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:

• [Ked2001]

• [Edix]

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)

Bases: ModuleElement

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

EXAMPLES:

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

coeff()

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

EXAMPLES:

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

coeffs(R=None)

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: 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)

coleman_integral(P, Q)

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: 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

extract_pow_y(k)

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

EXAMPLES:

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]

integrate(P, Q)

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: 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

max_pow_y()

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

EXAMPLES:

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

min_pow_y()

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

EXAMPLES:

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

reduce()

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: 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

reduce_fast(even_degree_only=False)

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: 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))

reduce_neg_y()

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

EXAMPLES:

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)

reduce_neg_y_fast(even_degree_only=False)

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

EXAMPLES:

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)

It leaves non-negative powers of \(y\) alone:

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)

reduce_neg_y_faster(even_degree_only=False)

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

EXAMPLES:

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)

reduce_pos_y()

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

EXAMPLES:

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)

reduce_pos_y_fast(even_degree_only=False)

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

EXAMPLES:

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)

class sage.schemes.hyperelliptic_curves.monsky_washnitzer.MonskyWashnitzerDifferentialRing(base_ring)

Bases: UniqueRepresentation, Module

A ring of Monsky–Washnitzer differentials over base_ring.

Element

alias of MonskyWashnitzerDifferential

Q()

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

EXAMPLES:

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

base_extend(R)

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

INPUT:

• R – ring

OUTPUT:

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

EXAMPLES:

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

change_ring(R)

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: 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

degree()

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: 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

dimension()

Return the dimension of self.

EXAMPLES:

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

frob_Q(p)

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: 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

frob_basis_elements(prec, p)

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: 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]

frob_invariant_differential(prec, p)

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: 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

helper_matrix()

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: 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]

invariant_differential()

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

EXAMPLES:

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

x_to_p(p)

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

EXAMPLES:

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

sage.schemes.hyperelliptic_curves.monsky_washnitzer.MonskyWashnitzerDifferentialRing_class

alias of MonskyWashnitzerDifferentialRing

class sage.schemes.hyperelliptic_curves.monsky_washnitzer.SpecialCubicQuotientRing(Q, laurent_series=False)

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: 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()

Get generators:

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

Coercions:

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

Create elements directly from polynomials:

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

Some arithmetic:

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

Retrieve coefficients (output is zero-padded):

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]]

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

alias of SpecialCubicQuotientRingElement

create_element(check, *args)

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 – bool (default True): whether to carry out coercion

EXAMPLES:

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

gens()

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: 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

one()

Return the unit of self.

EXAMPLES:

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

poly_ring()

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

EXAMPLES:

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

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

Bases: ModuleElement

An element of a SpecialCubicQuotientRing.

coeffs()

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: 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]]

scalar_multiply(scalar)

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: 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

shift(n)

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

EXAMPLES:

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

square()

Return the square of the element.

EXAMPLES:

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(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

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

Bases: ModuleElement

Element in the Hyperelliptic quotient ring.

EXAMPLES:

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

change_ring(R)

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

EXAMPLES:

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

coeffs(R=None)

Return the raw coefficients of this element.

INPUT:

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

OUTPUT:

• coeffs – a 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: 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)

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

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

A more complicated example:

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

Can also be used to construct elements:

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

diff()

Return the differential of self.

EXAMPLES:

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

extract_pow_y(k)

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

EXAMPLES:

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]

max_pow_y()

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

EXAMPLES:

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

min_pow_y()

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

EXAMPLES:

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

truncate_neg(n)

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

EXAMPLES:

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

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

Bases: UniqueRepresentation, Parent

The special hyperelliptic quotient ring.

Element

alias of SpecialHyperellipticQuotientElement

Q()

Return the defining polynomial of the underlying hyperelliptic curve.

EXAMPLES:

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

base_extend(R)

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

EXAMPLES:

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

change_ring(R)

Return the analog of self over the ring R.

EXAMPLES:

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

curve()

Return the underlying hyperelliptic curve.

EXAMPLES:

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

degree()

Return the degree of the underlying hyperelliptic curve.

EXAMPLES:

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

gens()

Return the generators of self

EXAMPLES:

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)

is_field(proof=True)

Return False as self is not a field.

EXAMPLES:

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

monomial(i, j, b=None)

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

EXAMPLES:

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

monomial_diff_coeffs(i, j)

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: 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))

monomial_diff_coeffs_matrices()

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

EXAMPLES:

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]
)

monsky_washnitzer()

Return the stored Monsky-Washnitzer differential ring.

EXAMPLES:

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'>

one()

Return the unit of self.

EXAMPLES:

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

prime()

Return the stored prime number \(p\).

EXAMPLES:

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

x()

Return the generator \(x\) of self

EXAMPLES:

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

y()

Return the generator \(y\) of self

EXAMPLES:

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

zero()

Return the zero of self.

EXAMPLES:

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

sage.schemes.hyperelliptic_curves.monsky_washnitzer.SpecialHyperellipticQuotientRing_class

alias of SpecialHyperellipticQuotientRing

sage.schemes.hyperelliptic_curves.monsky_washnitzer.adjusted_prec(p, prec)

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

sage.schemes.hyperelliptic_curves.monsky_washnitzer.frobenius_expansion_by_newton(Q, p, M)

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 – non-negative integer, as described above

EXAMPLES:

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)

sage.schemes.hyperelliptic_curves.monsky_washnitzer.frobenius_expansion_by_series(Q, p, M)

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 – non-negative integer, as described above

EXAMPLES:

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)

sage.schemes.hyperelliptic_curves.monsky_washnitzer.helper_matrix(Q)

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: 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]

sage.schemes.hyperelliptic_curves.monsky_washnitzer.lift(x)

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: # 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

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

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: 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]

But the result is only accurate to prec digits:

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

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

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

Try using the trace to speed up the calculation:

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

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

Check it works with only one digit of precision:

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]

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

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]

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: 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]

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: # 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

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

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

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

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: # 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)]

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

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)

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: # 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)]

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

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: 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)

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

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: 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]

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]

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

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

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]

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]

sage.schemes.hyperelliptic_curves.monsky_washnitzer.reduce_zero(Q, coeffs, offset, exact_form=None)

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: 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]

sage.schemes.hyperelliptic_curves.monsky_washnitzer.transpose_list(input)

INPUT:

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

OUTPUT:

• output – a list of lists such that output[i][j] = input[j][i]

EXAMPLES:

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]]