Computation of Frobenius matrix on Monsky-Washnitzer cohomology¶

The most interesting functions to be exported here are matrix_of_frobenius() and adjusted_prec().

Currently this code is limited to the case \(p \geq 5\) (no \(GF(p^n)\) for \(n > 1\)), and only handles the elliptic curve case (not more general hyperelliptic curves).

REFERENCES:

AUTHORS:

David Harvey and Robert Bradshaw: initial code developed at the 2006 MSRI graduate workshop, working with Jennifer Balakrishnan and Liang Xiao
David Harvey (2006-08): cleaned up, rewrote some chunks, lots more documentation, added Newton iteration method, added more complete ‘trace trick’, integrated better into Sage.
David Harvey (2007-02): added algorithm with sqrt(p) complexity (removed in May 2007 due to better C++ implementation)
Robert Bradshaw (2007-03): keep track of exact form in reduction algorithms
Robert Bradshaw (2007-04): generalization to hyperelliptic curves
Julian Rueth (2014-05-09): improved caching

class sage.schemes.hyperelliptic_curves.monsky_washnitzer.MonskyWashnitzerDifferential(parent, val, offset=0)[source]¶

Bases: ModuleElement

An element of the Monsky-Washnitzer ring of differentials, of the form \(F dx/2y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: x,y = C.monsky_washnitzer_gens()
sage: MW = C.invariant_differential().parent()
sage: MW(x)
x dx/2y
sage: MW(y)
y*1 dx/2y
sage: MW(x, 10)
y^10*x dx/2y

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> x,y = C.monsky_washnitzer_gens()
>>> MW = C.invariant_differential().parent()
>>> MW(x)
x dx/2y
>>> MW(y)
y*1 dx/2y
>>> MW(x, Integer(10))
y^10*x dx/2y

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
x,y = C.monsky_washnitzer_gens()
MW = C.invariant_differential().parent()
MW(x)
MW(y)
MW(x, 10)

coeff()[source]¶

Return \(A\), where this element is \(A dx/2y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: x,y = C.monsky_washnitzer_gens()
sage: w = C.invariant_differential()
sage: w
1 dx/2y
sage: w.coeff()
1
sage: (x*y*w).coeff()
y*x

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> x,y = C.monsky_washnitzer_gens()
>>> w = C.invariant_differential()
>>> w
1 dx/2y
>>> w.coeff()
1
>>> (x*y*w).coeff()
y*x

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
x,y = C.monsky_washnitzer_gens()
w = C.invariant_differential()
w
w.coeff()
(x*y*w).coeff()

coeffs(R=None)[source]¶

Used to obtain the raw coefficients of a differential, see SpecialHyperellipticQuotientElement.coeffs()

INPUT:

R – an (optional) base ring in which to cast the coefficients

OUTPUT: the raw coefficients of \(A\) where self is \(A dx/2y\)

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: x,y = C.monsky_washnitzer_gens()
sage: w = C.invariant_differential()
sage: w.coeffs()
([(1, 0, 0, 0, 0)], 0)
sage: (x*w).coeffs()
([(0, 1, 0, 0, 0)], 0)
sage: (y*w).coeffs()
([(0, 0, 0, 0, 0), (1, 0, 0, 0, 0)], 0)
sage: (y^-2*w).coeffs()
([(1, 0, 0, 0, 0), (0, 0, 0, 0, 0), (0, 0, 0, 0, 0)], -2)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> x,y = C.monsky_washnitzer_gens()
>>> w = C.invariant_differential()
>>> w.coeffs()
([(1, 0, 0, 0, 0)], 0)
>>> (x*w).coeffs()
([(0, 1, 0, 0, 0)], 0)
>>> (y*w).coeffs()
([(0, 0, 0, 0, 0), (1, 0, 0, 0, 0)], 0)
>>> (y**-Integer(2)*w).coeffs()
([(1, 0, 0, 0, 0), (0, 0, 0, 0, 0), (0, 0, 0, 0, 0)], -2)

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
x,y = C.monsky_washnitzer_gens()
w = C.invariant_differential()
w.coeffs()
(x*w).coeffs()
(y*w).coeffs()
(y^-2*w).coeffs()

coleman_integral(P, Q)[source]¶

Compute the definite integral of self from \(P\) to \(Q\).

INPUT:

\(P\), \(Q\) – two points on the underlying curve

OUTPUT: \(\int_P^Q \text{self}\)

EXAMPLES:

Sage

sage: K = pAdicField(5,7)
sage: E = EllipticCurve(K,[-31/3,-2501/108]) #11a
sage: P = E(K(14/3), K(11/2))
sage: w = E.invariant_differential()
sage: w.coleman_integral(P, 2*P)
O(5^6)

sage: Q = E([3,58332])
sage: w.coleman_integral(P,Q)
2*5 + 4*5^2 + 3*5^3 + 4*5^4 + 3*5^5 + O(5^6)
sage: w.coleman_integral(2*P,Q)
2*5 + 4*5^2 + 3*5^3 + 4*5^4 + 3*5^5 + O(5^6)
sage: (2*w).coleman_integral(P, Q) == 2*(w.coleman_integral(P, Q))
True

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(5),Integer(7))
>>> E = EllipticCurve(K,[-Integer(31)/Integer(3),-Integer(2501)/Integer(108)]) #11a
>>> P = E(K(Integer(14)/Integer(3)), K(Integer(11)/Integer(2)))
>>> w = E.invariant_differential()
>>> w.coleman_integral(P, Integer(2)*P)
O(5^6)

>>> Q = E([Integer(3),Integer(58332)])
>>> w.coleman_integral(P,Q)
2*5 + 4*5^2 + 3*5^3 + 4*5^4 + 3*5^5 + O(5^6)
>>> w.coleman_integral(Integer(2)*P,Q)
2*5 + 4*5^2 + 3*5^3 + 4*5^4 + 3*5^5 + O(5^6)
>>> (Integer(2)*w).coleman_integral(P, Q) == Integer(2)*(w.coleman_integral(P, Q))
True

Sage Live

K = pAdicField(5,7)
E = EllipticCurve(K,[-31/3,-2501/108]) #11a
P = E(K(14/3), K(11/2))
w = E.invariant_differential()
w.coleman_integral(P, 2*P)
Q = E([3,58332])
w.coleman_integral(P,Q)
w.coleman_integral(2*P,Q)
(2*w).coleman_integral(P, Q) == 2*(w.coleman_integral(P, Q))

extract_pow_y(k)[source]¶

Return the power of \(y\) in \(A\) where self is \(A dx/2y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = C.monsky_washnitzer_gens()
sage: A = y^5 - x*y^3
sage: A.extract_pow_y(5)
[1, 0, 0, 0, 0]
sage: (A * C.invariant_differential()).extract_pow_y(5)
[1, 0, 0, 0, 0]

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = C.monsky_washnitzer_gens()
>>> A = y**Integer(5) - x*y**Integer(3)
>>> A.extract_pow_y(Integer(5))
[1, 0, 0, 0, 0]
>>> (A * C.invariant_differential()).extract_pow_y(Integer(5))
[1, 0, 0, 0, 0]

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 3*x + 1)
x,y = C.monsky_washnitzer_gens()
A = y^5 - x*y^3
A.extract_pow_y(5)
(A * C.invariant_differential()).extract_pow_y(5)

integrate(P, Q)[source]¶

Compute the definite integral of self from \(P\) to \(Q\).

INPUT:

\(P\), \(Q\) – two points on the underlying curve

OUTPUT: \(\int_P^Q \text{self}\)

EXAMPLES:

Sage

sage: K = pAdicField(5,7)
sage: E = EllipticCurve(K,[-31/3,-2501/108]) #11a
sage: P = E(K(14/3), K(11/2))
sage: w = E.invariant_differential()
sage: w.coleman_integral(P, 2*P)
O(5^6)

sage: Q = E([3,58332])
sage: w.coleman_integral(P,Q)
2*5 + 4*5^2 + 3*5^3 + 4*5^4 + 3*5^5 + O(5^6)
sage: w.coleman_integral(2*P,Q)
2*5 + 4*5^2 + 3*5^3 + 4*5^4 + 3*5^5 + O(5^6)
sage: (2*w).coleman_integral(P, Q) == 2*(w.coleman_integral(P, Q))
True

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(5),Integer(7))
>>> E = EllipticCurve(K,[-Integer(31)/Integer(3),-Integer(2501)/Integer(108)]) #11a
>>> P = E(K(Integer(14)/Integer(3)), K(Integer(11)/Integer(2)))
>>> w = E.invariant_differential()
>>> w.coleman_integral(P, Integer(2)*P)
O(5^6)

>>> Q = E([Integer(3),Integer(58332)])
>>> w.coleman_integral(P,Q)
2*5 + 4*5^2 + 3*5^3 + 4*5^4 + 3*5^5 + O(5^6)
>>> w.coleman_integral(Integer(2)*P,Q)
2*5 + 4*5^2 + 3*5^3 + 4*5^4 + 3*5^5 + O(5^6)
>>> (Integer(2)*w).coleman_integral(P, Q) == Integer(2)*(w.coleman_integral(P, Q))
True

Sage Live

K = pAdicField(5,7)
E = EllipticCurve(K,[-31/3,-2501/108]) #11a
P = E(K(14/3), K(11/2))
w = E.invariant_differential()
w.coleman_integral(P, 2*P)
Q = E([3,58332])
w.coleman_integral(P,Q)
w.coleman_integral(2*P,Q)
(2*w).coleman_integral(P, Q) == 2*(w.coleman_integral(P, Q))

max_pow_y()[source]¶

Return the maximum power of \(y\) in \(A\) where self is \(A dx/2y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = C.monsky_washnitzer_gens()
sage: w = y^5 * C.invariant_differential()
sage: w.max_pow_y()
5
sage: w = (x^2*y^4 + y^5) * C.invariant_differential()
sage: w.max_pow_y()
5

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = C.monsky_washnitzer_gens()
>>> w = y**Integer(5) * C.invariant_differential()
>>> w.max_pow_y()
5
>>> w = (x**Integer(2)*y**Integer(4) + y**Integer(5)) * C.invariant_differential()
>>> w.max_pow_y()
5

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 3*x + 1)
x,y = C.monsky_washnitzer_gens()
w = y^5 * C.invariant_differential()
w.max_pow_y()
w = (x^2*y^4 + y^5) * C.invariant_differential()
w.max_pow_y()

min_pow_y()[source]¶

Return the minimum power of \(y\) in \(A\) where self is \(A dx/2y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = C.monsky_washnitzer_gens()
sage: w = y^5 * C.invariant_differential()
sage: w.min_pow_y()
5
sage: w = (x^2*y^4 + y^5) * C.invariant_differential()
sage: w.min_pow_y()
4

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = C.monsky_washnitzer_gens()
>>> w = y**Integer(5) * C.invariant_differential()
>>> w.min_pow_y()
5
>>> w = (x**Integer(2)*y**Integer(4) + y**Integer(5)) * C.invariant_differential()
>>> w.min_pow_y()
4

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 3*x + 1)
x,y = C.monsky_washnitzer_gens()
w = y^5 * C.invariant_differential()
w.min_pow_y()
w = (x^2*y^4 + y^5) * C.invariant_differential()
w.min_pow_y()

reduce()[source]¶

Use homology relations to find \(a\) and \(f\) such that this element is equal to \(a + df\), where \(a\) is given in terms of the \(x^i dx/2y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: x,y = C.monsky_washnitzer_gens()
sage: w = (y*x).diff()
sage: w.reduce()
(y*x, 0 dx/2y)

sage: w = x^4 * C.invariant_differential()
sage: w.reduce()
(1/5*y*1, 4/5*1 dx/2y)

sage: w = sum(QQ.random_element() * x^i * y^j
....:         for i in [0..4] for j in [-3..3]) * C.invariant_differential()
sage: f, a = w.reduce()
sage: f.diff() + a - w
0 dx/2y

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> x,y = C.monsky_washnitzer_gens()
>>> w = (y*x).diff()
>>> w.reduce()
(y*x, 0 dx/2y)

>>> w = x**Integer(4) * C.invariant_differential()
>>> w.reduce()
(1/5*y*1, 4/5*1 dx/2y)

>>> w = sum(QQ.random_element() * x**i * y**j
...         for i in (ellipsis_range(Integer(0),Ellipsis,Integer(4))) for j in (ellipsis_range(-Integer(3),Ellipsis,Integer(3)))) * C.invariant_differential()
>>> f, a = w.reduce()
>>> f.diff() + a - w
0 dx/2y

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
x,y = C.monsky_washnitzer_gens()
w = (y*x).diff()
w.reduce()
w = x^4 * C.invariant_differential()
w.reduce()
w = sum(QQ.random_element() * x^i * y^j
        for i in [0..4] for j in [-3..3]) * C.invariant_differential()
f, a = w.reduce()
f.diff() + a - w

reduce_fast(even_degree_only=False)[source]¶

Use homology relations to find \(a\) and \(f\) such that this element is equal to \(a + df\), where \(a\) is given in terms of the \(x^i dx/2y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^3 - 4*x + 4)
sage: x, y = E.monsky_washnitzer_gens()
sage: x.diff().reduce_fast()
(x, (0, 0))
sage: y.diff().reduce_fast()
(y*1, (0, 0))
sage: (y^-1).diff().reduce_fast()
((y^-1)*1, (0, 0))
sage: (y^-11).diff().reduce_fast()
((y^-11)*1, (0, 0))
sage: (x*y^2).diff().reduce_fast()
(y^2*x, (0, 0))

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(3) - Integer(4)*x + Integer(4))
>>> x, y = E.monsky_washnitzer_gens()
>>> x.diff().reduce_fast()
(x, (0, 0))
>>> y.diff().reduce_fast()
(y*1, (0, 0))
>>> (y**-Integer(1)).diff().reduce_fast()
((y^-1)*1, (0, 0))
>>> (y**-Integer(11)).diff().reduce_fast()
((y^-11)*1, (0, 0))
>>> (x*y**Integer(2)).diff().reduce_fast()
(y^2*x, (0, 0))

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^3 - 4*x + 4)
x, y = E.monsky_washnitzer_gens()
x.diff().reduce_fast()
y.diff().reduce_fast()
(y^-1).diff().reduce_fast()
(y^-11).diff().reduce_fast()
(x*y^2).diff().reduce_fast()

reduce_neg_y()[source]¶

Use homology relations to eliminate negative powers of \(y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = C.monsky_washnitzer_gens()
sage: (y^-1).diff().reduce_neg_y()
((y^-1)*1, 0 dx/2y)
sage: (y^-5*x^2+y^-1*x).diff().reduce_neg_y()
((y^-1)*x + (y^-5)*x^2, 0 dx/2y)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = C.monsky_washnitzer_gens()
>>> (y**-Integer(1)).diff().reduce_neg_y()
((y^-1)*1, 0 dx/2y)
>>> (y**-Integer(5)*x**Integer(2)+y**-Integer(1)*x).diff().reduce_neg_y()
((y^-1)*x + (y^-5)*x^2, 0 dx/2y)

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 3*x + 1)
x,y = C.monsky_washnitzer_gens()
(y^-1).diff().reduce_neg_y()
(y^-5*x^2+y^-1*x).diff().reduce_neg_y()

reduce_neg_y_fast(even_degree_only=False)[source]¶

Use homology relations to eliminate negative powers of \(y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x, y = E.monsky_washnitzer_gens()
sage: (y^-1).diff().reduce_neg_y_fast()
((y^-1)*1, 0 dx/2y)
sage: (y^-5*x^2+y^-1*x).diff().reduce_neg_y_fast()
((y^-1)*x + (y^-5)*x^2, 0 dx/2y)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x, y = E.monsky_washnitzer_gens()
>>> (y**-Integer(1)).diff().reduce_neg_y_fast()
((y^-1)*1, 0 dx/2y)
>>> (y**-Integer(5)*x**Integer(2)+y**-Integer(1)*x).diff().reduce_neg_y_fast()
((y^-1)*x + (y^-5)*x^2, 0 dx/2y)

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x, y = E.monsky_washnitzer_gens()
(y^-1).diff().reduce_neg_y_fast()
(y^-5*x^2+y^-1*x).diff().reduce_neg_y_fast()

It leaves nonnegative powers of \(y\) alone:

Sage

sage: y.diff()
(-3*1 + 5*x^4) dx/2y
sage: y.diff().reduce_neg_y_fast()
(0, (-3*1 + 5*x^4) dx/2y)

Python

>>> from sage.all import *
>>> y.diff()
(-3*1 + 5*x^4) dx/2y
>>> y.diff().reduce_neg_y_fast()
(0, (-3*1 + 5*x^4) dx/2y)

Sage Live

y.diff()
y.diff().reduce_neg_y_fast()

reduce_neg_y_faster(even_degree_only=False)[source]¶

Use homology relations to eliminate negative powers of \(y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = C.monsky_washnitzer_gens()
sage: (y^-1).diff().reduce_neg_y()
((y^-1)*1, 0 dx/2y)
sage: (y^-5*x^2+y^-1*x).diff().reduce_neg_y_faster()
((y^-1)*x + (y^-5)*x^2, 0 dx/2y)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = C.monsky_washnitzer_gens()
>>> (y**-Integer(1)).diff().reduce_neg_y()
((y^-1)*1, 0 dx/2y)
>>> (y**-Integer(5)*x**Integer(2)+y**-Integer(1)*x).diff().reduce_neg_y_faster()
((y^-1)*x + (y^-5)*x^2, 0 dx/2y)

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 3*x + 1)
x,y = C.monsky_washnitzer_gens()
(y^-1).diff().reduce_neg_y()
(y^-5*x^2+y^-1*x).diff().reduce_neg_y_faster()

reduce_pos_y()[source]¶

Use homology relations to eliminate positive powers of \(y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^3-4*x+4)
sage: x,y = C.monsky_washnitzer_gens()
sage: (y^2).diff().reduce_pos_y()
(y^2*1, 0 dx/2y)
sage: (y^2*x).diff().reduce_pos_y()
(y^2*x, 0 dx/2y)
sage: (y^92*x).diff().reduce_pos_y()
(y^92*x, 0 dx/2y)
sage: w = (y^3 + x).diff()
sage: w += w.parent()(x)
sage: w.reduce_pos_y_fast()
(y^3*1 + x, x dx/2y)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(3)-Integer(4)*x+Integer(4))
>>> x,y = C.monsky_washnitzer_gens()
>>> (y**Integer(2)).diff().reduce_pos_y()
(y^2*1, 0 dx/2y)
>>> (y**Integer(2)*x).diff().reduce_pos_y()
(y^2*x, 0 dx/2y)
>>> (y**Integer(92)*x).diff().reduce_pos_y()
(y^92*x, 0 dx/2y)
>>> w = (y**Integer(3) + x).diff()
>>> w += w.parent()(x)
>>> w.reduce_pos_y_fast()
(y^3*1 + x, x dx/2y)

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^3-4*x+4)
x,y = C.monsky_washnitzer_gens()
(y^2).diff().reduce_pos_y()
(y^2*x).diff().reduce_pos_y()
(y^92*x).diff().reduce_pos_y()
w = (y^3 + x).diff()
w += w.parent()(x)
w.reduce_pos_y_fast()

reduce_pos_y_fast(even_degree_only=False)[source]¶

Use homology relations to eliminate positive powers of \(y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^3 - 4*x + 4)
sage: x, y = E.monsky_washnitzer_gens()
sage: y.diff().reduce_pos_y_fast()
(y*1, 0 dx/2y)
sage: (y^2).diff().reduce_pos_y_fast()
(y^2*1, 0 dx/2y)
sage: (y^2*x).diff().reduce_pos_y_fast()
(y^2*x, 0 dx/2y)
sage: (y^92*x).diff().reduce_pos_y_fast()
(y^92*x, 0 dx/2y)
sage: w = (y^3 + x).diff()
sage: w += w.parent()(x)
sage: w.reduce_pos_y_fast()
(y^3*1 + x, x dx/2y)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(3) - Integer(4)*x + Integer(4))
>>> x, y = E.monsky_washnitzer_gens()
>>> y.diff().reduce_pos_y_fast()
(y*1, 0 dx/2y)
>>> (y**Integer(2)).diff().reduce_pos_y_fast()
(y^2*1, 0 dx/2y)
>>> (y**Integer(2)*x).diff().reduce_pos_y_fast()
(y^2*x, 0 dx/2y)
>>> (y**Integer(92)*x).diff().reduce_pos_y_fast()
(y^92*x, 0 dx/2y)
>>> w = (y**Integer(3) + x).diff()
>>> w += w.parent()(x)
>>> w.reduce_pos_y_fast()
(y^3*1 + x, x dx/2y)

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^3 - 4*x + 4)
x, y = E.monsky_washnitzer_gens()
y.diff().reduce_pos_y_fast()
(y^2).diff().reduce_pos_y_fast()
(y^2*x).diff().reduce_pos_y_fast()
(y^92*x).diff().reduce_pos_y_fast()
w = (y^3 + x).diff()
w += w.parent()(x)
w.reduce_pos_y_fast()

class sage.schemes.hyperelliptic_curves.monsky_washnitzer.MonskyWashnitzerDifferentialRing(base_ring)[source]¶

Bases: UniqueRepresentation, Module

A ring of Monsky–Washnitzer differentials over base_ring.

Element[source]¶: alias of MonskyWashnitzerDifferential

Q()[source]¶

Return \(Q(x)\) where the model of the underlying hyperelliptic curve of self is given by \(y^2 = Q(x)\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: MW = C.invariant_differential().parent()
sage: MW.Q()
x^5 - 4*x + 4

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> MW = C.invariant_differential().parent()
>>> MW.Q()
x^5 - 4*x + 4

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
MW = C.invariant_differential().parent()
MW.Q()

base_extend(R)[source]¶

Return a new differential ring which is self base-extended to \(R\).

INPUT:

R – ring

OUTPUT:

Self, base-extended to \(R\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: MW = C.invariant_differential().parent()
sage: MW.base_ring()
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = x^5 - 4*x + 4)
 over Rational Field
sage: MW.base_extend(Qp(5,5)).base_ring()                                   # needs sage.rings.padics
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = (1 + O(5^5))*x^5
            + (1 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5))*x + 4 + O(5^5))
 over 5-adic Field with capped relative precision 5

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> MW = C.invariant_differential().parent()
>>> MW.base_ring()
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = x^5 - 4*x + 4)
 over Rational Field
>>> MW.base_extend(Qp(Integer(5),Integer(5))).base_ring()                                   # needs sage.rings.padics
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = (1 + O(5^5))*x^5
            + (1 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5))*x + 4 + O(5^5))
 over 5-adic Field with capped relative precision 5

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
MW = C.invariant_differential().parent()
MW.base_ring()
MW.base_extend(Qp(5,5)).base_ring()                                   # needs sage.rings.padics

change_ring(R)[source]¶

Return a new differential ring which is self with the coefficient ring changed to \(R\).

INPUT:

R – ring of coefficients

OUTPUT: self with the coefficient ring changed to \(R\)

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: MW = C.invariant_differential().parent()
sage: MW.base_ring()
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = x^5 - 4*x + 4)
 over Rational Field
sage: MW.change_ring(Qp(5,5)).base_ring()                                   # needs sage.rings.padics
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = (1 + O(5^5))*x^5
            + (1 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5))*x + 4 + O(5^5))
 over 5-adic Field with capped relative precision 5

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> MW = C.invariant_differential().parent()
>>> MW.base_ring()
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = x^5 - 4*x + 4)
 over Rational Field
>>> MW.change_ring(Qp(Integer(5),Integer(5))).base_ring()                                   # needs sage.rings.padics
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = (1 + O(5^5))*x^5
            + (1 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5))*x + 4 + O(5^5))
 over 5-adic Field with capped relative precision 5

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
MW = C.invariant_differential().parent()
MW.base_ring()
MW.change_ring(Qp(5,5)).base_ring()                                   # needs sage.rings.padics

degree()[source]¶

Return the degree of \(Q(x)\), where the model of the underlying hyperelliptic curve of self is given by \(y^2 = Q(x)\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: MW = C.invariant_differential().parent()
sage: MW.Q()
x^5 - 4*x + 4
sage: MW.degree()
5

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> MW = C.invariant_differential().parent()
>>> MW.Q()
x^5 - 4*x + 4
>>> MW.degree()
5

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
MW = C.invariant_differential().parent()
MW.Q()
MW.degree()

dimension()[source]¶

Return the dimension of self.

EXAMPLES:

Sage

sage: # needs sage.rings.padics
sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: K = Qp(7,5)
sage: CK = C.change_ring(K)
sage: MW = CK.invariant_differential().parent()
sage: MW.dimension()
4

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> K = Qp(Integer(7),Integer(5))
>>> CK = C.change_ring(K)
>>> MW = CK.invariant_differential().parent()
>>> MW.dimension()
4

Sage Live

# needs sage.rings.padics
R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
K = Qp(7,5)
CK = C.change_ring(K)
MW = CK.invariant_differential().parent()
MW.dimension()

frob_Q(p)[source]¶

Return and cache \(Q(x^p)\), which is used in computing the image of \(y\) under a \(p\)-power lift of Frobenius to \(A^{\dagger}\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: MW = C.invariant_differential().parent()
sage: MW.frob_Q(3)
-(60-48*y^2+12*y^4-y^6)*1 + (192-96*y^2+12*y^4)*x - (192-48*y^2)*x^2 + 60*x^3
sage: MW.Q()(MW.x_to_p(3))                                                  # needs sage.rings.real_interval_field
-(60-48*y^2+12*y^4-y^6)*1 + (192-96*y^2+12*y^4)*x - (192-48*y^2)*x^2 + 60*x^3
sage: MW.frob_Q(11) is MW.frob_Q(11)
True

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> MW = C.invariant_differential().parent()
>>> MW.frob_Q(Integer(3))
-(60-48*y^2+12*y^4-y^6)*1 + (192-96*y^2+12*y^4)*x - (192-48*y^2)*x^2 + 60*x^3
>>> MW.Q()(MW.x_to_p(Integer(3)))                                                  # needs sage.rings.real_interval_field
-(60-48*y^2+12*y^4-y^6)*1 + (192-96*y^2+12*y^4)*x - (192-48*y^2)*x^2 + 60*x^3
>>> MW.frob_Q(Integer(11)) is MW.frob_Q(Integer(11))
True

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
MW = C.invariant_differential().parent()
MW.frob_Q(3)
MW.Q()(MW.x_to_p(3))                                                  # needs sage.rings.real_interval_field
MW.frob_Q(11) is MW.frob_Q(11)

frob_basis_elements(prec, p)[source]¶

Return the action of a \(p\)-power lift of Frobenius on the basis.

\[\{ dx/2y, x dx/2y, ..., x^{d-2} dx/2y \},\]

where \(d\) is the degree of the underlying hyperelliptic curve.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: prec = 1
sage: p = 5
sage: MW = C.invariant_differential().parent()
sage: MW.frob_basis_elements(prec, p)
[((92000*y^-14-74200*y^-12+32000*y^-10-8000*y^-8+1000*y^-6-50*y^-4)*1
  - (194400*y^-14-153600*y^-12+57600*y^-10-9600*y^-8+600*y^-6)*x
  + (204800*y^-14-153600*y^-12+38400*y^-10-3200*y^-8)*x^2
  - (153600*y^-14-76800*y^-12+9600*y^-10)*x^3
  + (63950*y^-14-18550*y^-12+1600*y^-10-400*y^-8+50*y^-6+5*y^-4)*x^4) dx/2y,
 (-(1391200*y^-14-941400*y^-12+302000*y^-10-76800*y^-8+14400*y^-6-1320*y^-4+30*y^-2)*1
  + (2168800*y^-14-1402400*y^-12+537600*y^-10-134400*y^-8+16800*y^-6-720*y^-4)*x
  - (1596800*y^-14-1433600*y^-12+537600*y^-10-89600*y^-8+5600*y^-6)*x^2
  + (1433600*y^-14-1075200*y^-12+268800*y^-10-22400*y^-8)*x^3
  - (870200*y^-14-445350*y^-12+63350*y^-10-3200*y^-8+600*y^-6-30*y^-4-5*y^-2)*x^4) dx/2y,
 ((19488000*y^-14-15763200*y^-12+4944400*y^-10-913800*y^-8+156800*y^-6-22560*y^-4+1480*y^-2-10)*1
  - (28163200*y^-14-18669600*y^-12+5774400*y^-10-1433600*y^-8+268800*y^-6-25440*y^-4+760*y^-2)*x
  + (15062400*y^-14-12940800*y^-12+5734400*y^-10-1433600*y^-8+179200*y^-6-8480*y^-4)*x^2
  - (12121600*y^-14-11468800*y^-12+4300800*y^-10-716800*y^-8+44800*y^-6)*x^3
  + (9215200*y^-14-6952400*y^-12+1773950*y^-10-165750*y^-8+5600*y^-6-720*y^-4+10*y^-2+5)*x^4) dx/2y,
 (-(225395200*y^-14-230640000*y^-12+91733600*y^-10-18347400*y^-8+2293600*y^-6-280960*y^-4+31520*y^-2-1480-10*y^2)*1
  + (338048000*y^-14-277132800*y^-12+89928000*y^-10-17816000*y^-8+3225600*y^-6-472320*y^-4+34560*y^-2-720)*x
  - (172902400*y^-14-141504000*y^-12+58976000*y^-10-17203200*y^-8+3225600*y^-6-314880*y^-4+11520*y^-2)*x^2
  + (108736000*y^-14-109760000*y^-12+51609600*y^-10-12902400*y^-8+1612800*y^-6-78720*y^-4)*x^3
  - (85347200*y^-14-82900000*y^-12+31251400*y^-10-5304150*y^-8+367350*y^-6-8480*y^-4+760*y^-2+10-5*y^2)*x^4) dx/2y]

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> prec = Integer(1)
>>> p = Integer(5)
>>> MW = C.invariant_differential().parent()
>>> MW.frob_basis_elements(prec, p)
[((92000*y^-14-74200*y^-12+32000*y^-10-8000*y^-8+1000*y^-6-50*y^-4)*1
  - (194400*y^-14-153600*y^-12+57600*y^-10-9600*y^-8+600*y^-6)*x
  + (204800*y^-14-153600*y^-12+38400*y^-10-3200*y^-8)*x^2
  - (153600*y^-14-76800*y^-12+9600*y^-10)*x^3
  + (63950*y^-14-18550*y^-12+1600*y^-10-400*y^-8+50*y^-6+5*y^-4)*x^4) dx/2y,
 (-(1391200*y^-14-941400*y^-12+302000*y^-10-76800*y^-8+14400*y^-6-1320*y^-4+30*y^-2)*1
  + (2168800*y^-14-1402400*y^-12+537600*y^-10-134400*y^-8+16800*y^-6-720*y^-4)*x
  - (1596800*y^-14-1433600*y^-12+537600*y^-10-89600*y^-8+5600*y^-6)*x^2
  + (1433600*y^-14-1075200*y^-12+268800*y^-10-22400*y^-8)*x^3
  - (870200*y^-14-445350*y^-12+63350*y^-10-3200*y^-8+600*y^-6-30*y^-4-5*y^-2)*x^4) dx/2y,
 ((19488000*y^-14-15763200*y^-12+4944400*y^-10-913800*y^-8+156800*y^-6-22560*y^-4+1480*y^-2-10)*1
  - (28163200*y^-14-18669600*y^-12+5774400*y^-10-1433600*y^-8+268800*y^-6-25440*y^-4+760*y^-2)*x
  + (15062400*y^-14-12940800*y^-12+5734400*y^-10-1433600*y^-8+179200*y^-6-8480*y^-4)*x^2
  - (12121600*y^-14-11468800*y^-12+4300800*y^-10-716800*y^-8+44800*y^-6)*x^3
  + (9215200*y^-14-6952400*y^-12+1773950*y^-10-165750*y^-8+5600*y^-6-720*y^-4+10*y^-2+5)*x^4) dx/2y,
 (-(225395200*y^-14-230640000*y^-12+91733600*y^-10-18347400*y^-8+2293600*y^-6-280960*y^-4+31520*y^-2-1480-10*y^2)*1
  + (338048000*y^-14-277132800*y^-12+89928000*y^-10-17816000*y^-8+3225600*y^-6-472320*y^-4+34560*y^-2-720)*x
  - (172902400*y^-14-141504000*y^-12+58976000*y^-10-17203200*y^-8+3225600*y^-6-314880*y^-4+11520*y^-2)*x^2
  + (108736000*y^-14-109760000*y^-12+51609600*y^-10-12902400*y^-8+1612800*y^-6-78720*y^-4)*x^3
  - (85347200*y^-14-82900000*y^-12+31251400*y^-10-5304150*y^-8+367350*y^-6-8480*y^-4+760*y^-2+10-5*y^2)*x^4) dx/2y]

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
prec = 1
p = 5
MW = C.invariant_differential().parent()
MW.frob_basis_elements(prec, p)

frob_invariant_differential(prec, p)[source]¶

Kedlaya’s algorithm allows us to calculate the action of Frobenius on the Monsky-Washnitzer cohomology. First we lift \(\phi\) to \(A^{\dagger}\) by setting

\[\phi(x) = x^p, \qquad\qquad \phi(y) = y^p \sqrt{1 + \frac{Q(x^p) - Q(x)^p}{Q(x)^p}}.\]

Pulling back the differential \(dx/2y\), we get

\[\phi^*(dx/2y) = px^{p-1} y(\phi(y))^{-1} dx/2y = px^{p-1} y^{1-p} \sqrt{1+ \frac{Q(x^p) - Q(x)^p}{Q(x)^p}} dx/2y.\]

Use Newton’s method to calculate the square root.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: prec = 2
sage: p = 7
sage: MW = C.invariant_differential().parent()
sage: MW.frob_invariant_differential(prec, p)
((67894400*y^-20-81198880*y^-18+40140800*y^-16-10035200*y^-14+1254400*y^-12-62720*y^-10)*1
 - (119503944*y^-20-116064242*y^-18+43753472*y^-16-7426048*y^-14+514304*y^-12-12544*y^-10+1568*y^-8-70*y^-6-7*y^-4)*x
 + (78905288*y^-20-61014016*y^-18+16859136*y^-16-2207744*y^-14+250880*y^-12-37632*y^-10+3136*y^-8-70*y^-6)*x^2
 - (39452448*y^-20-26148752*y^-18+8085490*y^-16-2007040*y^-14+376320*y^-12-37632*y^-10+1568*y^-8)*x^3
 + (21102144*y^-20-18120592*y^-18+8028160*y^-16-2007040*y^-14+250880*y^-12-12544*y^-10)*x^4) dx/2y

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> prec = Integer(2)
>>> p = Integer(7)
>>> MW = C.invariant_differential().parent()
>>> MW.frob_invariant_differential(prec, p)
((67894400*y^-20-81198880*y^-18+40140800*y^-16-10035200*y^-14+1254400*y^-12-62720*y^-10)*1
 - (119503944*y^-20-116064242*y^-18+43753472*y^-16-7426048*y^-14+514304*y^-12-12544*y^-10+1568*y^-8-70*y^-6-7*y^-4)*x
 + (78905288*y^-20-61014016*y^-18+16859136*y^-16-2207744*y^-14+250880*y^-12-37632*y^-10+3136*y^-8-70*y^-6)*x^2
 - (39452448*y^-20-26148752*y^-18+8085490*y^-16-2007040*y^-14+376320*y^-12-37632*y^-10+1568*y^-8)*x^3
 + (21102144*y^-20-18120592*y^-18+8028160*y^-16-2007040*y^-14+250880*y^-12-12544*y^-10)*x^4) dx/2y

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
prec = 2
p = 7
MW = C.invariant_differential().parent()
MW.frob_invariant_differential(prec, p)

helper_matrix()[source]¶

We use this to solve for the linear combination of \(x^i y^j\) needed to clear all terms with \(y^{j-1}\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: MW = C.invariant_differential().parent()
sage: MW.helper_matrix()
[ 256/2101  320/2101  400/2101  500/2101  625/2101]
[-625/8404  -64/2101  -80/2101 -100/2101 -125/2101]
[-125/2101 -625/8404  -64/2101  -80/2101 -100/2101]
[-100/2101 -125/2101 -625/8404  -64/2101  -80/2101]
[ -80/2101 -100/2101 -125/2101 -625/8404  -64/2101]

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> MW = C.invariant_differential().parent()
>>> MW.helper_matrix()
[ 256/2101  320/2101  400/2101  500/2101  625/2101]
[-625/8404  -64/2101  -80/2101 -100/2101 -125/2101]
[-125/2101 -625/8404  -64/2101  -80/2101 -100/2101]
[-100/2101 -125/2101 -625/8404  -64/2101  -80/2101]
[ -80/2101 -100/2101 -125/2101 -625/8404  -64/2101]

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
MW = C.invariant_differential().parent()
MW.helper_matrix()

invariant_differential()[source]¶

Return \(dx/2y\) as an element of self.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: MW = C.invariant_differential().parent()
sage: MW.invariant_differential()
1 dx/2y

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> MW = C.invariant_differential().parent()
>>> MW.invariant_differential()
1 dx/2y

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
MW = C.invariant_differential().parent()
MW.invariant_differential()

x_to_p(p)[source]¶

Return and cache \(x^p\), reduced via the relations coming from the defining polynomial of the hyperelliptic curve.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: MW = C.invariant_differential().parent()
sage: MW.x_to_p(3)
x^3
sage: MW.x_to_p(5)
-(4-y^2)*1 + 4*x
sage: MW.x_to_p(101) is MW.x_to_p(101)
True

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(5) - Integer(4)*x + Integer(4))
>>> MW = C.invariant_differential().parent()
>>> MW.x_to_p(Integer(3))
x^3
>>> MW.x_to_p(Integer(5))
-(4-y^2)*1 + 4*x
>>> MW.x_to_p(Integer(101)) is MW.x_to_p(Integer(101))
True

Sage Live

R.<x> = QQ['x']
C = HyperellipticCurve(x^5 - 4*x + 4)
MW = C.invariant_differential().parent()
MW.x_to_p(3)
MW.x_to_p(5)
MW.x_to_p(101) is MW.x_to_p(101)

sage.schemes.hyperelliptic_curves.monsky_washnitzer.MonskyWashnitzerDifferentialRing_class[source]¶: alias of MonskyWashnitzerDifferentialRing

class sage.schemes.hyperelliptic_curves.monsky_washnitzer.SpecialCubicQuotientRing(Q, laurent_series=False)[source]¶

Bases: UniqueRepresentation, Parent

Specialised class for representing the quotient ring \(R[x,T]/(T - x^3 - ax - b)\), where \(R\) is an arbitrary commutative base ring (in which 2 and 3 are invertible), \(a\) and \(b\) are elements of that ring.

Polynomials are represented internally in the form \(p_0 + p_1 x + p_2 x^2\) where the \(p_i\) are polynomials in \(T\). Multiplication of polynomials always reduces high powers of \(x\) (i.e. beyond \(x^2\)) to powers of \(T\).

Hopefully this ring is faster than a general quotient ring because it uses the special structure of this ring to speed multiplication (which is the dominant operation in the frobenius matrix calculation). I haven’t actually tested this theory though…

Todo

Eventually we will want to run this in characteristic 3, so we need to: (a) Allow \(Q(x)\) to contain an \(x^2\) term, and (b) Remove the requirement that 3 be invertible. Currently this is used in the Toom-Cook algorithm to speed multiplication.

EXAMPLES:

Sage

sage: B.<t> = PolynomialRing(Integers(125))
sage: R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
sage: R
SpecialCubicQuotientRing over Ring of integers modulo 125
with polynomial T = x^3 + 124*x + 94
sage: TestSuite(R).run()

Python

>>> from sage.all import *
>>> B = PolynomialRing(Integers(Integer(125)), names=('t',)); (t,) = B._first_ngens(1)
>>> R = monsky_washnitzer.SpecialCubicQuotientRing(t**Integer(3) - t + B(Integer(1)/Integer(4)))
>>> R
SpecialCubicQuotientRing over Ring of integers modulo 125
with polynomial T = x^3 + 124*x + 94
>>> TestSuite(R).run()

Sage Live

B.<t> = PolynomialRing(Integers(125))
R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
R
TestSuite(R).run()

Get generators:

Sage

sage: x, T = R.gens()
sage: x
(0) + (1)*x + (0)*x^2
sage: T
(T) + (0)*x + (0)*x^2

Python

>>> from sage.all import *
>>> x, T = R.gens()
>>> x
(0) + (1)*x + (0)*x^2
>>> T
(T) + (0)*x + (0)*x^2

Sage Live

x, T = R.gens()
x
T

Coercions:

Sage

sage: R(7)
(7) + (0)*x + (0)*x^2

Python

>>> from sage.all import *
>>> R(Integer(7))
(7) + (0)*x + (0)*x^2

Sage Live

R(7)

Create elements directly from polynomials:

Sage

sage: A = R.poly_ring()
sage: A
Univariate Polynomial Ring in T over Ring of integers modulo 125
sage: z = A.gen()
sage: R.create_element(z^2, z+1, 3)
(T^2) + (T + 1)*x + (3)*x^2

Python

>>> from sage.all import *
>>> A = R.poly_ring()
>>> A
Univariate Polynomial Ring in T over Ring of integers modulo 125
>>> z = A.gen()
>>> R.create_element(z**Integer(2), z+Integer(1), Integer(3))
(T^2) + (T + 1)*x + (3)*x^2

Sage Live

A = R.poly_ring()
A
z = A.gen()
R.create_element(z^2, z+1, 3)

Some arithmetic:

Sage

sage: x^3
(T + 31) + (1)*x + (0)*x^2
sage: 3 * x**15 * T**2 + x - T
(3*T^7 + 90*T^6 + 110*T^5 + 20*T^4 + 58*T^3 + 26*T^2 + 124*T) +
(15*T^6 + 110*T^5 + 35*T^4 + 63*T^2 + 1)*x +
(30*T^5 + 40*T^4 + 8*T^3 + 38*T^2)*x^2

Python

>>> from sage.all import *
>>> x**Integer(3)
(T + 31) + (1)*x + (0)*x^2
>>> Integer(3) * x**Integer(15) * T**Integer(2) + x - T
(3*T^7 + 90*T^6 + 110*T^5 + 20*T^4 + 58*T^3 + 26*T^2 + 124*T) +
(15*T^6 + 110*T^5 + 35*T^4 + 63*T^2 + 1)*x +
(30*T^5 + 40*T^4 + 8*T^3 + 38*T^2)*x^2

Sage Live

x^3
3 * x**15 * T**2 + x - T

Retrieve coefficients (output is zero-padded):

Sage

sage: x^10
(3*T^2 + 61*T + 8) + (T^3 + 93*T^2 + 12*T + 40)*x + (3*T^2 + 61*T + 9)*x^2
sage: (x^10).coeffs()
[[8, 61, 3, 0], [40, 12, 93, 1], [9, 61, 3, 0]]

Python

>>> from sage.all import *
>>> x**Integer(10)
(3*T^2 + 61*T + 8) + (T^3 + 93*T^2 + 12*T + 40)*x + (3*T^2 + 61*T + 9)*x^2
>>> (x**Integer(10)).coeffs()
[[8, 61, 3, 0], [40, 12, 93, 1], [9, 61, 3, 0]]

Sage Live

x^10
(x^10).coeffs()

Todo

write an example checking multiplication of these polynomials against Sage’s ordinary quotient ring arithmetic. I cannot seem to get the quotient ring stuff happening right now…

Element[source]¶: alias of SpecialCubicQuotientRingElement

create_element(check, *args)[source]¶

Create the element \(p_0 + p_1*x + p_2*x^2\), where the \(p_i\) are polynomials in \(T\).

INPUT:

p0, p1, p2 – coefficients; must be coercible into poly_ring()
check – boolean (default: True); whether to carry out coercion

EXAMPLES:

Sage

sage: B.<t> = PolynomialRing(Integers(125))
sage: R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
sage: A, z = R.poly_ring().objgen()
sage: R.create_element(z^2, z+1, 3)  # indirect doctest
(T^2) + (T + 1)*x + (3)*x^2

Python

>>> from sage.all import *
>>> B = PolynomialRing(Integers(Integer(125)), names=('t',)); (t,) = B._first_ngens(1)
>>> R = monsky_washnitzer.SpecialCubicQuotientRing(t**Integer(3) - t + B(Integer(1)/Integer(4)))
>>> A, z = R.poly_ring().objgen()
>>> R.create_element(z**Integer(2), z+Integer(1), Integer(3))  # indirect doctest
(T^2) + (T + 1)*x + (3)*x^2

Sage Live

B.<t> = PolynomialRing(Integers(125))
R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
A, z = R.poly_ring().objgen()
R.create_element(z^2, z+1, 3)  # indirect doctest

gens()[source]¶

Return a list [x, T] where x and T are the generators of the ring (as element of this ring).

Note

I have no idea if this is compatible with the usual Sage ‘gens’ interface.

EXAMPLES:

Sage

sage: B.<t> = PolynomialRing(Integers(125))
sage: R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
sage: x, T = R.gens()
sage: x
(0) + (1)*x + (0)*x^2
sage: T
(T) + (0)*x + (0)*x^2

Python

>>> from sage.all import *
>>> B = PolynomialRing(Integers(Integer(125)), names=('t',)); (t,) = B._first_ngens(1)
>>> R = monsky_washnitzer.SpecialCubicQuotientRing(t**Integer(3) - t + B(Integer(1)/Integer(4)))
>>> x, T = R.gens()
>>> x
(0) + (1)*x + (0)*x^2
>>> T
(T) + (0)*x + (0)*x^2

Sage Live

B.<t> = PolynomialRing(Integers(125))
R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
x, T = R.gens()
x
T

one()[source]¶

Return the unit of self.

EXAMPLES:

Sage

sage: B.<t> = PolynomialRing(Integers(125))
sage: R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
sage: R.one()
(1) + (0)*x + (0)*x^2

Python

>>> from sage.all import *
>>> B = PolynomialRing(Integers(Integer(125)), names=('t',)); (t,) = B._first_ngens(1)
>>> R = monsky_washnitzer.SpecialCubicQuotientRing(t**Integer(3) - t + B(Integer(1)/Integer(4)))
>>> R.one()
(1) + (0)*x + (0)*x^2

Sage Live

B.<t> = PolynomialRing(Integers(125))
R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
R.one()

poly_ring()[source]¶

Return the underlying polynomial ring in \(T\).

EXAMPLES:

Sage

sage: B.<t> = PolynomialRing(Integers(125))
sage: R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
sage: R.poly_ring()
Univariate Polynomial Ring in T over Ring of integers modulo 125

Python

>>> from sage.all import *
>>> B = PolynomialRing(Integers(Integer(125)), names=('t',)); (t,) = B._first_ngens(1)
>>> R = monsky_washnitzer.SpecialCubicQuotientRing(t**Integer(3) - t + B(Integer(1)/Integer(4)))
>>> R.poly_ring()
Univariate Polynomial Ring in T over Ring of integers modulo 125

Sage Live

B.<t> = PolynomialRing(Integers(125))
R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
R.poly_ring()

class sage.schemes.hyperelliptic_curves.monsky_washnitzer.SpecialCubicQuotientRingElement(parent, p0, p1, p2, check=True)[source]¶

Bases: ModuleElement

An element of a SpecialCubicQuotientRing.

coeffs()[source]¶

Return list of three lists of coefficients, corresponding to the \(x^0\), \(x^1\), \(x^2\) coefficients.

The lists are zero padded to the same length. The list entries belong to the base ring.

EXAMPLES:

Sage

sage: B.<t> = PolynomialRing(Integers(125))
sage: R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
sage: p = R.create_element(t, t^2 - 2, 3)
sage: p.coeffs()
[[0, 1, 0], [123, 0, 1], [3, 0, 0]]

Python

>>> from sage.all import *
>>> B = PolynomialRing(Integers(Integer(125)), names=('t',)); (t,) = B._first_ngens(1)
>>> R = monsky_washnitzer.SpecialCubicQuotientRing(t**Integer(3) - t + B(Integer(1)/Integer(4)))
>>> p = R.create_element(t, t**Integer(2) - Integer(2), Integer(3))
>>> p.coeffs()
[[0, 1, 0], [123, 0, 1], [3, 0, 0]]

Sage Live

B.<t> = PolynomialRing(Integers(125))
R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
p = R.create_element(t, t^2 - 2, 3)
p.coeffs()

scalar_multiply(scalar)[source]¶

Multiply this element by a scalar, i.e. just multiply each coefficient of \(x^j\) by the scalar.

INPUT:

scalar – either an element of base_ring, or an element of poly_ring

EXAMPLES:

Sage

sage: B.<t> = PolynomialRing(Integers(125))
sage: R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
sage: x, T = R.gens()
sage: f = R.create_element(2, t, t^2 - 3)
sage: f
(2) + (T)*x + (T^2 + 122)*x^2
sage: f.scalar_multiply(2)
(4) + (2*T)*x + (2*T^2 + 119)*x^2
sage: f.scalar_multiply(t)
(2*T) + (T^2)*x + (T^3 + 122*T)*x^2

Python

>>> from sage.all import *
>>> B = PolynomialRing(Integers(Integer(125)), names=('t',)); (t,) = B._first_ngens(1)
>>> R = monsky_washnitzer.SpecialCubicQuotientRing(t**Integer(3) - t + B(Integer(1)/Integer(4)))
>>> x, T = R.gens()
>>> f = R.create_element(Integer(2), t, t**Integer(2) - Integer(3))
>>> f
(2) + (T)*x + (T^2 + 122)*x^2
>>> f.scalar_multiply(Integer(2))
(4) + (2*T)*x + (2*T^2 + 119)*x^2
>>> f.scalar_multiply(t)
(2*T) + (T^2)*x + (T^3 + 122*T)*x^2

Sage Live

B.<t> = PolynomialRing(Integers(125))
R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
x, T = R.gens()
f = R.create_element(2, t, t^2 - 3)
f
f.scalar_multiply(2)
f.scalar_multiply(t)

shift(n)[source]¶

Return this element multiplied by \(T^n\).

EXAMPLES:

Sage

sage: B.<t> = PolynomialRing(Integers(125))
sage: R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
sage: f = R.create_element(2, t, t^2 - 3)
sage: f
(2) + (T)*x + (T^2 + 122)*x^2
sage: f.shift(1)
(2*T) + (T^2)*x + (T^3 + 122*T)*x^2
sage: f.shift(2)
(2*T^2) + (T^3)*x + (T^4 + 122*T^2)*x^2

Python

>>> from sage.all import *
>>> B = PolynomialRing(Integers(Integer(125)), names=('t',)); (t,) = B._first_ngens(1)
>>> R = monsky_washnitzer.SpecialCubicQuotientRing(t**Integer(3) - t + B(Integer(1)/Integer(4)))
>>> f = R.create_element(Integer(2), t, t**Integer(2) - Integer(3))
>>> f
(2) + (T)*x + (T^2 + 122)*x^2
>>> f.shift(Integer(1))
(2*T) + (T^2)*x + (T^3 + 122*T)*x^2
>>> f.shift(Integer(2))
(2*T^2) + (T^3)*x + (T^4 + 122*T^2)*x^2

Sage Live

B.<t> = PolynomialRing(Integers(125))
R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
f = R.create_element(2, t, t^2 - 3)
f
f.shift(1)
f.shift(2)

square()[source]¶

Return the square of the element.

EXAMPLES:

Sage

sage: B.<t> = PolynomialRing(Integers(125))
sage: R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
sage: x, T = R.gens()

Python

>>> from sage.all import *
>>> B = PolynomialRing(Integers(Integer(125)), names=('t',)); (t,) = B._first_ngens(1)
>>> R = monsky_washnitzer.SpecialCubicQuotientRing(t**Integer(3) - t + B(Integer(1)/Integer(4)))
>>> x, T = R.gens()

Sage Live

B.<t> = PolynomialRing(Integers(125))
R = monsky_washnitzer.SpecialCubicQuotientRing(t^3 - t + B(1/4))
x, T = R.gens()

Sage

sage: f = R.create_element(1 + 2*t + 3*t^2, 4 + 7*t + 9*t^2, 3 + 5*t + 11*t^2)
sage: f.square()
(73*T^5 + 16*T^4 + 38*T^3 + 39*T^2 + 70*T + 120)
+ (121*T^5 + 113*T^4 + 73*T^3 + 8*T^2 + 51*T + 61)*x
+ (18*T^4 + 60*T^3 + 22*T^2 + 108*T + 31)*x^2

Python

>>> from sage.all import *
>>> f = R.create_element(Integer(1) + Integer(2)*t + Integer(3)*t**Integer(2), Integer(4) + Integer(7)*t + Integer(9)*t**Integer(2), Integer(3) + Integer(5)*t + Integer(11)*t**Integer(2))
>>> f.square()
(73*T^5 + 16*T^4 + 38*T^3 + 39*T^2 + 70*T + 120)
+ (121*T^5 + 113*T^4 + 73*T^3 + 8*T^2 + 51*T + 61)*x
+ (18*T^4 + 60*T^3 + 22*T^2 + 108*T + 31)*x^2

Sage Live

f = R.create_element(1 + 2*t + 3*t^2, 4 + 7*t + 9*t^2, 3 + 5*t + 11*t^2)
f.square()

class sage.schemes.hyperelliptic_curves.monsky_washnitzer.SpecialHyperellipticQuotientElement(parent, val=0, offset=0, check=True)[source]¶

Bases: ModuleElement

Element in the Hyperelliptic quotient ring.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 36*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: MW = x.parent()
sage: MW(x + x**2 + y - 77)
-(77-y)*1 + x + x^2

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(36)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> MW = x.parent()
>>> MW(x + x**Integer(2) + y - Integer(77))
-(77-y)*1 + x + x^2

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 36*x + 1)
x,y = E.monsky_washnitzer_gens()
MW = x.parent()
MW(x + x**2 + y - 77)

change_ring(R)[source]¶

Return the same element after changing the base ring to \(R\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 36*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: MW = x.parent()
sage: z = MW(x + x**2 + y - 77)
sage: z.change_ring(AA).parent()                                            # needs sage.rings.number_field
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = x^5 - 36*x + 1)
over Algebraic Real Field

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(36)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> MW = x.parent()
>>> z = MW(x + x**Integer(2) + y - Integer(77))
>>> z.change_ring(AA).parent()                                            # needs sage.rings.number_field
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = x^5 - 36*x + 1)
over Algebraic Real Field

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 36*x + 1)
x,y = E.monsky_washnitzer_gens()
MW = x.parent()
z = MW(x + x**2 + y - 77)
z.change_ring(AA).parent()                                            # needs sage.rings.number_field

coeffs(R=None)[source]¶

Return the raw coefficients of this element.

INPUT:

R – an (optional) base-ring in which to cast the coefficients

OUTPUT:

coeffs – list of coefficients of powers of \(x\) for each power of \(y\)
n – an offset indicating the power of \(y\) of the first list element

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.coeffs()
([(0, 1, 0, 0, 0)], 0)
sage: y.coeffs()
([(0, 0, 0, 0, 0), (1, 0, 0, 0, 0)], 0)

sage: a = sum(n*x^n for n in range(5)); a
x + 2*x^2 + 3*x^3 + 4*x^4
sage: a.coeffs()
([(0, 1, 2, 3, 4)], 0)
sage: a.coeffs(Qp(7))                                                       # needs sage.rings.padics
([(0, 1 + O(7^20), 2 + O(7^20), 3 + O(7^20), 4 + O(7^20))], 0)
sage: (a*y).coeffs()
([(0, 0, 0, 0, 0), (0, 1, 2, 3, 4)], 0)
sage: (a*y^-2).coeffs()
([(0, 1, 2, 3, 4), (0, 0, 0, 0, 0), (0, 0, 0, 0, 0)], -2)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.coeffs()
([(0, 1, 0, 0, 0)], 0)
>>> y.coeffs()
([(0, 0, 0, 0, 0), (1, 0, 0, 0, 0)], 0)

>>> a = sum(n*x**n for n in range(Integer(5))); a
x + 2*x^2 + 3*x^3 + 4*x^4
>>> a.coeffs()
([(0, 1, 2, 3, 4)], 0)
>>> a.coeffs(Qp(Integer(7)))                                                       # needs sage.rings.padics
([(0, 1 + O(7^20), 2 + O(7^20), 3 + O(7^20), 4 + O(7^20))], 0)
>>> (a*y).coeffs()
([(0, 0, 0, 0, 0), (0, 1, 2, 3, 4)], 0)
>>> (a*y**-Integer(2)).coeffs()
([(0, 1, 2, 3, 4), (0, 0, 0, 0, 0), (0, 0, 0, 0, 0)], -2)

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.coeffs()
y.coeffs()
a = sum(n*x^n for n in range(5)); a
a.coeffs()
a.coeffs(Qp(7))                                                       # needs sage.rings.padics
(a*y).coeffs()
(a*y^-2).coeffs()

Note that the coefficient list is transposed compared to how they are stored and printed:

Sage

sage: a*y^-2
(y^-2)*x + (2*y^-2)*x^2 + (3*y^-2)*x^3 + (4*y^-2)*x^4

Python

>>> from sage.all import *
>>> a*y**-Integer(2)
(y^-2)*x + (2*y^-2)*x^2 + (3*y^-2)*x^3 + (4*y^-2)*x^4

Sage Live

a*y^-2

A more complicated example:

Sage

sage: a = x^20*y^-3 - x^11*y^2; a
(y^-3-4*y^-1+6*y-4*y^3+y^5)*1 - (12*y^-3-36*y^-1+36*y+y^2-12*y^3-2*y^4+y^6)*x
 + (54*y^-3-108*y^-1+54*y+6*y^2-6*y^4)*x^2 - (108*y^-3-108*y^-1+9*y^2)*x^3
 + (81*y^-3)*x^4
sage: raw, offset = a.coeffs()
sage: a.min_pow_y()
-3
sage: offset
-3
sage: raw
[(1, -12, 54, -108, 81),
 (0, 0, 0, 0, 0),
 (-4, 36, -108, 108, 0),
 (0, 0, 0, 0, 0),
 (6, -36, 54, 0, 0),
 (0, -1, 6, -9, 0),
 (-4, 12, 0, 0, 0),
 (0, 2, -6, 0, 0),
 (1, 0, 0, 0, 0),
 (0, -1, 0, 0, 0)]
sage: sum(c * x^i * y^(j+offset)
....:     for j, L in enumerate(raw) for i, c in enumerate(L)) == a
True

Python

>>> from sage.all import *
>>> a = x**Integer(20)*y**-Integer(3) - x**Integer(11)*y**Integer(2); a
(y^-3-4*y^-1+6*y-4*y^3+y^5)*1 - (12*y^-3-36*y^-1+36*y+y^2-12*y^3-2*y^4+y^6)*x
 + (54*y^-3-108*y^-1+54*y+6*y^2-6*y^4)*x^2 - (108*y^-3-108*y^-1+9*y^2)*x^3
 + (81*y^-3)*x^4
>>> raw, offset = a.coeffs()
>>> a.min_pow_y()
-3
>>> offset
-3
>>> raw
[(1, -12, 54, -108, 81),
 (0, 0, 0, 0, 0),
 (-4, 36, -108, 108, 0),
 (0, 0, 0, 0, 0),
 (6, -36, 54, 0, 0),
 (0, -1, 6, -9, 0),
 (-4, 12, 0, 0, 0),
 (0, 2, -6, 0, 0),
 (1, 0, 0, 0, 0),
 (0, -1, 0, 0, 0)]
>>> sum(c * x**i * y**(j+offset)
...     for j, L in enumerate(raw) for i, c in enumerate(L)) == a
True

Sage Live

a = x^20*y^-3 - x^11*y^2; a
raw, offset = a.coeffs()
a.min_pow_y()
offset
raw
sum(c * x^i * y^(j+offset)
    for j, L in enumerate(raw) for i, c in enumerate(L)) == a

Can also be used to construct elements:

Sage

sage: a.parent()(raw, offset) == a
True

Python

>>> from sage.all import *
>>> a.parent()(raw, offset) == a
True

Sage Live

a.parent()(raw, offset) == a

diff()[source]¶

Return the differential of self.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: (x + 3*y).diff()
(-(9-2*y)*1 + 15*x^4) dx/2y

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> (x + Integer(3)*y).diff()
(-(9-2*y)*1 + 15*x^4) dx/2y

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
(x + 3*y).diff()

extract_pow_y(k)[source]¶

Return the coefficients of \(y^k\) in self as a list.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: (x + 3*y + 9*x*y).extract_pow_y(1)
[3, 9, 0, 0, 0]

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> (x + Integer(3)*y + Integer(9)*x*y).extract_pow_y(Integer(1))
[3, 9, 0, 0, 0]

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
(x + 3*y + 9*x*y).extract_pow_y(1)

max_pow_y()[source]¶

Return the maximal degree of self with respect to \(y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: (x + 3*y).max_pow_y()
1

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> (x + Integer(3)*y).max_pow_y()
1

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
(x + 3*y).max_pow_y()

min_pow_y()[source]¶

Return the minimal degree of self with respect to \(y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: (x + 3*y).min_pow_y()
0

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> (x + Integer(3)*y).min_pow_y()
0

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
(x + 3*y).min_pow_y()

truncate_neg(n)[source]¶

Return self minus its terms of degree less than \(n\) wrt \(y\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: (x + 3*y + 7*x*2*y**4).truncate_neg(1)
3*y*1 + 14*y^4*x

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> (x + Integer(3)*y + Integer(7)*x*Integer(2)*y**Integer(4)).truncate_neg(Integer(1))
3*y*1 + 14*y^4*x

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
(x + 3*y + 7*x*2*y**4).truncate_neg(1)

class sage.schemes.hyperelliptic_curves.monsky_washnitzer.SpecialHyperellipticQuotientRing(Q, R=None, invert_y=True)[source]¶

Bases: UniqueRepresentation, Parent

The special hyperelliptic quotient ring.

Element[source]¶: alias of SpecialHyperellipticQuotientElement

Q()[source]¶

Return the defining polynomial of the underlying hyperelliptic curve.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5-2*x+1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().Q()
x^5 - 2*x + 1

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5)-Integer(2)*x+Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().Q()
x^5 - 2*x + 1

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5-2*x+1)
x,y = E.monsky_washnitzer_gens()
x.parent().Q()

base_extend(R)[source]¶

Return the base extension of self to the ring R if possible.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().base_extend(UniversalCyclotomicField())                    # needs sage.libs.gap
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = x^5 - 3*x + 1)
over Universal Cyclotomic Field
sage: x.parent().base_extend(ZZ)
Traceback (most recent call last):
...
TypeError: no such base extension

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().base_extend(UniversalCyclotomicField())                    # needs sage.libs.gap
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = x^5 - 3*x + 1)
over Universal Cyclotomic Field
>>> x.parent().base_extend(ZZ)
Traceback (most recent call last):
...
TypeError: no such base extension

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().base_extend(UniversalCyclotomicField())                    # needs sage.libs.gap
x.parent().base_extend(ZZ)

change_ring(R)[source]¶

Return the analog of self over the ring R.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().change_ring(ZZ)
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = x^5 - 3*x + 1)
over Integer Ring

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().change_ring(ZZ)
SpecialHyperellipticQuotientRing K[x,y,y^-1] / (y^2 = x^5 - 3*x + 1)
over Integer Ring

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().change_ring(ZZ)

curve()[source]¶

Return the underlying hyperelliptic curve.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().curve()
Hyperelliptic Curve over Rational Field defined by y^2 = x^5 - 3*x + 1

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().curve()
Hyperelliptic Curve over Rational Field defined by y^2 = x^5 - 3*x + 1

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().curve()

degree()[source]¶

Return the degree of the underlying hyperelliptic curve.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().degree()
5

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().degree()
5

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().degree()

gens()[source]¶

Return the generators of self.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().gens()
(x, y*1)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().gens()
(x, y*1)

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().gens()

is_field(proof=True)[source]¶

Return False as self is not a field.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().is_field()
False

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().is_field()
False

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().is_field()

monomial(i, j, b=None)[source]¶

Return \(b y^j x^i\), computed quickly.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().monomial(4,5)
y^5*x^4

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().monomial(Integer(4),Integer(5))
y^5*x^4

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().monomial(4,5)

monomial_diff_coeffs(i, j)[source]¶

Compute coefficients of the basis representation of \(d(x^iy^j)\).

The key here is that the formula for \(d(x^iy^j)\) is messy in terms of \(i\), but varies nicely with \(j\).

\[d(x^iy^j) = y^{j-1} (2ix^{i-1}y^2 + j (A_i(x) + B_i(x)y^2)) \frac{dx}{2y},\]

where \(A,B\) have degree at most \(n-1\) for each \(i\). Pre-compute \(A_i, B_i\) for each \(i\) the “hard” way, and the rest are easy.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().monomial_diff_coeffs(2,3)
((0, -15, 36, 0, 0), (0, 19, 0, 0, 0))

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().monomial_diff_coeffs(Integer(2),Integer(3))
((0, -15, 36, 0, 0), (0, 19, 0, 0, 0))

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().monomial_diff_coeffs(2,3)

monomial_diff_coeffs_matrices()[source]¶

Compute tables of coefficients of the basis representation of \(d(x^iy^j)\) for small \(i\), \(j\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().monomial_diff_coeffs_matrices()
(
[0 5 0 0 0]  [0 2 0 0 0]
[0 0 5 0 0]  [0 0 4 0 0]
[0 0 0 5 0]  [0 0 0 6 0]
[0 0 0 0 5]  [0 0 0 0 8]
[0 0 0 0 0], [0 0 0 0 0]
)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().monomial_diff_coeffs_matrices()
(
[0 5 0 0 0]  [0 2 0 0 0]
[0 0 5 0 0]  [0 0 4 0 0]
[0 0 0 5 0]  [0 0 0 6 0]
[0 0 0 0 5]  [0 0 0 0 8]
[0 0 0 0 0], [0 0 0 0 0]
)

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().monomial_diff_coeffs_matrices()

monsky_washnitzer()[source]¶

Return the stored Monsky-Washnitzer differential ring.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: type(x.parent().monsky_washnitzer())
<class 'sage.schemes.hyperelliptic_curves.monsky_washnitzer.MonskyWashnitzerDifferentialRing_with_category'>

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> type(x.parent().monsky_washnitzer())
<class 'sage.schemes.hyperelliptic_curves.monsky_washnitzer.MonskyWashnitzerDifferentialRing_with_category'>

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
type(x.parent().monsky_washnitzer())

one()[source]¶

Return the unit of self.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().one()
1

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().one()
1

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().one()

prime()[source]¶

Return the stored prime number \(p\).

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().prime() is None
True

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().prime() is None
True

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().prime() is None

x()[source]¶

Return the generator \(x\) of self.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().x()
x

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().x()
x

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().x()

y()[source]¶

Return the generator \(y\) of self.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().y()
y*1

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().y()
y*1

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().y()

zero()[source]¶

Return the zero of self.

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: E = HyperellipticCurve(x^5 - 3*x + 1)
sage: x,y = E.monsky_washnitzer_gens()
sage: x.parent().zero()
0

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(3)*x + Integer(1))
>>> x,y = E.monsky_washnitzer_gens()
>>> x.parent().zero()
0

Sage Live

R.<x> = QQ['x']
E = HyperellipticCurve(x^5 - 3*x + 1)
x,y = E.monsky_washnitzer_gens()
x.parent().zero()

sage.schemes.hyperelliptic_curves.monsky_washnitzer.SpecialHyperellipticQuotientRing_class[source]¶: alias of SpecialHyperellipticQuotientRing

sage.schemes.hyperelliptic_curves.monsky_washnitzer.adjusted_prec(p, prec)[source]¶

Compute how much precision is required in matrix_of_frobenius to get an answer correct to prec \(p\)-adic digits.

The issue is that the algorithm used in matrix_of_frobenius() sometimes performs divisions by \(p\), so precision is lost during the algorithm.

The estimate returned by this function is based on Kedlaya’s result (Lemmas 2 and 3 of [Ked2001]), which implies that if we start with \(M\) \(p\)-adic digits, the total precision loss is at most \(1 + \lfloor \log_p(2M-3) \rfloor\) \(p\)-adic digits. (This estimate is somewhat less than the amount you would expect by naively counting the number of divisions by \(p\).)

INPUT:

p – a prime p >= 5
prec – integer; desired output precision, prec >= 1

OUTPUT: adjusted precision (usually slightly more than prec)

EXAMPLES:

Sage

sage: from sage.schemes.hyperelliptic_curves.monsky_washnitzer import adjusted_prec
sage: adjusted_prec(5,2)
3

Python

>>> from sage.all import *
>>> from sage.schemes.hyperelliptic_curves.monsky_washnitzer import adjusted_prec
>>> adjusted_prec(Integer(5),Integer(2))
3

Sage Live

from sage.schemes.hyperelliptic_curves.monsky_washnitzer import adjusted_prec
adjusted_prec(5,2)

sage.schemes.hyperelliptic_curves.monsky_washnitzer.frobenius_expansion_by_newton(Q, p, M)[source]¶

Compute the action of Frobenius on \(dx/y\) and on \(x dx/y\), using Newton’s method (as suggested in Kedlaya’s paper [Ked2001]).

(This function does not yet use the cohomology relations - that happens afterwards in the “reduction” step.)

More specifically, it finds \(F_0\) and \(F_1\) in the quotient ring \(R[x, T]/(T - Q(x))\), such that

\[F( dx/y) = T^{-r} F0 dx/y, \text{\ and\ } F(x dx/y) = T^{-r} F1 dx/y\]

where

\[r = ( (2M-3)p - 1 )/2.\]

(Here \(T\) is \(y^2 = z^{-2}\), and \(R\) is the coefficient ring of \(Q\).)

\(F_0\) and \(F_1\) are computed in the SpecialCubicQuotientRing associated to \(Q\), so all powers of \(x^j\) for \(j \geq 3\) are reduced to powers of \(T\).

INPUT:

Q – cubic polynomial of the form \(Q(x) = x^3 + ax + b\), whose coefficient ring is a \(Z/(p^M)Z\)-algebra
p – residue characteristic of the \(p\)-adic field
M – \(p\)-adic precision of the coefficient ring (this will be used to determine the number of Newton iterations)

OUTPUT:

F0, F1 – elements of SpecialCubicQuotientRing(Q), as described above
r – nonnegative integer, as described above

EXAMPLES:

Sage

sage: from sage.schemes.hyperelliptic_curves.monsky_washnitzer import frobenius_expansion_by_newton
sage: R.<x> = Integers(5^3)['x']
sage: Q = x^3 - x + R(1/4)
sage: frobenius_expansion_by_newton(Q,5,3)
((25*T^5 + 75*T^3 + 100*T^2 + 100*T + 100) + (5*T^6 + 80*T^5 + 100*T^3
+ 25*T + 50)*x + (55*T^5 + 50*T^4 + 75*T^3 + 25*T^2 + 25*T + 25)*x^2,
(5*T^8 + 15*T^7 + 95*T^6 + 10*T^5 + 25*T^4 + 25*T^3 + 100*T^2 + 50)
+ (65*T^7 + 55*T^6 + 70*T^5 + 100*T^4 + 25*T^2 + 100*T)*x
+ (15*T^6 + 115*T^5 + 75*T^4 + 100*T^3 + 50*T^2 + 75*T + 75)*x^2, 7)

Python

>>> from sage.all import *
>>> from sage.schemes.hyperelliptic_curves.monsky_washnitzer import frobenius_expansion_by_newton
>>> R = Integers(Integer(5)**Integer(3))['x']; (x,) = R._first_ngens(1)
>>> Q = x**Integer(3) - x + R(Integer(1)/Integer(4))
>>> frobenius_expansion_by_newton(Q,Integer(5),Integer(3))
((25*T^5 + 75*T^3 + 100*T^2 + 100*T + 100) + (5*T^6 + 80*T^5 + 100*T^3
+ 25*T + 50)*x + (55*T^5 + 50*T^4 + 75*T^3 + 25*T^2 + 25*T + 25)*x^2,
(5*T^8 + 15*T^7 + 95*T^6 + 10*T^5 + 25*T^4 + 25*T^3 + 100*T^2 + 50)
+ (65*T^7 + 55*T^6 + 70*T^5 + 100*T^4 + 25*T^2 + 100*T)*x
+ (15*T^6 + 115*T^5 + 75*T^4 + 100*T^3 + 50*T^2 + 75*T + 75)*x^2, 7)

Sage Live

from sage.schemes.hyperelliptic_curves.monsky_washnitzer import frobenius_expansion_by_newton
R.<x> = Integers(5^3)['x']
Q = x^3 - x + R(1/4)
frobenius_expansion_by_newton(Q,5,3)

sage.schemes.hyperelliptic_curves.monsky_washnitzer.frobenius_expansion_by_series(Q, p, M)[source]¶

Compute the action of Frobenius on \(dx/y\) and on \(x dx/y\), using a series expansion.

(This function computes the same thing as frobenius_expansion_by_newton(), using a different method. Theoretically the Newton method should be asymptotically faster, when the precision gets large. However, in practice, this functions seems to be marginally faster for moderate precision, so I’m keeping it here until I figure out exactly why it is faster.)

(This function does not yet use the cohomology relations - that happens afterwards in the “reduction” step.)

More specifically, it finds F0 and F1 in the quotient ring \(R[x, T]/(T - Q(x))\), such that \(F( dx/y) = T^{-r} F0 dx/y\), and \(F(x dx/y) = T^{-r} F1 dx/y\) where \(r = ( (2M-3)p - 1 )/2\). (Here \(T\) is \(y^2 = z^{-2}\), and \(R\) is the coefficient ring of \(Q\).)

\(F_0\) and \(F_1\) are computed in the SpecialCubicQuotientRing associated to \(Q\), so all powers of \(x^j\) for \(j \geq 3\) are reduced to powers of \(T\).

It uses the sum

\[F0 = \sum_{k=0}^{M-2} \binom{-1/2}{k} p x^{p-1} E^k T^{(M-2-k)p}\]

and

\[ \begin{align}\begin{aligned} F1 = x^p F0,\\where `E = Q(x^p) - Q(x)^p`.\end{aligned}\end{align} \]

INPUT:

Q – cubic polynomial of the form \(Q(x) = x^3 + ax + b\), whose coefficient ring is a \(\ZZ/(p^M)\ZZ\) -algebra
p – residue characteristic of the \(p\)-adic field
M – \(p\)-adic precision of the coefficient ring (this will be used to determine the number of terms in the series)

OUTPUT:

F0, F1 – elements of SpecialCubicQuotientRing(Q), as described above
r – nonnegative integer, as described above

EXAMPLES:

Sage

sage: from sage.schemes.hyperelliptic_curves.monsky_washnitzer import frobenius_expansion_by_series
sage: R.<x> = Integers(5^3)['x']
sage: Q = x^3 - x + R(1/4)
sage: frobenius_expansion_by_series(Q,5,3)                                      # needs sage.libs.pari
((25*T^5 + 75*T^3 + 100*T^2 + 100*T + 100) + (5*T^6 + 80*T^5 + 100*T^3
+ 25*T + 50)*x + (55*T^5 + 50*T^4 + 75*T^3 + 25*T^2 + 25*T + 25)*x^2,
(5*T^8 + 15*T^7 + 95*T^6 + 10*T^5 + 25*T^4 + 25*T^3 + 100*T^2 + 50)
+ (65*T^7 + 55*T^6 + 70*T^5 + 100*T^4 + 25*T^2 + 100*T)*x
+ (15*T^6 + 115*T^5 + 75*T^4 + 100*T^3 + 50*T^2 + 75*T + 75)*x^2, 7)

Python

>>> from sage.all import *
>>> from sage.schemes.hyperelliptic_curves.monsky_washnitzer import frobenius_expansion_by_series
>>> R = Integers(Integer(5)**Integer(3))['x']; (x,) = R._first_ngens(1)
>>> Q = x**Integer(3) - x + R(Integer(1)/Integer(4))
>>> frobenius_expansion_by_series(Q,Integer(5),Integer(3))                                      # needs sage.libs.pari
((25*T^5 + 75*T^3 + 100*T^2 + 100*T + 100) + (5*T^6 + 80*T^5 + 100*T^3
+ 25*T + 50)*x + (55*T^5 + 50*T^4 + 75*T^3 + 25*T^2 + 25*T + 25)*x^2,
(5*T^8 + 15*T^7 + 95*T^6 + 10*T^5 + 25*T^4 + 25*T^3 + 100*T^2 + 50)
+ (65*T^7 + 55*T^6 + 70*T^5 + 100*T^4 + 25*T^2 + 100*T)*x
+ (15*T^6 + 115*T^5 + 75*T^4 + 100*T^3 + 50*T^2 + 75*T + 75)*x^2, 7)

Sage Live

from sage.schemes.hyperelliptic_curves.monsky_washnitzer import frobenius_expansion_by_series
R.<x> = Integers(5^3)['x']
Q = x^3 - x + R(1/4)
frobenius_expansion_by_series(Q,5,3)                                      # needs sage.libs.pari

sage.schemes.hyperelliptic_curves.monsky_washnitzer.helper_matrix(Q)[source]¶

Compute the (constant) matrix used to calculate the linear combinations of the \(d(x^i y^j)\) needed to eliminate the negative powers of \(y\) in the cohomology (i.e., in reduce_negative()).

INPUT:

Q – cubic polynomial

EXAMPLES:

Sage

sage: t = polygen(QQ,'t')
sage: from sage.schemes.hyperelliptic_curves.monsky_washnitzer import helper_matrix
sage: helper_matrix(t**3-4*t-691)
[     64/12891731  -16584/12891731 4297329/12891731]
[   6219/12891731     -32/12891731    8292/12891731]
[    -24/12891731    6219/12891731     -32/12891731]

Python

>>> from sage.all import *
>>> t = polygen(QQ,'t')
>>> from sage.schemes.hyperelliptic_curves.monsky_washnitzer import helper_matrix
>>> helper_matrix(t**Integer(3)-Integer(4)*t-Integer(691))
[     64/12891731  -16584/12891731 4297329/12891731]
[   6219/12891731     -32/12891731    8292/12891731]
[    -24/12891731    6219/12891731     -32/12891731]

Sage Live

t = polygen(QQ,'t')
from sage.schemes.hyperelliptic_curves.monsky_washnitzer import helper_matrix
helper_matrix(t**3-4*t-691)

sage.schemes.hyperelliptic_curves.monsky_washnitzer.lift(x)[source]¶

Try to call x.lift(), presumably from the \(p\)-adics to \(\ZZ\).

If this fails, it assumes the input is a power series, and tries to lift it to a power series over \(\QQ\).

This function is just a very kludgy solution to the problem of trying to make the reduction code (below) work over both \(\ZZ_p\) and \(\ZZ_p[[t]]\).

EXAMPLES:

Sage

sage: # needs sage.rings.padics
sage: from sage.schemes.hyperelliptic_curves.monsky_washnitzer import lift
sage: l = lift(Qp(13)(131)); l
131
sage: l.parent()
Integer Ring
sage: x = PowerSeriesRing(Qp(17),'x').gen()
sage: l = lift(4 + 5*x + 17*x**6); l
4 + 5*t + 17*t^6
sage: l.parent()
Power Series Ring in t over Rational Field

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> from sage.schemes.hyperelliptic_curves.monsky_washnitzer import lift
>>> l = lift(Qp(Integer(13))(Integer(131))); l
131
>>> l.parent()
Integer Ring
>>> x = PowerSeriesRing(Qp(Integer(17)),'x').gen()
>>> l = lift(Integer(4) + Integer(5)*x + Integer(17)*x**Integer(6)); l
4 + 5*t + 17*t^6
>>> l.parent()
Power Series Ring in t over Rational Field

Sage Live

# needs sage.rings.padics
from sage.schemes.hyperelliptic_curves.monsky_washnitzer import lift
l = lift(Qp(13)(131)); l
l.parent()
x = PowerSeriesRing(Qp(17),'x').gen()
l = lift(4 + 5*x + 17*x**6); l
l.parent()

sage.schemes.hyperelliptic_curves.monsky_washnitzer.matrix_of_frobenius(Q, p, M, trace=None, compute_exact_forms=False)[source]¶

Compute the matrix of Frobenius on Monsky-Washnitzer cohomology, with respect to the basis \((dx/y, x dx/y)\).

INPUT:

Q – cubic polynomial \(Q(x) = x^3 + ax + b\) defining an elliptic curve \(E\) by \(y^2 = Q(x)\). The coefficient ring of \(Q\) should be a \(\ZZ/(p^M)\ZZ\)-algebra in which the matrix of frobenius will be constructed.
p – prime >= 5 for which E has good reduction
M – integer >= 2; \(p\) -adic precision of the coefficient ring
trace – (optional) the trace of the matrix, if known in advance. This is easy to compute because it is just the \(a_p\) of the curve. If the trace is supplied, matrix_of_frobenius will use it to speed the computation (i.e. we know the determinant is \(p\), so we have two conditions, so really only column of the matrix needs to be computed. it is actually a little more complicated than that, but that’s the basic idea.) If trace=None, then both columns will be computed independently, and you can get a strong indication of correctness by verifying the trace afterwards.

Warning

THE RESULT WILL NOT NECESSARILY BE CORRECT TO M p-ADIC DIGITS. If you want prec digits of precision, you need to use the function adjusted_prec(), and then you need to reduce the answer mod \(p^{\mathrm{prec}}\) at the end.

OUTPUT:

\(2 \times 2\) matrix of Frobenius acting on Monsky-Washnitzer cohomology, with entries in the coefficient ring of Q.

EXAMPLES:

A simple example:

Sage

sage: p = 5
sage: prec = 3
sage: M = monsky_washnitzer.adjusted_prec(p, prec); M
4
sage: R.<x> = PolynomialRing(Integers(p**M))
sage: A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
sage: A
[340  62]
[ 70 533]

Python

>>> from sage.all import *
>>> p = Integer(5)
>>> prec = Integer(3)
>>> M = monsky_washnitzer.adjusted_prec(p, prec); M
4
>>> R = PolynomialRing(Integers(p**M), names=('x',)); (x,) = R._first_ngens(1)
>>> A = monsky_washnitzer.matrix_of_frobenius(x**Integer(3) - x + R(Integer(1)/Integer(4)), p, M)
>>> A
[340  62]
[ 70 533]

Sage Live

p = 5
prec = 3
M = monsky_washnitzer.adjusted_prec(p, prec); M
R.<x> = PolynomialRing(Integers(p**M))
A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
A

But the result is only accurate to prec digits:

Sage

sage: B = A.change_ring(Integers(p**prec))
sage: B
[90 62]
[70 33]

Python

>>> from sage.all import *
>>> B = A.change_ring(Integers(p**prec))
>>> B
[90 62]
[70 33]

Sage Live

B = A.change_ring(Integers(p**prec))
B

Check trace (123 = -2 mod 125) and determinant:

Sage

sage: B.det()
5
sage: B.trace()
123
sage: EllipticCurve([-1, 1/4]).ap(5)
-2

Python

>>> from sage.all import *
>>> B.det()
5
>>> B.trace()
123
>>> EllipticCurve([-Integer(1), Integer(1)/Integer(4)]).ap(Integer(5))
-2

Sage Live

B.det()
B.trace()
EllipticCurve([-1, 1/4]).ap(5)

Try using the trace to speed up the calculation:

Sage

sage: A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4),
....:                                           p, M, -2)
sage: A
[ 90  62]
[320 533]

Python

>>> from sage.all import *
>>> A = monsky_washnitzer.matrix_of_frobenius(x**Integer(3) - x + R(Integer(1)/Integer(4)),
...                                           p, M, -Integer(2))
>>> A
[ 90  62]
[320 533]

Sage Live

A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4),
                                          p, M, -2)
A

Hmmm… it looks different, but that’s because the trace of our first answer was only -2 modulo \(5^3\), not -2 modulo \(5^5\). So the right answer is:

Sage

sage: A.change_ring(Integers(p**prec))
[90 62]
[70 33]

Python

>>> from sage.all import *
>>> A.change_ring(Integers(p**prec))
[90 62]
[70 33]

Sage Live

A.change_ring(Integers(p**prec))

Check it works with only one digit of precision:

Sage

sage: p = 5
sage: prec = 1
sage: M = monsky_washnitzer.adjusted_prec(p, prec)
sage: R.<x> = PolynomialRing(Integers(p**M))
sage: A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
sage: A.change_ring(Integers(p))
[0 2]
[0 3]

Python

>>> from sage.all import *
>>> p = Integer(5)
>>> prec = Integer(1)
>>> M = monsky_washnitzer.adjusted_prec(p, prec)
>>> R = PolynomialRing(Integers(p**M), names=('x',)); (x,) = R._first_ngens(1)
>>> A = monsky_washnitzer.matrix_of_frobenius(x**Integer(3) - x + R(Integer(1)/Integer(4)), p, M)
>>> A.change_ring(Integers(p))
[0 2]
[0 3]

Sage Live

p = 5
prec = 1
M = monsky_washnitzer.adjusted_prec(p, prec)
R.<x> = PolynomialRing(Integers(p**M))
A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
A.change_ring(Integers(p))

Here is an example that is particularly badly conditioned for using the trace trick:

Sage

sage: # needs sage.libs.pari
sage: p = 11
sage: prec = 3
sage: M = monsky_washnitzer.adjusted_prec(p, prec)
sage: R.<x> = PolynomialRing(Integers(p**M))
sage: A = monsky_washnitzer.matrix_of_frobenius(x^3 + 7*x + 8, p, M)
sage: A.change_ring(Integers(p**prec))
[1144  176]
[ 847  185]

Python

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> p = Integer(11)
>>> prec = Integer(3)
>>> M = monsky_washnitzer.adjusted_prec(p, prec)
>>> R = PolynomialRing(Integers(p**M), names=('x',)); (x,) = R._first_ngens(1)
>>> A = monsky_washnitzer.matrix_of_frobenius(x**Integer(3) + Integer(7)*x + Integer(8), p, M)
>>> A.change_ring(Integers(p**prec))
[1144  176]
[ 847  185]

Sage Live

# needs sage.libs.pari
p = 11
prec = 3
M = monsky_washnitzer.adjusted_prec(p, prec)
R.<x> = PolynomialRing(Integers(p**M))
A = monsky_washnitzer.matrix_of_frobenius(x^3 + 7*x + 8, p, M)
A.change_ring(Integers(p**prec))

The problem here is that the top-right entry is divisible by 11, and the bottom-left entry is divisible by \(11^2\). So when you apply the trace trick, neither \(F(dx/y)\) nor \(F(x dx/y)\) is enough to compute the whole matrix to the desired precision, even if you try increasing the target precision by one. Nevertheless, matrix_of_frobenius knows how to get the right answer by evaluating \(F((x+1) dx/y)\) instead:

Sage

sage: A = monsky_washnitzer.matrix_of_frobenius(x^3 + 7*x + 8, p, M, -2)
sage: A.change_ring(Integers(p**prec))
[1144  176]
[ 847  185]

Python

>>> from sage.all import *
>>> A = monsky_washnitzer.matrix_of_frobenius(x**Integer(3) + Integer(7)*x + Integer(8), p, M, -Integer(2))
>>> A.change_ring(Integers(p**prec))
[1144  176]
[ 847  185]

Sage Live

A = monsky_washnitzer.matrix_of_frobenius(x^3 + 7*x + 8, p, M, -2)
A.change_ring(Integers(p**prec))

The running time is about O(p*prec**2) (times some logarithmic factors), so it is feasible to run on fairly large primes, or precision (or both?!?!):

Sage

sage: # long time, needs sage.libs.pari
sage: p = 10007
sage: prec = 2
sage: M = monsky_washnitzer.adjusted_prec(p, prec)
sage: R.<x> = PolynomialRing(Integers(p**M))
sage: A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
sage: B = A.change_ring(Integers(p**prec)); B
[74311982 57996908]
[95877067 25828133]
sage: B.det()
10007
sage: B.trace()
66
sage: EllipticCurve([-1, 1/4]).ap(10007)
66

Python

>>> from sage.all import *
>>> # long time, needs sage.libs.pari
>>> p = Integer(10007)
>>> prec = Integer(2)
>>> M = monsky_washnitzer.adjusted_prec(p, prec)
>>> R = PolynomialRing(Integers(p**M), names=('x',)); (x,) = R._first_ngens(1)
>>> A = monsky_washnitzer.matrix_of_frobenius(x**Integer(3) - x + R(Integer(1)/Integer(4)), p, M)
>>> B = A.change_ring(Integers(p**prec)); B
[74311982 57996908]
[95877067 25828133]
>>> B.det()
10007
>>> B.trace()
66
>>> EllipticCurve([-Integer(1), Integer(1)/Integer(4)]).ap(Integer(10007))
66

Sage Live

# long time, needs sage.libs.pari
p = 10007
prec = 2
M = monsky_washnitzer.adjusted_prec(p, prec)
R.<x> = PolynomialRing(Integers(p**M))
A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
B = A.change_ring(Integers(p**prec)); B
B.det()
B.trace()
EllipticCurve([-1, 1/4]).ap(10007)

Sage

sage: # long time, needs sage.libs.pari
sage: p = 5
sage: prec = 300
sage: M = monsky_washnitzer.adjusted_prec(p, prec)
sage: R.<x> = PolynomialRing(Integers(p**M))
sage: A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
sage: B = A.change_ring(Integers(p**prec))
sage: B.det()
5
sage: -B.trace()
2
sage: EllipticCurve([-1, 1/4]).ap(5)
-2

Python

>>> from sage.all import *
>>> # long time, needs sage.libs.pari
>>> p = Integer(5)
>>> prec = Integer(300)
>>> M = monsky_washnitzer.adjusted_prec(p, prec)
>>> R = PolynomialRing(Integers(p**M), names=('x',)); (x,) = R._first_ngens(1)
>>> A = monsky_washnitzer.matrix_of_frobenius(x**Integer(3) - x + R(Integer(1)/Integer(4)), p, M)
>>> B = A.change_ring(Integers(p**prec))
>>> B.det()
5
>>> -B.trace()
2
>>> EllipticCurve([-Integer(1), Integer(1)/Integer(4)]).ap(Integer(5))
-2

Sage Live

# long time, needs sage.libs.pari
p = 5
prec = 300
M = monsky_washnitzer.adjusted_prec(p, prec)
R.<x> = PolynomialRing(Integers(p**M))
A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
B = A.change_ring(Integers(p**prec))
B.det()
-B.trace()
EllipticCurve([-1, 1/4]).ap(5)

Let us check consistency of the results for a range of precisions:

Sage

sage: # long time, needs sage.libs.pari
sage: p = 5
sage: max_prec = 60
sage: M = monsky_washnitzer.adjusted_prec(p, max_prec)
sage: R.<x> = PolynomialRing(Integers(p**M))
sage: A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
sage: A = A.change_ring(Integers(p**max_prec))
sage: result = []
sage: for prec in range(1, max_prec):
....:     M = monsky_washnitzer.adjusted_prec(p, prec)
....:     R.<x> = PolynomialRing(Integers(p^M),'x')
....:     B = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
....:     B = B.change_ring(Integers(p**prec))
....:     result.append(B == A.change_ring(Integers(p**prec)))
sage: result == [True] * (max_prec - 1)
True

Python

>>> from sage.all import *
>>> # long time, needs sage.libs.pari
>>> p = Integer(5)
>>> max_prec = Integer(60)
>>> M = monsky_washnitzer.adjusted_prec(p, max_prec)
>>> R = PolynomialRing(Integers(p**M), names=('x',)); (x,) = R._first_ngens(1)
>>> A = monsky_washnitzer.matrix_of_frobenius(x**Integer(3) - x + R(Integer(1)/Integer(4)), p, M)
>>> A = A.change_ring(Integers(p**max_prec))
>>> result = []
>>> for prec in range(Integer(1), max_prec):
...     M = monsky_washnitzer.adjusted_prec(p, prec)
...     R = PolynomialRing(Integers(p**M),'x', names=('x',)); (x,) = R._first_ngens(1)
...     B = monsky_washnitzer.matrix_of_frobenius(x**Integer(3) - x + R(Integer(1)/Integer(4)), p, M)
...     B = B.change_ring(Integers(p**prec))
...     result.append(B == A.change_ring(Integers(p**prec)))
>>> result == [True] * (max_prec - Integer(1))
True

Sage Live

# long time, needs sage.libs.pari
p = 5
max_prec = 60
M = monsky_washnitzer.adjusted_prec(p, max_prec)
R.<x> = PolynomialRing(Integers(p**M))
A = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
A = A.change_ring(Integers(p**max_prec))
result = []
for prec in range(1, max_prec):
    M = monsky_washnitzer.adjusted_prec(p, prec)
    R.<x> = PolynomialRing(Integers(p^M),'x')
    B = monsky_washnitzer.matrix_of_frobenius(x^3 - x + R(1/4), p, M)
    B = B.change_ring(Integers(p**prec))
    result.append(B == A.change_ring(Integers(p**prec)))
result == [True] * (max_prec - 1)

The remaining examples discuss what happens when you take the coefficient ring to be a power series ring; i.e. in effect you’re looking at a family of curves.

The code does in fact work…

Sage

sage: # needs sage.libs.pari
sage: p = 11
sage: prec = 3
sage: M = monsky_washnitzer.adjusted_prec(p, prec)
sage: S.<t> = PowerSeriesRing(Integers(p**M), default_prec=4)
sage: a = 7 + t + 3*t^2
sage: b = 8 - 6*t + 17*t^2
sage: R.<x> = PolynomialRing(S)
sage: Q = x**3 + a*x + b
sage: A = monsky_washnitzer.matrix_of_frobenius(Q, p, M)            # long time
sage: B = A.change_ring(PowerSeriesRing(Integers(p**prec), 't',     # long time
....:                                   default_prec=4)); B
[1144 + 264*t + 841*t^2 + 1025*t^3 + O(t^4)  176 + 1052*t + 216*t^2 + 523*t^3 + O(t^4)]
[   847 + 668*t + 81*t^2 + 424*t^3 + O(t^4)   185 + 341*t + 171*t^2 + 642*t^3 + O(t^4)]

Python

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> p = Integer(11)
>>> prec = Integer(3)
>>> M = monsky_washnitzer.adjusted_prec(p, prec)
>>> S = PowerSeriesRing(Integers(p**M), default_prec=Integer(4), names=('t',)); (t,) = S._first_ngens(1)
>>> a = Integer(7) + t + Integer(3)*t**Integer(2)
>>> b = Integer(8) - Integer(6)*t + Integer(17)*t**Integer(2)
>>> R = PolynomialRing(S, names=('x',)); (x,) = R._first_ngens(1)
>>> Q = x**Integer(3) + a*x + b
>>> A = monsky_washnitzer.matrix_of_frobenius(Q, p, M)            # long time
>>> B = A.change_ring(PowerSeriesRing(Integers(p**prec), 't',     # long time
...                                   default_prec=Integer(4))); B
[1144 + 264*t + 841*t^2 + 1025*t^3 + O(t^4)  176 + 1052*t + 216*t^2 + 523*t^3 + O(t^4)]
[   847 + 668*t + 81*t^2 + 424*t^3 + O(t^4)   185 + 341*t + 171*t^2 + 642*t^3 + O(t^4)]

Sage Live

# needs sage.libs.pari
p = 11
prec = 3
M = monsky_washnitzer.adjusted_prec(p, prec)
S.<t> = PowerSeriesRing(Integers(p**M), default_prec=4)
a = 7 + t + 3*t^2
b = 8 - 6*t + 17*t^2
R.<x> = PolynomialRing(S)
Q = x**3 + a*x + b
A = monsky_washnitzer.matrix_of_frobenius(Q, p, M)            # long time
B = A.change_ring(PowerSeriesRing(Integers(p**prec), 't',     # long time
                                  default_prec=4)); B

The trace trick should work for power series rings too, even in the badly-conditioned case. Unfortunately I do not know how to compute the trace in advance, so I am not sure exactly how this would help. Also, I suspect the running time will be dominated by the expansion, so the trace trick will not really speed things up anyway. Another problem is that the determinant is not always p:

Sage

sage: B.det()                                               # long time
11 + 484*t^2 + 451*t^3 + O(t^4)

Python

>>> from sage.all import *
>>> B.det()                                               # long time
11 + 484*t^2 + 451*t^3 + O(t^4)

Sage Live

B.det()                                               # long time

However, it appears that the determinant always has the property that if you substitute t - 11t, you do get the constant series p (mod p**prec). Similarly for the trace. And since the parameter only really makes sense when it is divisible by p anyway, perhaps this is not a problem after all.

sage.schemes.hyperelliptic_curves.monsky_washnitzer.matrix_of_frobenius_hyperelliptic(Q, p=None, prec=None, M=None)[source]¶

Compute the matrix of Frobenius on Monsky-Washnitzer cohomology, with respect to the basis \((dx/2y, x dx/2y, ...x^{d-2} dx/2y)\), where \(d\) is the degree of \(Q\).

INPUT:

Q – monic polynomial \(Q(x)\)
p – prime \(\geq 5\) for which \(E\) has good reduction
prec – (optional) \(p\)-adic precision of the coefficient ring
M – (optional) adjusted \(p\)-adic precision of the coefficient ring

OUTPUT:

\((d-1)\) x \((d-1)\) matrix \(M\) of Frobenius on Monsky-Washnitzer cohomology, and list of differentials {f_i } such that

\[\phi^* (x^i dx/2y) = df_i + M[i]*vec(dx/2y, ..., x^{d-2} dx/2y)\]

EXAMPLES:

Sage

sage: # needs sage.rings.padics
sage: p = 5
sage: prec = 3
sage: R.<x> = QQ['x']
sage: A,f = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(x^5 - 2*x + 3, p, prec)
sage: A
[            4*5 + O(5^3)       5 + 2*5^2 + O(5^3) 2 + 3*5 + 2*5^2 + O(5^3)     2 + 5 + 5^2 + O(5^3)]
[      3*5 + 5^2 + O(5^3)             3*5 + O(5^3)             4*5 + O(5^3)         2 + 5^2 + O(5^3)]
[    4*5 + 4*5^2 + O(5^3)     3*5 + 2*5^2 + O(5^3)       5 + 3*5^2 + O(5^3)     2*5 + 2*5^2 + O(5^3)]
[            5^2 + O(5^3)       5 + 4*5^2 + O(5^3)     4*5 + 3*5^2 + O(5^3)             2*5 + O(5^3)]

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> p = Integer(5)
>>> prec = Integer(3)
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> A,f = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(x**Integer(5) - Integer(2)*x + Integer(3), p, prec)
>>> A
[            4*5 + O(5^3)       5 + 2*5^2 + O(5^3) 2 + 3*5 + 2*5^2 + O(5^3)     2 + 5 + 5^2 + O(5^3)]
[      3*5 + 5^2 + O(5^3)             3*5 + O(5^3)             4*5 + O(5^3)         2 + 5^2 + O(5^3)]
[    4*5 + 4*5^2 + O(5^3)     3*5 + 2*5^2 + O(5^3)       5 + 3*5^2 + O(5^3)     2*5 + 2*5^2 + O(5^3)]
[            5^2 + O(5^3)       5 + 4*5^2 + O(5^3)     4*5 + 3*5^2 + O(5^3)             2*5 + O(5^3)]

Sage Live

# needs sage.rings.padics
p = 5
prec = 3
R.<x> = QQ['x']
A,f = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(x^5 - 2*x + 3, p, prec)
A

sage.schemes.hyperelliptic_curves.monsky_washnitzer.reduce_all(Q, p, coeffs, offset, compute_exact_form=False)[source]¶

Apply cohomology relations to reduce all terms to a linear combination of \(dx/y\) and \(x dx/y\).

INPUT:

Q – cubic polynomial
coeffs – list of length 3 lists. The \(i\)-th list [a, b, c] represents \(y^{2(i - offset)} (a + bx + cx^2) dx/y\).
offset – nonnegative integer

OUTPUT:

A, B – pair such that the input differential is cohomologous to (A + Bx) dx/y

Note

The algorithm operates in-place, so the data in coeffs is destroyed.

EXAMPLES:

Sage

sage: R.<x> = Integers(5^3)['x']
sage: Q = x^3 - x + R(1/4)
sage: coeffs = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
sage: coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
sage: monsky_washnitzer.reduce_all(Q, 5, coeffs, 1)
(21, 106)

Python

>>> from sage.all import *
>>> R = Integers(Integer(5)**Integer(3))['x']; (x,) = R._first_ngens(1)
>>> Q = x**Integer(3) - x + R(Integer(1)/Integer(4))
>>> coeffs = [[Integer(1), Integer(2), Integer(3)], [Integer(4), Integer(5), Integer(6)], [Integer(7), Integer(8), Integer(9)]]
>>> coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
>>> monsky_washnitzer.reduce_all(Q, Integer(5), coeffs, Integer(1))
(21, 106)

Sage Live

R.<x> = Integers(5^3)['x']
Q = x^3 - x + R(1/4)
coeffs = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
monsky_washnitzer.reduce_all(Q, 5, coeffs, 1)

sage.schemes.hyperelliptic_curves.monsky_washnitzer.reduce_negative(Q, p, coeffs, offset, exact_form=None)[source]¶

Apply cohomology relations to incorporate negative powers of \(y\) into the \(y^0\) term.

INPUT:

p – prime
Q – cubic polynomial
coeffs – list of length 3 lists. The \(i\)-th list [a, b, c] represents \(y^{2(i - offset)} (a + bx + cx^2) dx/y\).
offset – nonnegative integer

OUTPUT:

The reduction is performed in-place. The output is placed in coeffs[offset]. Note that coeffs[i] will be meaningless for i offset after this function is finished.

EXAMPLES:

Sage

sage: R.<x> = Integers(5^3)['x']
sage: Q = x^3 - x + R(1/4)
sage: coeffs = [[10, 15, 20], [1, 2, 3], [4, 5, 6], [7, 8, 9]]
sage: coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
sage: monsky_washnitzer.reduce_negative(Q, 5, coeffs, 3)
sage: coeffs[3]
[28, 52, 9]

Python

>>> from sage.all import *
>>> R = Integers(Integer(5)**Integer(3))['x']; (x,) = R._first_ngens(1)
>>> Q = x**Integer(3) - x + R(Integer(1)/Integer(4))
>>> coeffs = [[Integer(10), Integer(15), Integer(20)], [Integer(1), Integer(2), Integer(3)], [Integer(4), Integer(5), Integer(6)], [Integer(7), Integer(8), Integer(9)]]
>>> coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
>>> monsky_washnitzer.reduce_negative(Q, Integer(5), coeffs, Integer(3))
>>> coeffs[Integer(3)]
[28, 52, 9]

Sage Live

R.<x> = Integers(5^3)['x']
Q = x^3 - x + R(1/4)
coeffs = [[10, 15, 20], [1, 2, 3], [4, 5, 6], [7, 8, 9]]
coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
monsky_washnitzer.reduce_negative(Q, 5, coeffs, 3)
coeffs[3]

Sage

sage: R.<x> = Integers(7^3)['x']
sage: Q = x^3 - x + R(1/4)
sage: coeffs = [[7, 14, 21], [1, 2, 3], [4, 5, 6], [7, 8, 9]]
sage: coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
sage: monsky_washnitzer.reduce_negative(Q, 7, coeffs, 3)
sage: coeffs[3]
[245, 332, 9]

Python

>>> from sage.all import *
>>> R = Integers(Integer(7)**Integer(3))['x']; (x,) = R._first_ngens(1)
>>> Q = x**Integer(3) - x + R(Integer(1)/Integer(4))
>>> coeffs = [[Integer(7), Integer(14), Integer(21)], [Integer(1), Integer(2), Integer(3)], [Integer(4), Integer(5), Integer(6)], [Integer(7), Integer(8), Integer(9)]]
>>> coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
>>> monsky_washnitzer.reduce_negative(Q, Integer(7), coeffs, Integer(3))
>>> coeffs[Integer(3)]
[245, 332, 9]

Sage Live

R.<x> = Integers(7^3)['x']
Q = x^3 - x + R(1/4)
coeffs = [[7, 14, 21], [1, 2, 3], [4, 5, 6], [7, 8, 9]]
coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
monsky_washnitzer.reduce_negative(Q, 7, coeffs, 3)
coeffs[3]

sage.schemes.hyperelliptic_curves.monsky_washnitzer.reduce_positive(Q, p, coeffs, offset, exact_form=None)[source]¶

Apply cohomology relations to incorporate positive powers of \(y\) into the \(y^0\) term.

INPUT:

Q – cubic polynomial
coeffs – list of length 3 lists. The \(i\)-th list [a, b, c] represents \(y^{2(i - offset)} (a + bx + cx^2) dx/y\).
offset – nonnegative integer

OUTPUT:

The reduction is performed in-place. The output is placed in coeffs[offset]. Note that coeffs[i] will be meaningless for i offset after this function is finished.

EXAMPLES:

Sage

sage: R.<x> = Integers(5^3)['x']
sage: Q = x^3 - x + R(1/4)

Python

>>> from sage.all import *
>>> R = Integers(Integer(5)**Integer(3))['x']; (x,) = R._first_ngens(1)
>>> Q = x**Integer(3) - x + R(Integer(1)/Integer(4))

Sage Live

R.<x> = Integers(5^3)['x']
Q = x^3 - x + R(1/4)

Sage

sage: coeffs = [[1, 2, 3], [10, 15, 20]]
sage: coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
sage: monsky_washnitzer.reduce_positive(Q, 5, coeffs, 0)
sage: coeffs[0]
[16, 102, 88]

Python

>>> from sage.all import *
>>> coeffs = [[Integer(1), Integer(2), Integer(3)], [Integer(10), Integer(15), Integer(20)]]
>>> coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
>>> monsky_washnitzer.reduce_positive(Q, Integer(5), coeffs, Integer(0))
>>> coeffs[Integer(0)]
[16, 102, 88]

Sage Live

coeffs = [[1, 2, 3], [10, 15, 20]]
coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
monsky_washnitzer.reduce_positive(Q, 5, coeffs, 0)
coeffs[0]

Sage

sage: coeffs = [[9, 8, 7], [10, 15, 20]]
sage: coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
sage: monsky_washnitzer.reduce_positive(Q, 5, coeffs, 0)
sage: coeffs[0]
[24, 108, 92]

Python

>>> from sage.all import *
>>> coeffs = [[Integer(9), Integer(8), Integer(7)], [Integer(10), Integer(15), Integer(20)]]
>>> coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
>>> monsky_washnitzer.reduce_positive(Q, Integer(5), coeffs, Integer(0))
>>> coeffs[Integer(0)]
[24, 108, 92]

Sage Live

coeffs = [[9, 8, 7], [10, 15, 20]]
coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
monsky_washnitzer.reduce_positive(Q, 5, coeffs, 0)
coeffs[0]

sage.schemes.hyperelliptic_curves.monsky_washnitzer.reduce_zero(Q, coeffs, offset, exact_form=None)[source]¶

Apply cohomology relation to incorporate \(x^2 y^0\) term into \(x^0 y^0\) and \(x^1 y^0\) terms.

INPUT:

Q – cubic polynomial
coeffs – list of length 3 lists. The \(i\)-th list [a, b, c] represents \(y^{2(i - offset)} (a + bx + cx^2) dx/y\).
offset – nonnegative integer

OUTPUT:

The reduction is performed in-place. The output is placed in coeffs[offset]. This method completely ignores coeffs[i] for i != offset.

EXAMPLES:

Sage

sage: R.<x> = Integers(5^3)['x']
sage: Q = x^3 - x + R(1/4)
sage: coeffs = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
sage: coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
sage: monsky_washnitzer.reduce_zero(Q, coeffs, 1)
sage: coeffs[1]
[6, 5, 0]

Python

>>> from sage.all import *
>>> R = Integers(Integer(5)**Integer(3))['x']; (x,) = R._first_ngens(1)
>>> Q = x**Integer(3) - x + R(Integer(1)/Integer(4))
>>> coeffs = [[Integer(1), Integer(2), Integer(3)], [Integer(4), Integer(5), Integer(6)], [Integer(7), Integer(8), Integer(9)]]
>>> coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
>>> monsky_washnitzer.reduce_zero(Q, coeffs, Integer(1))
>>> coeffs[Integer(1)]
[6, 5, 0]

Sage Live

R.<x> = Integers(5^3)['x']
Q = x^3 - x + R(1/4)
coeffs = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
coeffs = [[R.base_ring()(a) for a in row] for row in coeffs]
monsky_washnitzer.reduce_zero(Q, coeffs, 1)
coeffs[1]

sage.schemes.hyperelliptic_curves.monsky_washnitzer.transpose_list(input)[source]¶

INPUT:

input – list of lists, each list of the same length

OUTPUT: list of lists such that output[i][j] = input[j][i]

EXAMPLES:

Sage

sage: from sage.schemes.hyperelliptic_curves.monsky_washnitzer import transpose_list
sage: L = [[1, 2], [3, 4], [5, 6]]
sage: transpose_list(L)
[[1, 3, 5], [2, 4, 6]]

Python

>>> from sage.all import *
>>> from sage.schemes.hyperelliptic_curves.monsky_washnitzer import transpose_list
>>> L = [[Integer(1), Integer(2)], [Integer(3), Integer(4)], [Integer(5), Integer(6)]]
>>> transpose_list(L)
[[1, 3, 5], [2, 4, 6]]

Sage Live

from sage.schemes.hyperelliptic_curves.monsky_washnitzer import transpose_list
L = [[1, 2], [3, 4], [5, 6]]
transpose_list(L)