Let be an elliptic curve defined over the -adic numbers . Suppose that has multiplicative reduction, i.e. that the -invariant of has negative valuation, say . Then there exists a parameter in of valuation such that the points of defined over the algebraic closure are in bijection with . More precisely there exists the series and such that the curve is isomorphic to over (or over if the reduction is split multiplicative). There is -adic analytic map from to this curve with kernel . Points of good reduction correspond to points of valuation in . See chapter V of [Sil2] for more details.
REFERENCES :
GTM 151, Springer 1994.
AUTHORS:
Bases: sage.structure.sage_object.SageObject
Tate’s -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 .
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 :
Returns the value of the -adic Eisenstein series of weight 2 evaluated on the elliptic curve having split multiplicative reduction.
INPUT:
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)
Returns the mysterious -invariant associated to an elliptic curve with split multiplicative reduction. One instance where this constant appears is in the exceptional case of the -adic Birch and Swinnerton-Dyer conjecture as formulated in [MTT]. See [Col] for a detailed discussion.
INPUT:
REFERENCES:
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)
Returns the -adic elliptic curve of the form . This curve with split multiplicative reduction is isomorphic to the given curve over the algebraic closure of .
INPUT:
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
Returns 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
Given a point in the formal group of the elliptic curve with split multiplicative reduction, this produces an element in mapped to the point by the Tate parametrisation. The algorithm return the unique such element in .
INPUT:
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(5^20))
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))
Returns 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
Returns the canonical -adic height function on the original curve.
INPUT:
OUTPUT:
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(5^8)
Check that it is a quadratic function:
sage: h(3*P)-3^2*h(P)
O(5^8)
Computes the canonical -adic regulator on the extended Mordell-Weil group as in [MTT] (with the correction of [Wer] and sign convention in [SW].) The -adic Birch and Swinnerton-Dyer conjecture predicts that this value appears in the formula for the leading term of the -adic L-function.
INPUT:
REFERENCES:
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 + O(5^20)
Returns the Tate parameter such that the curve is isomorphic over the algebraic closure of to the curve .
INPUT:
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)
Given an element in , this computes its image on the original curve under the -adic uniformisation of .
INPUT:
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(5^20))
Here is how one gets a 4-torsion point on over :
sage: R = Qp(5,10)
sage: i = R(-1).sqrt()
sage: T = eq.parametrisation_onto_original_curve(i); T
(2 + 3*5 + 4*5^2 + 2*5^3 + 5^4 + 4*5^5 + 2*5^7 + 5^8 + 5^9 + O(5^10) :
3*5 + 5^2 + 5^4 + 3*5^5 + 3*5^7 + 2*5^8 + 4*5^9 + O(5^10) : 1 + O(5^20))
sage: 4*T
(0 : 1 + O(5^20) : 0)
Given an element in , this computes its image on the Tate curve under the -adic uniformisation of .
INPUT:
EXAMPLES:
sage: eq = EllipticCurve('130a1').tate_curve(5)
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(5^20))
Returns the residual characteristic .
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