Compute non-archimedean Darmon points

## Project description

(for the full documentation, please see http://mmasdeu.github.io/darmonpoints/doc/html/)

## What is this?

The **darmonpoints** package can compute many different types of what is known as Darmon points. These are known as *Stark-Heegner* points in some literature, and originated in [Darmon]. Subsequent generalizations were introduced by [Greenberg] and [Trifkovic]. This has been generalized by [GMS1] to elliptic curves defined over number fields of arbitrary signature. Darmon points are attached to triples (F,E,K), where F is a number field, E/F is an elliptic curve defined over F, and K/F is a quadratic extension. These triples must satisfy certain conditions for Darmon points to be attached to them. The article [GMS1] contains an overview of all of this. We include also a variation used in [KP].

The **darmonpoints** package can also compute equations for some elliptic curves E/F defined over number fields F, as long as certain conditions are satisfied. Namely:

- F has narrow class number 1.
- if N is the conductor of the elliptic curve, it must admit a factorization of the form N = PDM, where:
- P, D and M are relative coprime.
- P is a prime ideal of F of prime norm.
- D is the discriminant of a quaternion algebra over F which is split at only one infinite place.

Finally, we include the module *padicperiods*, which allows for the computation of p-adic periods attached to two-dimensional components of the cohomology of the same arithmetic groups, and which has allowed us to find the corresponding abelian surfaces in some cases (see [GM]).

The full documentation can be found at http://mmasdeu.github.io/darmonpoints/doc/html/

## Installation

Installation of the *darmonpoints* package has been greatly simplified, thanks to Matthias Köppe “Sample Sage” (https://github.com/mkoeppe/sage_sample). For most operations one *does need* to have **Magma** (https://magma.maths.usyd.edu.au/magma/) installed, although we do hope that in the future Sage will include the required functionality.

We include for convenience the dependency Magma package *KleinianGroups* by Aurel Page, the original of which can be found at http://www.normalesup.org/~page/Recherche/Logiciels/KleinianGroups/KleinianGroups-1.0.tar.gz.

To install the package use the following command:

sage -pip install --user --upgrade darmonpoints

If you rather install the cutting-edge version from the github repository (which is likely to be broken), then use the following command:

sage -pip install --user --upgrade -v git+https://github.com/mmasdeu/darmonpoints.git

## Basic usage

The files `darmonpoints.py`, `findcurve.py` and `padicperiods.py` contain the high level routines from which show how to use the package, although one can use parts of the package in other ways if one feels adventurous. Here are some sample calculations that one can try:

sage: from darmonpoints.darmonpoints import darmon_point

A classical Darmon (a.k.a. Stark-Heegner) point. The following will perform a 7-adic calculation to precision 7^20, to find a point over the real quadratic field of discriminant 41 for the elliptic curve

`35a1`:sage: darmon_point(7,EllipticCurve('35a1'),41,20)

A quaternionic Darmon (a.k.a. Greenberg) point:

sage: darmon_point(13,EllipticCurve('78a1'),5,20)

A Darmon point for a curve over a field of mixed signature:

sage: F.<r> = NumberField(x^3 - x^2 - x + 2) sage: E = EllipticCurve([-r -1, -r, -r - 1,-r - 1, 0]) sage: N = E.conductor() sage: P = F.ideal(r^2 - 2*r - 1) sage: beta = -3*r^2 + 9*r - 6 sage: darmon_point(P,E,beta,20)

We can also *discover* equations of curves!

We first find a curve over the rationals. The following command will find a curve of conductor 30, using a 5-adic calculation with precision of 5^20, and the quaternion algebra of discriminant 6:

sage: from darmonpoints.findcurve import find_curve sage: find_curve(5,6,30,20)

This constructs the curve with equation:

y^2 + x*y + y = x^3 + x + 2

Now for a curve defined over a real quadratic field. Note that here we must specify which place will ramify in the quaternion algebra:

sage: from darmonpoints.findcurve import find_curve sage: F.<r> = QuadraticField(5) sage: P = F.ideal(3/2*r + 1/2) sage: D = F.ideal(3) sage: find_curve(P,D,P*D,30,ramification_at_infinity = F.real_places()[:1])

This returns something like:

y^2 + (1/2*r-1/2)*x*y = x^3 + (1/2*r+1/2)*x^2 + (285/2*r-793/2)*x + (3153/2*r-7689/2)

A curve over a cubic field of mixed signature:

sage: from darmonpoints.findcurve import find_curve sage: F.<r> = NumberField(x^3 -3) sage: P = F.ideal(r-2) sage: D = F.ideal(r-1) sage: find_curve(P,D,P*D,30)

This should return an elliptic curve like this:

y^2 + r*x*y + (r+1)*y = x^3 + (-575*r^2-829*r-1195)*x + (-13327*r^2-19221*r-27721)

[Darmon] | H.Darmon. Integration on Hp x H and arithmetic applications. Annals of Math. |

[Greenberg] | M.Greenberg. Stark-Heegner points and the cohomology of quaternionic Shimura varieties. Duke Math. |

[GM] | X.Guitart, M.Masdeu. Periods of modular GL2-type abelian varieties and p-adic integration. Experimental Mathematics. |

[GMS1] | (1, 2) X.Guitart, M.Masdeu, M.H.Sengun. Darmon points on elliptic curves over number fields of arbitrary signature. Proc. LMS. |

[GMS2] | X.Guitart, M.Masdeu, M.H.Sengun. Uniformization of modular elliptic curves via p-adic methods. Journal of Algebra. |

[KP] | A.Pacetti, D.Kohen (with an appendix by M.Masdeu) On Heegner points for primes of additive reduction ramifying in the base field. Transactions of the AMS. |

[Trifkovic] | M.Trifkovic. Stark-Heegner points on elliptic curves defined over imaginary quadratic fields. Duke Math. |

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