Applications of the Square Sieve to a Conjecture of Lang and Trotter for a Pair of Elliptic Curves Over the Rationals

Abstract

Let E be an elliptic curve over Q. Let p be a prime of good reduction for E. Then, for a prime p = ℓ, the Frobenius automorphism associated to p (unique up to conjugation) acts on the ℓ-adic Tate module of E. The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is independent of ℓ. Its splitting field is called the Frobenius field of E at p. Let E 1 and E 2 be two elliptic curves defined over Q that are non-isogenous over Q and also without complex multiplication over Q. In analogy with the well-known Lang-Trotter conjecture for a single elliptic curve, it is natural to consider the asymptotic behaviour of the function that counts the number of primes p ≤ x such that the Frobenius fields of E 1 and E 2 at p coincide. In this short note, using Heath-Brown's square sieve, we provide both conditional (upon the Generalized Riemann Hypothesis) and unconditional upper bounds. To V. Kumar Murty: On the occasion of his sixtieth birthdayConjecture 5. 1 [5, Conjecture 1] (Pair of elliptic curves and equal Frobenius fields) Let E 1 and E 2 be two elliptic curves over the rationals, and both without complex multiplication over Q. Then, E 1 is not isogenous to E 2 over Q if and only ifRemark 3. Suppose E 1 and E 2 are isogenous over Q, hence over some number field L. By extending L if necessary, we can assume that L is Galois over Q. Then, for any prime p ∤ N 1 N 2 and that splits in L, a p (E 1 ) = a p (E 2 ). Thus, from the Chebotarev density theorem, it follows that S(E 1 , E 2 ; x) grows at least as much as

Quantitative upper bounds related to an isogeny criterion for elliptic curves

“…The argument highlighted by Serre in [4, 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 [1] 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 .…”

“…For a prime ideal 𝔭 ∈ ∑ 𝐾 , we denote by N 𝐾 (𝔭) its norm in 𝐾∕ℚ. We say that 𝐾 satisfies the 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).…”

“…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 nontrivial 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.…”