Quantitative upper bounds related to an isogeny criterion for elliptic curves (Q6593640)

Consider two elliptic curves \( E_1 \) and \( E_2 \) defined over a number field \( K \), both without complex multiplication. Let \(\mathfrak{p}\) be a prime ideal of \( K \) where both \( E_1 \) and \( E_2 \) have good reduction. Denote the norm of \(\mathfrak{p}\) by \(\text{N}_{K}(\mathfrak{p})\).\N\NFor each \( j \in \{1, 2\} \), the Frobenius trace \( a_{\mathfrak{p}}(E_j) \) is given by \( \text{N}_{K}(\mathfrak{p}) + 1 - \#\widetilde{E}_j(\mathbb{F}_{\mathfrak{p}})\) where \( \widetilde{E}_j \) is \( E_j \) modulo \(\mathfrak{p}\). Let \(\pi_{\mathfrak{p}}(E_j)\) be a root of the polynomial \(X^2 - a_{\mathfrak{p}}(E_j)X + \text{N}_{K}(\mathfrak{p}) \in \mathbb{Z}[X].\) The Frobenius field associated with \( E_j \) is defined as \( \mathbb{Q}(\pi_{\mathfrak{p}}(E_j)) \). Now, define the counting function\N\[\N{\mathcal{F}}_{E_1, E_2}(x) := \#\{ \mathfrak{p} : \text{N}_{K}(\mathfrak{p}) \leq x, \; E_1, E_2 \text{ have good reduction at } \mathfrak{p}, \; \mathbb{Q}(\pi_{\mathfrak{p}}(E_1)) = \mathbb{Q}(\pi_{\mathfrak{p}}(E_2)) \}.\N\]\N\N\textit{M. Kulkarni} et al. [J. Number Theory 164, 87--102 (2016; Zbl 1402.11082)] prove the following isogeny criterion: \( E_1 \) and \( E_2 \) are not potentially isogenous if and only if\N\[\N\mathcal{F}_{E_1, E_2}(x) = o\left(\frac{x}{\log x}\right).\N\]\NThey further mention the conjecture that \( E_1 \) and \( E_2 \) are not potentially isogenous if and only if\N\[\N\mathcal{F}_{E_1, E_2}(x) \ll_{E_1, E_2, K} \log\log x.\N\]\NSee also [\textit{S. Baier} and \textit{V. M. Patankar}, Springer Proc. Math. Stat. 251, 39--57 (2018; Zbl 1475.11170)], p. 42. The paper under review establishes the following results towards this conjecture, using effective versions of the Chebotarev's density theorem. Suppose \( E_1 \) and \( E_2 \) are not potentially isogenous. Then,\N\begin{itemize}\N\item[1.]\N\[\N\mathcal{F}_{E_1, E_2}(x) \ll_{E_1, E_2, K} \frac{x (\log \log x)^{\frac{1}{9}}}{(\log x)^{\frac{19}{18}}};\N\]\N\item[2.]\N\[\N\mathcal{F}_{E_1, E_2}(x) \ll_{E_1, E_2, K} \frac{x^{\frac{6}{7}}}{(\log x)^{\frac{5}{7}}},\N\]\Nunder the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH);\N\item[3.]\N\[\N\mathcal{F}_{E_1, E_2}(x) \ll_{E_1, E_2, K} x^{\frac{2}{3}} (\log x)^{\frac{1}{3}},\N\]\Nunder 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.\N\end{itemize}\NThese results improve the previously known bounds (see [\textit{S. Baier} and \textit{V. M. Patankar}, Springer Proc. Math. Stat. 251, 39--57 (2018; Zbl 1475.11170)]).