Modular algorithms for Gross–Stark units and Stark–Heegner points

Abstract

In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form whose spectral coefficients are p p -adic logarithms of Gross–Stark units and Stark–Heegner points. Here we describe how this construction gives rise to a practical algorithm for explicitly computing these logarithms to specified precision, and how to recover the exact values of the Gross–Stark units and Stark–Heegner points from them. Key tools are overconvergent modular forms, reduction theory of quadratic forms and Newton polygons. As an application, we tabulate Gross–Stark units in narrow Hilbert class fields of real quadratic fields with discriminants up to 10000 10000 , for primes less than 20 20 , as well as Stark–Heegner points on elliptic curves.