Computing Gross-Stark units from constant terms of Hilbert modular forms

Description

This repository contains code for computing Gross-Stark units and Stark-Heegner points using the diagonal restriction derivative from DPV2, as described in D-J23.

Getting Started

Dependencies

• Some version of magma

Installing

• Simply clone (or copy) the repository into your favourite folder.

Executing programme

• To load the code, move to the directory containing main.m, run magma, and then

  AttachSpec("spec");
  

  This imports the main function GSUnit, which takes as input an indefinite quadratic form $F$ with discriminant $D$, a prime $p$ inert in $\mathbb{Q}(\sqrt{D})$ and an integer parameter $m$, the number of terms computed of the modular form whose constant term contains the Gross-Stark unit.

  TODO: decide this parameter automatically and only ask for pprec.

One can then compute the Gross-Stark attached to a quadratic form $Q$ as follows:

Q := QuadraticForms(33)!<-2,3,3>;
p := 5;
GSUnit(Q,p,30);

Similarly, we can find Stark-Heegner points:

Q := QuadraticForms(24)!<-2,4,1>;
p := 11;
SHPoints(Q,p,30);

• The file examples.m contains functions for tabulating Gross-Stark units, and data for primes less than $20$ and discriminants less than $10000$ can be found in the directory called data.

Author

Håvard Damm-Johnsen

License

This project is licensed under the GPL License - see the LICENSE.md file for details

Acknowledgments

This is based on code by Jan Vonk and Alan Lauder, as well as their papers (with and without collaborators).