Computing L-functions and Gross-Stark units using Hilbert Eisenstein series

Introduction

Computation of algebraic invariants attached to real quadratic fields, using algorithms from D-J23​, building on DPV1 and DPV2, implemented in SageMath. For corresponding code in Magma, see the DRD repository.

Usage

Navigate to the main directory, run

load("main.sage")

This loads the necessary files using sage’s attach command.

To run tests to make sure everything works as intended, you can then type in

run_tests()

Examples

Compute diagonal restriction derivative

Let $D$ be a positive fundamental discriminant, and let $p$ be a prime inert in $\mathbb{Q}(\sqrt D)$.

The diagonal restriction derivative from DPV2, a certain overconvergent $p$-adic modular form of tame level $1$, can be computed using the following function:

F = BinaryQF([3,3,-5])
p = 7
number_of_terms = 20
diagonal_restriction_derivative(F,p,number_of_terms)

Compute Gross-Stark units

The function GS_unit(D,p) computes the Gross–Stark units attached to a positive fundamental discriminant $D$ and a prime $p$ inert in $\mathbb Q(\sqrt D)$. This is done by recognising the $p$-adic exponential of the constant term of the diagonal restriction derivative as an algebraic number.

GS_unit(221,3)

By bumping the precision, it is also possible to find units for much bigger primes than in magma. This is probably due to our slightly different algdep implementation.

GS_unit(24,41,100,30)
# returns 41*x^2 + 46*x + 41

Zagier reduction of quadratic forms

The file src/quadforms.sage implements Zagier’s reduction theory for indefinite binary quadratic forms, and in particular uses it to compute the automorph, meaning the stabiliser of a BQF in $\mathrm{SL}_2(\mathbb Z)$, and special values of associated partial $ζ$-functions.

F = BinaryQF([3,3,-5])

F.automorph()
# [ 17 -15]
# [ -9   8]

F.is_Zagier_reduced()
# False

F.Zagier_reduce()
# x^2 + 9*x*y + 3*y^2

F.Zagier_reduced_cycle()
# ([x^2 + 9*x*y + 3*y^2, 3*x^2 + 9*x*y + y^2], [9, 3])

On compatibility with magma code

• The function diagonal_restriction_derivative has been tested against the magma code in the drd repository. The (finite approximation to) the diagonal restriction copy-pasted from sage is found by magma in the correct space of overconvergent forms, and the constant term gives rise to a GS unit in the usual way. (Tested for $D = 33$, $p = 5$, $Q = 2x^2 + 3xy - 3y^2$.)

Todo:

• [X] Detach $p$-adic precision from number of terms of diagonal restriction derivative, and make intelligent choice automatically
• [ ] Implement algorithms to compute Stark–Heegner points using sage’s existing $p$-adic uniformisation framework, see this. This requires in particular porting the ordinary projection routines from Lauder/Vonk.
• [X] Add doctests, examples (can do this using Pyment, available from pip.)