Thetanullwerte: From periods to good equations

Abstract

We will show the utility of the classical Jacobi Thetanullwerte for the description of certain period lattices of elliptic curves, providing equations with good arithmetical properties. These equations will be the starting point for the construction of families of elliptic curves with everywhere good reduction. IntroductionThe determination of an equation for an elliptic curve associated to a given complex torus is a common task in number theory. The common solution follows from the uniformization of the torus by means of the Weierstrass ℘, ℘ ′ functions. Unfortunately, such a direct approach is not always satisfactory, especially if one desires an equation with good arithmetical properties.A very basic example illustrates this phenomenon: let us consider the complex torus C/Z[i]. We may compute numerically the common Weierstrass coefficients and find While the second one seems to vanish clearly, the first one is difficult to recognize, since it turns to be the transcendental number4Γ(3/4) 4 . On the other hand, it is well-known that the equation Y 2 = X 3 − X gives an algebraic model of this torus, with good reduction outside 2.We will explain here a general method to find such nice equations for elliptic curves. The key ingredient will be the classical Jacobi theta functions. The models provided have very good arithmetical properties. For instance, they are not far from having good reduction everywhere.2000 Mathematics Subject Classification. 11G05, 14K25.