The number of elliptic curves over \(\mathbb{Q}\) with conductor \(N\)
Let \(\varepsilon>0\) and \(S\) be a finite set of rational primes. Put \(M= \prod_{p\in S}p\). The authors prove that there is a constant \(k_1(\varepsilon)\), independent of the set \(S\), such that the number of elliptic curves defined over \(\mathbb{Q}\) with good reduction outside \(S\) is at most \(k_1 (\varepsilon) M^{1/2+\varepsilon}\). In particular, there is a constant \(k_2(\varepsilon)\) such that the number of elliptic curves defined over \(\mathbb{Q}\) with conductor \(N\) is bounded by \(k_2(\varepsilon) N^{1/2+\varepsilon}\). An interesting consequence of this result is that there is a constant \(k_3(\varepsilon)\) such that for all integers \(N\geq 1\), the number of \(\mathbb{Q}\)-rational elliptic factors of the modular Jacobian \(J_0(N)\) is at most \(k_3(\varepsilon) N^{1/2+\varepsilon}\). Furthermore, the authors prove, using standard conjectures, that their bound on the number of elliptic curves defined over \(\mathbb{Q}\) with good reduction outside \(S\) can be improved to \(k_4 M^{k_5/\log\log M}\), where \(k_4\) and \(k_5\) are absolute constants.
elliptic factors of the modular Jacobian
scientific article; zbMATH DE number 980701
scientific article