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

TL;DR

This paper presents a practical algorithm for computing Gross-Stark units and Stark-Heegner points using modular forms, enabling explicit calculations and tabulation of these special algebraic numbers in number fields.

Contribution

It introduces a novel computational method leveraging overconvergent modular forms and reduction theory to explicitly determine Gross-Stark units and Stark-Heegner points.

Findings

Successfully computed and tabulated Brumer-Stark units in various real quadratic fields.

Developed an algorithm that recovers exact algebraic values from p-adic logarithms.

Demonstrated the effectiveness of the method on fields with discriminants up to 10000.

Abstract

In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form whose spectral coefficients are $p$-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 Brumer-Stark units in narrow Hilbert class fields of real quadratic fields with discriminants up to $10000$, for primes less than $20$, as well as Stark-Heegner points on elliptic curves.