The density of elliptic curves having a global minimal Weierstrass equation
Let \(E\) be an elliptic curve defined over some number field \(K\). Over each completion \(K_v\) of \(K\), \(E\) has a model for which its discriminant \(\Delta_v\) has minimal valuation. The minimal discriminant of \(E/K\) is then defined to be \(D_{E/K} = \prod {\mathfrak p}_v^{v(\Delta_v)}\), where the product is over all finite places \(v\) of \(K\), and where \({\mathfrak p}_v\) is the prime ideal associated to \(v\). We say that a Weierstrass equation for \(E\) is minimal over \(K\) if \((\Delta) = D_{E/K}\). Let \(x = u_v2 x_v + r_v\), \(y = u_v^3 y_v + s_v u_v^2 x_v + t_v\) be the change of coordinates that transforms a given Weierstrass equation of \(E/K\) into a minimal model over \(K_v\). Then the Weierstrass class \(C_{E/K}\) of \(E/K\) is defined to te the ideal class of the ideal \({\mathfrak a} = \prod {\mathfrak p}_v^{-v(u_v)}\). It is known that \(E\) has a global minimal Weierstrass model if and only if \(C_{E/K}\) is trivial. In this article, the author shows that elliptic curves \(E/K\) with given Weierstrass class \(C_{E/K}\) have a density that can be expressed in terms of the partial Dedekind zeta functions for ideal classes. As a special case, she shows the following: if all primes above \(2\) and all ramified primes above \(3\) in \(K\) are principal, then the density of elliptic curves over \(K\) with a global minimal equation is \(\zeta_K(C_0,10)/\zeta_K(10)\), where \(C_0\) is the class of principal ideals. Applying her results to \(K = \mathbb Q(\sqrt{65}\,)\), the density comes out as a rational number \(\approx \frac12 + 10^{-6.234}\), whereas for \(K = \mathbb Q(\sqrt{85}\,)\) it is \(\approx 1 - 10^{-4.469}\).
scientific article; zbMATH DE number 2136918