Darmon Points¶

For installation and basic usage instructions please see the main Github repository (https://github.com/mmasdeu/darmonpoints).

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:
1. P, D and M are relative coprime.
2. P is a prime ideal of F of prime norm.
3. 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]).

[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.

This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.

Darmon Points¶

High level functionality¶

Arithmetic Groups¶

Cohomology and Homology¶

Integration pairing¶

Overconvergent distributions¶

Internals¶

Indices and Tables¶