# Stark-Heegner points via overconvergent modular symbols

Developed by Henri Darmon and Robert Pollack.

To install the shp package it is best if you have a Linux or Unix machine with Magma already installed. To download and install the package, follow these steps.

2. Type

tar -xf shp.tar

at the Unix prompt. This creates a new directory called shp_package and completes the installation.

# 2. Using the shp package

Instructions for using the shp Package, as well as the mathematics underlying the programs, can be found in the article

H. Darmon and R. Pollack, The efficient calculation of Stark-Heegner points via overconvergent modular symbols, to appear.

Here is a sample Magma dialogue which can be executed after invoking Magma from within the shp subdirectory of shp_package. (User input has been entered in boldface.)

>
>
>
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The maximal accuracy currently available is 400 11 -adic digits
100

-------------------------------------------------------------------
You are now set up to perform Stark-Heegner point calculations on
E = Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
working over Qp = the 11-adic field mod 11^100

To compute the Stark-Heegner points of discriminant D, type
HP,P,hD := stark_heegner_points(E,D,Qp);
from the Magma command prompt.
The first few values of D you can try are: [ 8, 13, 17, 21, 24 ]
-------------------------------------------------------------------
> HP,P,hD:= stark_heegner_points(E,8,Qp);
--> Computing the Stark-Heegner points of discriminant 8 over the Elliptic
Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
The calculation is being done in 11-adic field mod 11^100

The discriminant D = 8 has class number 1
1 Computing point attached to the binary quadratic form <1,2,-1>
Sum of the Stark-Heegner points (over C_p) =
(-68903061699111350920591685860448183881321656000192332165732387760774926047761\ 538384700579748729263222996 + O(11^100) :
172257654247778377301479214651120459703304140000480830414330969401937315119\ 40384596175144937182315805751*w + 68903061699111350920591685860448183881321\ 656000192332165732387760774926047761538384700579748729263223000 + O(11^100)
: 1 + O(11^100))
This p-adic point is close to the global element [
9/2,
1/8*(7*s - 4),
1
]
(9/2 : 1/8*(7*s - 4) : 1) is indeed a global point on E(K).

The polynomial satisfied by the x-ccordinates of the Stark-Heegner points is
x - 9/2

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>

> HP,P,hD:= stark_heegner_points(E,101,Qp);
--> Computing the Stark-Heegner points of discriminant 101 over the EllipticCurve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
The calculation is being done in 11-adic field mod 11^100

The discriminant D = 101 has class number 1

1 Computing point attached to the binary quadratic form <1,9,-5>

Sum of the Stark-Heegner points (over C_p) =
(583746317230061216315886136468232240701573432686974474692584171215769760262158\ 14798368433233779413593884 + O(11^100) :
-11749410862925460842271745708724612497101068471386977180472245534964827015\ 001096220235157941536451534318*w + 6890306169911135092059168586044818388132\ 1656000192332165732387760774926047761538384700579748729263223000 + O(11^100): 1 + O(11^100))
This p-adic point is close to the global element [
1081624136644692539667084685116849/246846541822770321447579971520100,
1/3878292595549843673832197996141240094552431551000*(4503481327176251972713\ 25875616860240657045635493*s - 19391462977749218369160989980706200472762157\ 75500),
1
]
(1081624136644692539667084685116849/246846541822770321447579971520100 :
1/3878292595549843673832197996141240094552431551000*(45034813271762519727132587\5616860240657045635493*s - 1939146297774921836916098998070620047276215775500) :
1) is indeed a global point on E(K).

The polynomial satisfied by the x-ccordinates of the Stark-Heegner points is
x - 1081624136644692539667084685116849/246846541822770321447579971520100

----------------------------------------------------------------

>
>
>

-----------------------------------------------------------------------
The maximal accuracy currently available is 100 37 -adic digits
40

-------------------------------------------------------------------
You are now set up to perform Stark-Heegner point calculations on
E = Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
working over Qp = the 37-adic field mod 37^40

To compute the Stark-Heegner points of discriminant D, type
HP,P,hD := stark_heegner_points(E,D,Qp);
from the Magma command prompt.
The first few values of D you can try are: [ 5, 8, 13, 17, 20 ]
-------------------------------------------------------------------
> HP,P,hD := stark_heegner_points(E,401,Qp);
--> Computing the Stark-Heegner points of discriminant 401 over the EllipticCurve defined by y^2 + y = x^3 - x over Rational Field
The calculation is being done in 37-adic field mod 37^40

The discriminant D = 401 has class number 5

1 Computing point attached to the binary quadratic form <1,19,-10>
2 Computing point attached to the binary quadratic form <-2,19,5>
3 Computing point attached to the binary quadratic form <4,15,-11>
4 Computing point attached to the binary quadratic form <-4,17,7>
5 Computing point attached to the binary quadratic form <2,17,-14>

Sum of the Stark-Heegner points (over C_p) = (O(37^40) : -1 + O(37^40) : 1 + O(37^40))
This p-adic point is close to the global element [ 0, -1, 1 ]
(0 : -1 : 1) is indeed a global point on E(K).

The polynomial satisfied by the x-ccordinates of the Stark-Heegner points is
x^5 - 73/9*x^4 + 1195/81*x^3 - 173/81*x^2 - 976/81*x + 527/81

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