Tate’s parametrisation of \(p\)-adic curves with multiplicative reduction¶
Let \(E\) be an elliptic curve defined over the \(p\)-adic numbers \(\QQ_p\). Suppose that \(E\) has multiplicative reduction, i.e. that the \(j\)-invariant of \(E\) has negative valuation, say \(n\). Then there exists a parameter \(q\) in \(\ZZ_p\) of valuation \(n\) such that the points of \(E\) defined over the algebraic closure \(\bar{\QQ}_p\) are in bijection with \(\bar{\QQ}_p^{\times}\,/\, q^{\ZZ}\). More precisely there exists the series \(s_4(q)\) and \(s_6(q)\) such that the \(y^2+x y = x^3 + s_4(q) x+s_6(q)\) curve is isomorphic to \(E\) over \(\bar{\QQ}_p\) (or over \(\QQ_p\) if the reduction is split multiplicative). There is a \(p\)-adic analytic map from \(\bar{\QQ}^{\times}_p\) to this curve with kernel \(q^{\ZZ}\). Points of good reduction correspond to points of valuation \(0\) in \(\bar{\QQ}^{\times}_p\).
See chapter V of [Sil1994] for more details.
AUTHORS:
- Chris Wuthrich (23/05/2007): first version 
- William Stein (2007-05-29): added some examples; editing. 
- Chris Wuthrich (04/09): reformatted docstrings. 
- class sage.schemes.elliptic_curves.ell_tate_curve.TateCurve(E, p)[source]¶
- Bases: - SageObject- Tate’s \(p\)-adic uniformisation of an elliptic curve with multiplicative reduction. - Note - Some of the methods of this Tate curve only work when the reduction is split multiplicative over \(\QQ_p\). - EXAMPLES: - sage: e = EllipticCurve('130a1') sage: eq = e.tate_curve(5); eq 5-adic Tate curve associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 - 33*x + 68 over Rational Field sage: eq == loads(dumps(eq)) True - >>> from sage.all import * >>> e = EllipticCurve('130a1') >>> eq = e.tate_curve(Integer(5)); eq 5-adic Tate curve associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 - 33*x + 68 over Rational Field >>> eq == loads(dumps(eq)) True - REFERENCES: [Sil1994] - E2(prec=20)[source]¶
- Return the value of the \(p\)-adic Eisenstein series of weight 2 evaluated on the elliptic curve having split multiplicative reduction. - INPUT: - prec– the \(p\)-adic precision (default: 20)
 - EXAMPLES: - sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.E2(prec=10) 4 + 2*5^2 + 2*5^3 + 5^4 + 2*5^5 + 5^7 + 5^8 + 2*5^9 + O(5^10) sage: T = EllipticCurve('14').tate_curve(7) sage: T.E2(30) 2 + 4*7 + 7^2 + 3*7^3 + 6*7^4 + 5*7^5 + 2*7^6 + 7^7 + 5*7^8 + 6*7^9 + 5*7^10 + 2*7^11 + 6*7^12 + 4*7^13 + 3*7^15 + 5*7^16 + 4*7^17 + 4*7^18 + 2*7^20 + 7^21 + 5*7^22 + 4*7^23 + 4*7^24 + 3*7^25 + 6*7^26 + 3*7^27 + 6*7^28 + O(7^30) - >>> from sage.all import * >>> eq = EllipticCurve('130a1').tate_curve(Integer(5)) >>> eq.E2(prec=Integer(10)) 4 + 2*5^2 + 2*5^3 + 5^4 + 2*5^5 + 5^7 + 5^8 + 2*5^9 + O(5^10) >>> T = EllipticCurve('14').tate_curve(Integer(7)) >>> T.E2(Integer(30)) 2 + 4*7 + 7^2 + 3*7^3 + 6*7^4 + 5*7^5 + 2*7^6 + 7^7 + 5*7^8 + 6*7^9 + 5*7^10 + 2*7^11 + 6*7^12 + 4*7^13 + 3*7^15 + 5*7^16 + 4*7^17 + 4*7^18 + 2*7^20 + 7^21 + 5*7^22 + 4*7^23 + 4*7^24 + 3*7^25 + 6*7^26 + 3*7^27 + 6*7^28 + O(7^30) 
 - L_invariant(prec=20)[source]¶
- Return the mysterious \(\mathcal{L}\)-invariant associated to an elliptic curve with split multiplicative reduction. - One instance where this constant appears is in the exceptional case of the \(p\)-adic Birch and Swinnerton-Dyer conjecture as formulated in [MTT1986]. See [Col2004] for a detailed discussion. - INPUT: - prec– the \(p\)-adic precision (default: 20)
 - EXAMPLES: - sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.L_invariant(prec=10) 5^3 + 4*5^4 + 2*5^5 + 2*5^6 + 2*5^7 + 3*5^8 + 5^9 + O(5^10) - >>> from sage.all import * >>> eq = EllipticCurve('130a1').tate_curve(Integer(5)) >>> eq.L_invariant(prec=Integer(10)) 5^3 + 4*5^4 + 2*5^5 + 2*5^6 + 2*5^7 + 3*5^8 + 5^9 + O(5^10) 
 - curve(prec=20)[source]¶
- Return the \(p\)-adic elliptic curve of the form \(y^2+x y = x^3 + s_4 x+s_6\). - This curve with split multiplicative reduction is isomorphic to the given curve over the algebraic closure of \(\QQ_p\). - INPUT: - prec– the \(p\)-adic precision (default: 20)
 - EXAMPLES: - sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.curve(prec=5) Elliptic Curve defined by y^2 + (1+O(5^5))*x*y = x^3 + (2*5^4+5^5+2*5^6+5^7+3*5^8+O(5^9))*x + (2*5^3+5^4+2*5^5+5^7+O(5^8)) over 5-adic Field with capped relative precision 5 - >>> from sage.all import * >>> eq = EllipticCurve('130a1').tate_curve(Integer(5)) >>> eq.curve(prec=Integer(5)) Elliptic Curve defined by y^2 + (1+O(5^5))*x*y = x^3 + (2*5^4+5^5+2*5^6+5^7+3*5^8+O(5^9))*x + (2*5^3+5^4+2*5^5+5^7+O(5^8)) over 5-adic Field with capped relative precision 5 
 - is_split()[source]¶
- Return - Trueif the given elliptic curve has split multiplicative reduction.- EXAMPLES: - sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.is_split() True sage: eq = EllipticCurve('37a1').tate_curve(37) sage: eq.is_split() False - >>> from sage.all import * >>> eq = EllipticCurve('130a1').tate_curve(Integer(5)) >>> eq.is_split() True >>> eq = EllipticCurve('37a1').tate_curve(Integer(37)) >>> eq.is_split() False 
 - lift(P, prec=20)[source]¶
- Given a point \(P\) in the formal group of the elliptic curve \(E\) with split multiplicative reduction, this produces an element \(u\) in \(\QQ_p^{\times}\) mapped to the point \(P\) by the Tate parametrisation. The algorithm return the unique such element in \(1+p\ZZ_p\). - INPUT: - P– a point on the elliptic curve
- prec– the \(p\)-adic precision (default: 20)
 - EXAMPLES: - sage: e = EllipticCurve('130a1') sage: eq = e.tate_curve(5) sage: P = e([-6,10]) sage: l = eq.lift(12*P, prec=10); l 1 + 4*5 + 5^3 + 5^4 + 4*5^5 + 5^6 + 5^7 + 4*5^8 + 5^9 + O(5^10) - >>> from sage.all import * >>> e = EllipticCurve('130a1') >>> eq = e.tate_curve(Integer(5)) >>> P = e([-Integer(6),Integer(10)]) >>> l = eq.lift(Integer(12)*P, prec=Integer(10)); l 1 + 4*5 + 5^3 + 5^4 + 4*5^5 + 5^6 + 5^7 + 4*5^8 + 5^9 + O(5^10) - Now we map the lift l back and check that it is indeed right.: - sage: eq.parametrisation_onto_original_curve(l) (4*5^-2 + 2*5^-1 + 4*5 + 3*5^3 + 5^4 + 2*5^5 + 4*5^6 + O(5^7) : 2*5^-3 + 5^-1 + 4 + 4*5 + 5^2 + 3*5^3 + 4*5^4 + O(5^6) : 1 + O(5^10)) sage: e5 = e.change_ring(Qp(5,9)) sage: e5(12*P) (4*5^-2 + 2*5^-1 + 4*5 + 3*5^3 + 5^4 + 2*5^5 + 4*5^6 + O(5^7) : 2*5^-3 + 5^-1 + 4 + 4*5 + 5^2 + 3*5^3 + 4*5^4 + O(5^6) : 1 + O(5^9)) - >>> from sage.all import * >>> eq.parametrisation_onto_original_curve(l) (4*5^-2 + 2*5^-1 + 4*5 + 3*5^3 + 5^4 + 2*5^5 + 4*5^6 + O(5^7) : 2*5^-3 + 5^-1 + 4 + 4*5 + 5^2 + 3*5^3 + 4*5^4 + O(5^6) : 1 + O(5^10)) >>> e5 = e.change_ring(Qp(Integer(5),Integer(9))) >>> e5(Integer(12)*P) (4*5^-2 + 2*5^-1 + 4*5 + 3*5^3 + 5^4 + 2*5^5 + 4*5^6 + O(5^7) : 2*5^-3 + 5^-1 + 4 + 4*5 + 5^2 + 3*5^3 + 4*5^4 + O(5^6) : 1 + O(5^9)) 
 - original_curve()[source]¶
- Return the elliptic curve the Tate curve was constructed from. - EXAMPLES: - sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.original_curve() Elliptic Curve defined by y^2 + x*y + y = x^3 - 33*x + 68 over Rational Field - >>> from sage.all import * >>> eq = EllipticCurve('130a1').tate_curve(Integer(5)) >>> eq.original_curve() Elliptic Curve defined by y^2 + x*y + y = x^3 - 33*x + 68 over Rational Field 
 - padic_height(prec=20)[source]¶
- Return the canonical \(p\)-adic height function on the original curve. - INPUT: - prec– the \(p\)-adic precision (default: 20)
 - OUTPUT: a function that can be evaluated on rational points of \(E\) - EXAMPLES: - sage: e = EllipticCurve('130a1') sage: eq = e.tate_curve(5) sage: h = eq.padic_height(prec=10) sage: P = e.gens()[0] sage: h(P) 2*5^-1 + 1 + 2*5 + 2*5^2 + 3*5^3 + 3*5^6 + 5^7 + O(5^9) - >>> from sage.all import * >>> e = EllipticCurve('130a1') >>> eq = e.tate_curve(Integer(5)) >>> h = eq.padic_height(prec=Integer(10)) >>> P = e.gens()[Integer(0)] >>> h(P) 2*5^-1 + 1 + 2*5 + 2*5^2 + 3*5^3 + 3*5^6 + 5^7 + O(5^9) - Check that it is a quadratic function: - sage: h(3*P)-3^2*h(P) O(5^9) - >>> from sage.all import * >>> h(Integer(3)*P)-Integer(3)**Integer(2)*h(P) O(5^9) 
 - padic_regulator(prec=20)[source]¶
- Compute the canonical \(p\)-adic regulator on the extended Mordell-Weil group as in [MTT1986] (with the correction of [Wer1998] and sign convention in [SW2013].) - The \(p\)-adic Birch and Swinnerton-Dyer conjecture predicts that this value appears in the formula for the leading term of the \(p\)-adic \(L\)-function. - INPUT: - prec– the \(p\)-adic precision (default: 20)
 - EXAMPLES: - sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.padic_regulator() 2*5^-1 + 1 + 2*5 + 2*5^2 + 3*5^3 + 3*5^6 + 5^7 + 3*5^9 + 3*5^10 + 3*5^12 + 4*5^13 + 3*5^15 + 2*5^16 + 3*5^18 + 4*5^19 + 4*5^20 + 3*5^21 + 4*5^22 + O(5^23) - >>> from sage.all import * >>> eq = EllipticCurve('130a1').tate_curve(Integer(5)) >>> eq.padic_regulator() 2*5^-1 + 1 + 2*5 + 2*5^2 + 3*5^3 + 3*5^6 + 5^7 + 3*5^9 + 3*5^10 + 3*5^12 + 4*5^13 + 3*5^15 + 2*5^16 + 3*5^18 + 4*5^19 + 4*5^20 + 3*5^21 + 4*5^22 + O(5^23) 
 - parameter(prec=20)[source]¶
- Return the Tate parameter \(q\) such that the curve is isomorphic over the algebraic closure of \(\QQ_p\) to the curve \(\QQ_p^{\times}/q^{\ZZ}\). - INPUT: - prec– the \(p\)-adic precision (default: 20)
 - EXAMPLES: - sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.parameter(prec=5) 3*5^3 + 3*5^4 + 2*5^5 + 2*5^6 + 3*5^7 + O(5^8) - >>> from sage.all import * >>> eq = EllipticCurve('130a1').tate_curve(Integer(5)) >>> eq.parameter(prec=Integer(5)) 3*5^3 + 3*5^4 + 2*5^5 + 2*5^6 + 3*5^7 + O(5^8) 
 - parametrisation_onto_original_curve(u, prec=None)[source]¶
- Given an element \(u\) in \(\QQ_p^{\times}\), this computes its image on the original curve under the \(p\)-adic uniformisation of \(E\). - INPUT: - u– a nonzero \(p\)-adic number
- prec– the \(p\)-adic precision; default is the relative precision of- u, otherwise 20
 - EXAMPLES: - sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.parametrisation_onto_original_curve(1+5+5^2+O(5^10)) (4*5^-2 + 4*5^-1 + 4 + 2*5^3 + 3*5^4 + 2*5^6 + O(5^7) : 3*5^-3 + 5^-2 + 4*5^-1 + 1 + 4*5 + 5^2 + 3*5^5 + O(5^6) : 1 + O(5^10)) sage: eq.parametrisation_onto_original_curve(1+5+5^2+O(5^10), prec=20) Traceback (most recent call last): ... ValueError: requested more precision than the precision of u - >>> from sage.all import * >>> eq = EllipticCurve('130a1').tate_curve(Integer(5)) >>> eq.parametrisation_onto_original_curve(Integer(1)+Integer(5)+Integer(5)**Integer(2)+O(Integer(5)**Integer(10))) (4*5^-2 + 4*5^-1 + 4 + 2*5^3 + 3*5^4 + 2*5^6 + O(5^7) : 3*5^-3 + 5^-2 + 4*5^-1 + 1 + 4*5 + 5^2 + 3*5^5 + O(5^6) : 1 + O(5^10)) >>> eq.parametrisation_onto_original_curve(Integer(1)+Integer(5)+Integer(5)**Integer(2)+O(Integer(5)**Integer(10)), prec=Integer(20)) Traceback (most recent call last): ... ValueError: requested more precision than the precision of u - Here is how one gets a 4-torsion point on \(E\) over \(\QQ_5\): - sage: R = Qp(5,30) sage: i = R(-1).sqrt() sage: T = eq.parametrisation_onto_original_curve(i, prec=30); T (2 + 3*5 + 4*5^2 + 2*5^3 + 5^4 + 4*5^5 + 2*5^7 + 5^8 + 5^9 + 5^12 + 3*5^13 + 3*5^14 + 5^15 + 4*5^17 + 5^18 + 3*5^19 + 2*5^20 + 4*5^21 + 5^22 + 3*5^23 + 3*5^24 + 4*5^25 + 3*5^26 + 3*5^27 + 3*5^28 + 3*5^29 + O(5^30) : 3*5 + 5^2 + 5^4 + 3*5^5 + 3*5^7 + 2*5^8 + 4*5^9 + 5^10 + 2*5^11 + 4*5^13 + 2*5^14 + 4*5^15 + 4*5^16 + 3*5^17 + 2*5^18 + 4*5^20 + 2*5^21 + 2*5^22 + 4*5^23 + 4*5^24 + 4*5^25 + 5^26 + 3*5^27 + 2*5^28 + O(5^30) : 1 + O(5^30)) sage: 4*T (0 : 1 + O(5^30) : 0) - >>> from sage.all import * >>> R = Qp(Integer(5),Integer(30)) >>> i = R(-Integer(1)).sqrt() >>> T = eq.parametrisation_onto_original_curve(i, prec=Integer(30)); T (2 + 3*5 + 4*5^2 + 2*5^3 + 5^4 + 4*5^5 + 2*5^7 + 5^8 + 5^9 + 5^12 + 3*5^13 + 3*5^14 + 5^15 + 4*5^17 + 5^18 + 3*5^19 + 2*5^20 + 4*5^21 + 5^22 + 3*5^23 + 3*5^24 + 4*5^25 + 3*5^26 + 3*5^27 + 3*5^28 + 3*5^29 + O(5^30) : 3*5 + 5^2 + 5^4 + 3*5^5 + 3*5^7 + 2*5^8 + 4*5^9 + 5^10 + 2*5^11 + 4*5^13 + 2*5^14 + 4*5^15 + 4*5^16 + 3*5^17 + 2*5^18 + 4*5^20 + 2*5^21 + 2*5^22 + 4*5^23 + 4*5^24 + 4*5^25 + 5^26 + 3*5^27 + 2*5^28 + O(5^30) : 1 + O(5^30)) >>> Integer(4)*T (0 : 1 + O(5^30) : 0) 
 - parametrisation_onto_tate_curve(u, prec=None)[source]¶
- Given an element \(u\) in \(\QQ_p^{\times}\), this computes its image on the Tate curve under the \(p\)-adic uniformisation of \(E\). - INPUT: - u– a nonzero \(p\)-adic number
- prec– the \(p\)-adic precision; default is the relative precision of- u, otherwise 20
 - EXAMPLES: - sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.parametrisation_onto_tate_curve(1+5+5^2+O(5^10), prec=10) (5^-2 + 4*5^-1 + 1 + 2*5 + 3*5^2 + 2*5^5 + 3*5^6 + O(5^7) : 4*5^-3 + 2*5^-1 + 4 + 2*5 + 3*5^4 + 2*5^5 + O(5^6) : 1 + O(5^10)) sage: eq.parametrisation_onto_tate_curve(1+5+5^2+O(5^10)) (5^-2 + 4*5^-1 + 1 + 2*5 + 3*5^2 + 2*5^5 + 3*5^6 + O(5^7) : 4*5^-3 + 2*5^-1 + 4 + 2*5 + 3*5^4 + 2*5^5 + O(5^6) : 1 + O(5^10)) sage: eq.parametrisation_onto_tate_curve(1+5+5^2+O(5^10), prec=20) Traceback (most recent call last): ... ValueError: requested more precision than the precision of u - >>> from sage.all import * >>> eq = EllipticCurve('130a1').tate_curve(Integer(5)) >>> eq.parametrisation_onto_tate_curve(Integer(1)+Integer(5)+Integer(5)**Integer(2)+O(Integer(5)**Integer(10)), prec=Integer(10)) (5^-2 + 4*5^-1 + 1 + 2*5 + 3*5^2 + 2*5^5 + 3*5^6 + O(5^7) : 4*5^-3 + 2*5^-1 + 4 + 2*5 + 3*5^4 + 2*5^5 + O(5^6) : 1 + O(5^10)) >>> eq.parametrisation_onto_tate_curve(Integer(1)+Integer(5)+Integer(5)**Integer(2)+O(Integer(5)**Integer(10))) (5^-2 + 4*5^-1 + 1 + 2*5 + 3*5^2 + 2*5^5 + 3*5^6 + O(5^7) : 4*5^-3 + 2*5^-1 + 4 + 2*5 + 3*5^4 + 2*5^5 + O(5^6) : 1 + O(5^10)) >>> eq.parametrisation_onto_tate_curve(Integer(1)+Integer(5)+Integer(5)**Integer(2)+O(Integer(5)**Integer(10)), prec=Integer(20)) Traceback (most recent call last): ... ValueError: requested more precision than the precision of u 
 - prime()[source]¶
- Return the residual characteristic \(p\). - EXAMPLES: - sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.original_curve() Elliptic Curve defined by y^2 + x*y + y = x^3 - 33*x + 68 over Rational Field sage: eq.prime() 5 - >>> from sage.all import * >>> eq = EllipticCurve('130a1').tate_curve(Integer(5)) >>> eq.original_curve() Elliptic Curve defined by y^2 + x*y + y = x^3 - 33*x + 68 over Rational Field >>> eq.prime() 5