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: 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 - >>> 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 - coeff()[source]¶
- 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 - >>> 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 
 - 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 - selfis \(A dx/2y\)- 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) - >>> 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) 
 - coleman_integral(P, Q)[source]¶
- Compute the definite integral of - selffrom \(P\) 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 - >>> 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 
 - extract_pow_y(k)[source]¶
- Return the power of \(y\) in \(A\) where - selfis \(A dx/2y\).- 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] - >>> 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] 
 - integrate(P, Q)[source]¶
- Compute the definite integral of - selffrom \(P\) 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 - >>> 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 
 - max_pow_y()[source]¶
- Return the maximum power of \(y\) in \(A\) where - selfis \(A dx/2y\).- 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 - >>> 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 
 - min_pow_y()[source]¶
- Return the minimum power of \(y\) in \(A\) where - selfis \(A dx/2y\).- 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 - >>> 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 
 - 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: 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 - >>> 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 
 - 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: 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)) - >>> 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)) 
 - reduce_neg_y()[source]¶
- 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) - >>> 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) 
 - reduce_neg_y_fast(even_degree_only=False)[source]¶
- 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) - >>> 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) - It leaves nonnegative 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) - >>> 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) 
 - reduce_neg_y_faster(even_degree_only=False)[source]¶
- 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) - >>> 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) 
 - reduce_pos_y()[source]¶
- 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) - >>> 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) 
 - reduce_pos_y_fast(even_degree_only=False)[source]¶
- 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) - >>> 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) 
 
- 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 - selfis 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 - >>> 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 
 - base_extend(R)[source]¶
- Return a new differential ring which is - selfbase-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 - >>> 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 
 - change_ring(R)[source]¶
- Return a new differential ring which is - selfwith the coefficient ring changed to \(R\).- INPUT: - R– ring of coefficients
 - OUTPUT: - selfwith 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 - >>> 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 
 - degree()[source]¶
- Return the degree of \(Q(x)\), where the model of the underlying hyperelliptic curve of - selfis 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 - >>> 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 
 - dimension()[source]¶
- 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 - >>> 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 
 - 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: 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 - >>> 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 
 - 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: 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] - >>> 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] 
 - 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: 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 - >>> 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 
 - 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: 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] - >>> 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] 
 - invariant_differential()[source]¶
- 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 - >>> 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 
 - 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: 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 - >>> 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.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: 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() - >>> 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() - Get generators: - sage: x, T = R.gens() sage: x (0) + (1)*x + (0)*x^2 sage: T (T) + (0)*x + (0)*x^2 - >>> from sage.all import * >>> x, T = R.gens() >>> x (0) + (1)*x + (0)*x^2 >>> T (T) + (0)*x + (0)*x^2 - Coercions: - sage: R(7) (7) + (0)*x + (0)*x^2 - >>> from sage.all import * >>> R(Integer(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 - >>> 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 - 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 - >>> 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 - 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]] - >>> 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]] - 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: 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 - >>> 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 
 - gens()[source]¶
- Return (x, T) where x and T are the generators of the ring (as elements of this 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: x (0) + (1)*x + (0)*x^2 sage: T (T) + (0)*x + (0)*x^2 - >>> 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 
 - one()[source]¶
- 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 - >>> 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 
 - poly_ring()[source]¶
- 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 - >>> 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 
 
- 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: 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]] - >>> 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]] 
 - 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: 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 - >>> 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 
 - shift(n)[source]¶
- 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 - >>> 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 
 - square()[source]¶
- 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() - >>> 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: 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 - >>> 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 
 
- 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: 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 - >>> 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 - change_ring(R)[source]¶
- 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 - >>> 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 
 - 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: 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) - >>> 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) - 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 - >>> 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 - 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 - >>> 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 - Can also be used to construct elements: - sage: a.parent()(raw, offset) == a True - >>> from sage.all import * >>> a.parent()(raw, offset) == a True 
 - diff()[source]¶
- 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 - >>> 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 
 - extract_pow_y(k)[source]¶
- Return the coefficients of \(y^k\) in - selfas 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] - >>> 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] 
 - max_pow_y()[source]¶
- Return the maximal degree of - selfwith 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 - >>> 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 
 - min_pow_y()[source]¶
- Return the minimal degree of - selfwith 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 - >>> 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 
 - truncate_neg(n)[source]¶
- Return - selfminus 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 - >>> 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 
 
- 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: 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 - >>> 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 
 - base_extend(R)[source]¶
- Return the base extension of - selfto the ring- Rif 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 - >>> 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 
 - change_ring(R)[source]¶
- Return the analog of - selfover 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 - >>> 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 
 - curve()[source]¶
- 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 - >>> 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 
 - degree()[source]¶
- 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 - >>> 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 
 - gens()[source]¶
- 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) - >>> 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) 
 - is_field(proof=True)[source]¶
- Return - Falseas- selfis 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 - >>> 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 
 - monomial(i, j, b=None)[source]¶
- 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 - >>> 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 
 - 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: 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)) - >>> 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)) 
 - monomial_diff_coeffs_matrices()[source]¶
- 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] ) - >>> 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] ) 
 - monsky_washnitzer()[source]¶
- 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'> - >>> 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'> 
 - one()[source]¶
- 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 - >>> 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 
 - prime()[source]¶
- 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 - >>> 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 
 - x()[source]¶
- 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 - >>> 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 
 - y()[source]¶
- 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 - >>> 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 
 - zero()[source]¶
- 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 - >>> 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.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_frobeniusto 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 - >>> from sage.all import * >>> from sage.schemes.hyperelliptic_curves.monsky_washnitzer import adjusted_prec >>> adjusted_prec(Integer(5),Integer(2)) 3 
- 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 - SpecialCubicQuotientRingassociated 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: 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) - >>> 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.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 - SpecialCubicQuotientRingassociated 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: 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) - >>> 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.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: 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] - >>> 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.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: # 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 - >>> 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.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: 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] - >>> 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] - But the result is only accurate to - precdigits:- sage: B = A.change_ring(Integers(p**prec)) sage: B [90 62] [70 33] - >>> from sage.all import * >>> B = A.change_ring(Integers(p**prec)) >>> 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 - >>> from sage.all import * >>> B.det() 5 >>> B.trace() 123 >>> EllipticCurve([-Integer(1), Integer(1)/Integer(4)]).ap(Integer(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] - >>> 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] - 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] - >>> from sage.all import * >>> 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] - >>> 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] - 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] - >>> 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] - 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_frobeniusknows 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] - >>> 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] - 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 - >>> 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: # 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 - >>> 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 - 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 - >>> 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 - 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)] - >>> 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)] - 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) - >>> from sage.all import * >>> 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)[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: # 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)] - >>> 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.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: 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) - >>> 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.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: 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] - >>> 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: 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] - >>> 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.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: R.<x> = Integers(5^3)['x'] sage: Q = x^3 - x + R(1/4) - >>> 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: 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] - >>> 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: 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] - >>> 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.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: 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] - >>> 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.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: 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]] - >>> 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]]