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# 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)

  Bases: `sage.structure.sage_object.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

  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 +
        (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`()

    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

  `padic_height`(prec=20)

    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: 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)

  `padic_regulator`(prec=20)

    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)
        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)

  `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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