2024/04/18 by Cojocaru, Alina Carmen, Hinz, Auden, Wang, Tian
#11G05 #11G20 #11N05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2404.12466
For E1 and E2 elliptic curves defined over a number field K, without complex multiplication, we consider the function FE1, E2(x) counting non-zero prime ideals \mathfrakp of the ring of integers of K, of good reduction for E1 and E2, of norm at most x, and for which the Frobenius fields ℚ(π_\mathfrakp(E1)) and ℚ(π_\mathfrakp(E2)) are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that E1 and E2 are not potentially isogenous if and only if FE1, E2(x) = o ((x)/(log x)), we investigate the growth in x of FE1, E2(x). We prove that if E1 and E2 are not potentially isogenous, then there exist positive constants κ(E1, E2, K), κ'(E1, E2, K), and κ''(E1, E2, K) such that the following bounds hold: (i) FE1, E2(x) < κ(E1, E2, K) \frac x (loglog x)(1)/(9) (log x)(19)/(18); (ii) FE1, E2(x) < κ'(E1, E2, K) \frac x(6)/(7) (log x)(5)/(7) under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) FE1, E2(x) < κ''(E1, E2, K) x(2)/(3) (log x)(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.