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 ${\mathcal{F}}_{E_1,E_2}(x)$ 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 ${\mathcal{F}}_{E_1,E_2}(x)=o\left(\frac{x}{\log x}\right)$, 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 $\kappa (E_1,E_2,K)$, ${\kappa }^{\prime }(E_1,E_2,K)$, and ${\kappa }^{\prime \prime }(E_1,E_2,K)$ 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) ${\mathcal{F}}_{E_1,E_2}(x)<{\kappa }^{\prime }(E_1,E_2,K)\frac{x^{\frac{6}{7}}}{(\log x{)}^{\frac{5}{7}}}$ under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) ${\mathcal{F}}_{E_1,E_2}(x)<{\kappa }^{\prime \prime }(E_1,E_2,K)x^{\frac{2}{3}}(\log x{)}^{\frac{1}{3}}$ 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.