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)¶
Bases:
sage.structure.sage_object.SageObjectTate’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
REFERENCES: [Sil1994]
- E2(prec=20)¶
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 is 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)
- L_invariant(prec=20)¶
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 is 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)
- curve(prec=20)¶
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 is 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
- is_split()¶
Return True if 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
- lift(P, prec=20)¶
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 is 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)
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))
- original_curve()¶
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
- padic_height(prec=20)¶
Return the canonical \(p\)-adic height function on the original curve.
INPUT:
prec– the \(p\)-adic precision, default is 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)
Check that it is a quadratic function:
sage: h(3*P)-3^2*h(P) O(5^9)
- padic_regulator(prec=20)¶
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 is 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)
- parameter(prec=20)¶
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 is 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)
- parametrisation_onto_original_curve(u, prec=None)¶
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 non-zero \(p\)-adic number.prec– the \(p\)-adic precision, default is the relative precision ofuotherwise 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
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)
- parametrisation_onto_tate_curve(u, prec=None)¶
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 non-zero \(p\)-adic number.prec– the \(p\)-adic precision, default is the relative precision ofuotherwise 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
- prime()¶
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