Quantitative upper bounds related to an isogeny criterion for elliptic curves

Abstract

For $E_1$ and $E_2$ elliptic curves defined over a number field $K$, without complex multiplication, we consider the function $F_{E_1,E_2}\left(x\right)$ counting nonzero prime ideals $p$ 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 $Q\left({\pi }_p\left(E_1\right)\right)$ and $Q\left({\pi }_p\left(E_2\right)\right)$ 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 $F_{E_1,E_2}\left(x\right)=o\left(\frac{x}{\log x}\right)$, we investigate the growth in $x$ of $F_{E_1,E_2}\left(x\right)$. We prove that if $E_1$ and $E_2$ are not potentially isogenous, then there exist positive constants $\kappa (E_1,E_2,K)$, ${\kappa }^{\prime }(E_1,E_2,K)$, and ${\kappa }^{\prime \prime }(E_1,E_2,K)$ such that the following bounds hold: (i) $F_{E_1,E_2}\left(x\right)<\kappa (E_1,E_2,K)\frac{x{\left(\log \log x\right)}^{\frac{1}{9}}}{{\left(\log x\right)}^{\frac{19}{18}}}$; (ii) $F_{E_1,E_2}\left(x\right)<{\kappa }^{\prime }(E_1,E_2,K)\frac{x^{\frac{6}{7}}}{{\left(\log x\right)}^{\frac{5}{7}}}$ under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) $F_{E_1,E_2}\left(x\right)<{\kappa }^{\prime \prime }(E_1,E_2,K)x^{\frac{2}{3}}{\left(\log x\right)}^{\frac{1}{3}}$ 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.