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 $$p$$-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 [Sil2] for more details.

REFERENCES :

• [Sil2] Silverman Joseph, Advanced Topics in the Arithmetic of Elliptic Curves,

GTM 151, Springer 1994.

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)

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
True


REFERENCES :

• [Sil2] Silverman Joseph, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, Springer 1994.
E2(prec=20)

Returns 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)

Returns 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 [MTT]. See [Col] for a detailed discussion.

INPUT:

• prec - the $$p$$-adic precision, default is 20.

REFERENCES:

• [MTT] B. Mazur, J. Tate, and J. Teitelbaum, On $$p$$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones mathematicae 84, (1986), 1-48.
• [Col] Pierre Colmez, Invariant $$\mathcal{L}$$ et derivees de valeurs propores de Frobenius, preprint, 2004.

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)

Returns 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 +
Field with capped relative precision 5

is_split()

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

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(5^20))
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))

original_curve()

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 $$p$$-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: 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 $$p$$-adic regulator on the extended Mordell-Weil group as in [MTT] (with the correction of [Wer] and sign convention in [SW].) 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.

REFERENCES:

• [MTT] B. Mazur, J. Tate, and J. Teitelbaum, On $$p$$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones mathematicae 84, (1986), 1-48.
• [Wer] Annette Werner, Local heights on abelian varieties and rigid analytic unifomization, Doc. Math. 3 (1998), 301-319.
• [SW] William Stein and Christian Wuthrich, Computations About Tate-Shafarevich Groups using Iwasawa theory, preprint 2009.

EXAMPLES:

sage: eq = EllipticCurve('130a1').tate_curve(5)
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)

parameter(prec=20)

Returns 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=20)

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 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(5^20))


Here is how one gets a 4-torsion point on $$E$$ over $$\QQ_5$$:

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)

parametrisation_onto_tate_curve(u, prec=20)

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 20.

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))

prime()

Returns 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


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Formal groups of elliptic curves