37.9 Formal groups of elliptic curves

Module: sage.schemes.elliptic_curves.formal_group

Formal groups of elliptic curves.

Author Log:

Class: EllipticCurveFormalGroup

class EllipticCurveFormalGroup
The formal group associated to an elliptic curve.
EllipticCurveFormalGroup( self, E)

Functions: curve,$ \,$ differential,$ \,$ group_law,$ \,$ inverse,$ \,$ log,$ \,$ mult_by_n,$ \,$ sigma,$ \,$ w,$ \,$ x,$ \,$ y

curve( self)

The elliptic curve this formal group is associated to.

sage: E = EllipticCurve("37a")
sage: F = E.formal_group()
sage: F.curve()
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field

differential( self, [prec=20])

Returns the power series $ f(t) = 1 + \cdots$ such that $ f(t) dt$ is the usual invariant differential $ dx/(2y + a_1 x + a_3)$ .

Input:

prec
- nonnegative integer, answer will be returned $ O(t^{\var{prec}})$

Output: a power series with given precision

DETAILS: Return the formal series

$\displaystyle f(t) = 1 + a_1 t + ({a_1}^2 + a_2) t^2 + \cdots
$

to precision $ O(t^prec)$ of page 113 of [Silverman AEC1].

The result is cached, and a cached version is returned if possible.

WARNING: The resulting series will have precision prec, but its parent PowerSeriesRing will have default precision 20 (or whatever the default default is).

sage: EllipticCurve([-1, 1/4]).formal_group().differential(15)
 1 - 2*t^4 + 3/4*t^6 + 6*t^8 - 5*t^10 - 305/16*t^12 + 105/4*t^14 + O(t^15)
sage: EllipticCurve(Integers(53), [-1, 1/4]).formal_group().differential(15)
 1 + 51*t^4 + 14*t^6 + 6*t^8 + 48*t^10 + 24*t^12 + 13*t^14 + O(t^15)

Author: David Harvey (2006-09-10): factored out of log

group_law( self, [prec=10])

The formal group law.

Input:

prec
- integer

Output: a power series with given precision in ZZ[[ ZZ[['t1']], 't2']]

DETAILS: Return the formal power series

$\displaystyle F(t1, t2) = t1 + t2 - a1 t1 t2 - \cdots
$

to precision $ O(t^prec)$ of page 115 of [Silverman AEC1].

The result is cached, and a cached version is returned if possible.

WARNING: The resulting power series will have precision prec, but its parent PowerSeriesRing will have default precision 20 (or whatever the default default is).

Author: Nick Alexander: minor fixes, docstring

sage: e = EllipticCurve([1, 2])
sage: F = e.formal_group().group_law(5); F
 t1 + O(t1^5) + (1 - 2*t1^4 + O(t1^5))*t2 + (-4*t1^3 + O(t1^5))*t2^2 +
(-4*t1^2 - 30*t1^4 + O(t1^5))*t2^3 + (-2*t1 - 30*t1^3 + O(t1^5))*t2^4 +
O(t2^5)
sage: i = e.formal_group().inverse(5)
sage: Fx = F.base_base_extend(i.parent())
sage: Fx (i.parent().gen()) (i)
 O(t^5)

Let's ensure caching with changed precision is working:

sage: e.formal_group().group_law(4)
 t1 + O(t1^4) + (1 + O(t1^4))*t2 + (-4*t1^3 + O(t1^4))*t2^2 + (-4*t1^2 +
O(t1^4))*t2^3 + O(t2^4)

inverse( self, [prec=20])

The formal group inverse law i(t), which satisfies F(t, i(t)) = 0.

Input:

prec
- integer

Output: a power series with given precision

DETAILS: Return the formal power series

$\displaystyle i(t) = - t + a_1 t^2 + \cdots
$

to precision $ O(t^prec)$ of page 114 of [Silverman AEC1].

The result is cached, and a cached version is returned if possible.

WARNING: The resulting power series will have precision prec, but its parent PowerSeriesRing will have default precision 20 (or whatever the default default is).

sage: e = EllipticCurve([1, 2])
sage: F = e.formal_group().group_law(5)
sage: i = e.formal_group().inverse(5)
sage: Fx = F.base_base_extend(i.parent())
sage: Fx (i) (i.parent().gen())
O(t^5)

log( self, [prec=20])

Returns the power series $ f(t) = t + \cdots$ which is an isomorphism to the additive formal group.

Generally this only makes sense in characteristic zero, although the terms before $ t^p$ may work in characteristic $ p$ .

Input:

prec
- nonnegative integer

Output: a power series with given precision

sage: EllipticCurve([-1, 1/4]).formal_group().log(15)
 t - 2/5*t^5 + 3/28*t^7 + 2/3*t^9 - 5/11*t^11 - 305/208*t^13 + O(t^15)

Author: David Harvey (2006-09-10): rewrote to use differential

mult_by_n( self, n, [prec=10])

The formal 'multiplication by n' endomorphism $ [n]$ .

Input:

prec
- integer

Output: a power series with given precision

DETAILS: Return the formal power series

$\displaystyle [n](t) = n t + \cdots
$

to precision $ O(t^prec)$ of Proposition 2.3 of [Silverman AEC1].

WARNING: The resulting power series will have precision prec, but its parent PowerSeriesRing will have default precision 20 (or whatever the default default is).

Author Log:

sage: e = EllipticCurve([1, 2, 3, 4, 6])
sage: e.formal_group().mult_by_n(0, 5)
 O(t^5)
sage: e.formal_group().mult_by_n(1, 5)
 t + O(t^5)

We verify an identity of low degree:

sage: none = e.formal_group().mult_by_n(-1, 5)
sage: two = e.formal_group().mult_by_n(2, 5)
sage: ntwo = e.formal_group().mult_by_n(-2, 5)
sage: ntwo - none(two)
 O(t^5)
sage: ntwo - two(none)
 O(t^5)

It's quite fast:

sage: E = EllipticCurve("37a"); F = E.formal_group()
sage: F.mult_by_n(100, 20)
100*t - 49999950*t^4 + 3999999960*t^5 + 14285614285800*t^7 -
2999989920000150*t^8 + 133333325333333400*t^9 - 3571378571674999800*t^10 +
1402585362624965454000*t^11 - 146666057066712847999500*t^12 +
5336978000014213190385000*t^13 - 519472790950932256570002000*t^14 +
93851927683683567270392002800*t^15 - 6673787211563812368630730325175*t^16 +
320129060335050875009191524993000*t^17 -
45670288869783478472872833214986000*t^18 +
5302464956134111125466184947310391600*t^19 + O(t^20)

w( self, [prec=20])

The formal group power series w.

Input:

prec
- integer

Output: a power series with given precision

DETAILS: Return the formal power series

$\displaystyle w(t) = t^3 + a_1 t^4 + (a_2 + a_1^2) t^5 + \cdots
$

to precision $ O(t^prec)$ of Proposition IV.1.1 of [Silverman AEC1]. This is the formal expansion of $ w = -1/y$ about the formal parameter $ t = -x/y$ at $ \infty$ .

The result is cached, and a cached version is returned if possible.

WARNING: The resulting power series will have precision prec, but its parent PowerSeriesRing will have default precision 20 (or whatever the default default is).

ALGORITHM: Uses Newton's method to solve the elliptic curve equation at the origin. Complexity is roughly $ O(M(n))$ where $ n$ is the precision and $ M(n)$ is the time required to multiply polynomials of length $ n$ over the coefficient ring of $ E$ .

Author: David Harvey (2006-09-09): modified to use Newton's method instead of a recurrence formula.

sage: e = EllipticCurve([0, 0, 1, -1, 0])
sage: e.formal_group().w(10)
 t^3 + t^6 - t^7 + 2*t^9 + O(t^10)

Check that caching works:

sage: e = EllipticCurve([3, 2, -4, -2, 5])
sage: e.formal_group().w(20)
 t^3 + 3*t^4 + 11*t^5 + 35*t^6 + 101*t^7 + 237*t^8 + 312*t^9 - 949*t^10 -
10389*t^11 - 57087*t^12 - 244092*t^13 - 865333*t^14 - 2455206*t^15 -
4366196*t^16 + 6136610*t^17 + 109938783*t^18 + 688672497*t^19 + O(t^20)
sage: e.formal_group().w(7)
 t^3 + 3*t^4 + 11*t^5 + 35*t^6 + O(t^7)
sage: e.formal_group().w(35)
 t^3 + 3*t^4 + 11*t^5 + 35*t^6 + 101*t^7 + 237*t^8 + 312*t^9 - 949*t^10 -
10389*t^11 - 57087*t^12 - 244092*t^13 - 865333*t^14 - 2455206*t^15 -
4366196*t^16 + 6136610*t^17 + 109938783*t^18 + 688672497*t^19 +
3219525807*t^20 + 12337076504*t^21 + 38106669615*t^22 + 79452618700*t^23 -
33430470002*t^24 - 1522228110356*t^25 - 10561222329021*t^26 -
52449326572178*t^27 - 211701726058446*t^28 - 693522772940043*t^29 -
1613471639599050*t^30 - 421817906421378*t^31 + 23651687753515182*t^32 +
181817896829144595*t^33 + 950887648021211163*t^34 + O(t^35)

x( self, [prec=20])

Return the formal series $ x(t) = t/w(t)$ in terms of the local parameter $ t = -x/y$ at infinity.

Input:

prec
- integer

Output: a laurent series with given precision

DETAILS: Return the formal series

$\displaystyle x(t) = t^{-2} - a_1 t^{-1} - a_2 - a_3 t - \cdots
$

to precision $ O(t^prec)$ of page 113 of [Silverman AEC1].

WARNING: The resulting series will have precision prec, but its parent PowerSeriesRing will have default precision 20 (or whatever the default default is).

sage: EllipticCurve([0, 0, 1, -1, 0]).formal_group().x(10)
 t^-2 - t + t^2 - t^4 + 2*t^5 - t^6 - 2*t^7 + 6*t^8 - 6*t^9 + O(t^10)

y( self, [prec=20])

Return the formal series $ y(t) = -1/w(t)$ in terms of the local parameter $ t = -x/y$ at infinity.

Input:

prec
- integer

Output: a laurent series with given precision

DETAILS: Return the formal series

$\displaystyle y(t) = - t^{-3} + a_1 t^{-2} + a_2 t + a_3 + \cdots
$

to precision $ O(t^prec)$ of page 113 of [Silverman AEC1].

The result is cached, and a cached version is returned if possible.

WARNING: The resulting series will have precision prec, but its parent PowerSeriesRing will have default precision 20 (or whatever the default default is).

sage: EllipticCurve([0, 0, 1, -1, 0]).formal_group().y(10)
 -t^-3 + 1 - t + t^3 - 2*t^4 + t^5 + 2*t^6 - 6*t^7 + 6*t^8 + 3*t^9 +
O(t^10)

Special Functions: __init__,$ \,$ _repr_

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