 42 .
The argument highlighted by Serre in[CoFoMu05, p. 1174] is based on a direct application of a conditional upper bound version of the Chebotarev density theorem in the setting of an infinite Galois extension of K defined by the ℓ-adic Galois representations of E 1 and E 2 , for a suitably chosen rational prime ℓ.The proofs given by Baier and Patankar in[BaPa18] are based on indirect applications of conditional and unconditional effective asymptotic versions of the Chebotarev density theorem, via the square sieve, in the setting of a finite Galois extension of Q defined by the residual modulo ℓ 1 ℓ 2 Galois representations of E 1 and E 2 , for distinct suitably chosen rational primes ℓ 1 and ℓ 2 .The main goal of this paper is to improve the current upper bounds for F E1,E2 (x), as follows: unconditionally; under the Generalized Riemann Hypothesis for Dedekind zeta functions; under the Generalized Riemann Hypothesis for Dedekind zeta functions, Artin's Holomorphy Conjecture for the Artin L-functions of number field extensions, and a Pair Correlation Conjecture regarding the zeros of the Artin L-functions setting of Theorem 3 and invoking [MaWa23, Corollary 1.2, p. 3] instead of [Lo16, Lemma 7.1, p. 409] in the proofs of parts (ii) and (iii), our proof leads to the following more explicit bounds.