Tate’s parametrisation of $p$-adic curves with multiplicative reduction

Let $E$ be an elliptic curve defined over the $p$-adic numbers ${\mathbf{Q}}_p$. 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 ${\mathbf{Z}}_p$ of valuation $n$ such that the points of $E$ defined over the algebraic closure ${\overline{\mathbf{Q}}}_p$ are in bijection with ${\overline{\mathbf{Q}}}_p^{\times }/q^{\mathbf{Z}}$. More precisely there exists the series $s_4(q)$ and $s_6(q)$ such that the $y^2+xy=x^3+s_4(q)x+s_6(q)$ curve is isomorphic to $E$ over ${\overline{\mathbf{Q}}}_p$ (or over ${\mathbf{Q}}_p$ if the reduction is split multiplicative). There is a $p$-adic analytic map from ${\overline{\mathbf{Q}}}_p^{\times }$ to this curve with kernel $q^{\mathbf{Z}}$. Points of good reduction correspond to points of valuation $0$ in ${\overline{\mathbf{Q}}}_p^{\times }$.

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: 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 ${\mathbf{Q}}_p$.

EXAMPLES:

Sage

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

Sage Live

e = EllipticCurve('130a1')
eq = e.tate_curve(5); eq
eq == loads(dumps(eq))

Python

>>> 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)

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

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)

Sage Live

eq = EllipticCurve('130a1').tate_curve(5)
eq.E2(prec=10)
T = EllipticCurve('14').tate_curve(7)
T.E2(30)

Python

>>> 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)

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

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)

Sage Live

eq = EllipticCurve('130a1').tate_curve(5)
eq.L_invariant(prec=10)

Python

>>> 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)

Return the $p$-adic elliptic curve of the form $y^2+xy=x^3+s_{4}x+s_6$.

This curve with split multiplicative reduction is isomorphic to the given curve over the algebraic closure of ${\mathbf{Q}}_p$.

INPUT:

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

EXAMPLES:

Sage

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

Sage Live

eq = EllipticCurve('130a1').tate_curve(5)
eq.curve(prec=5)

Python

>>> 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()

Return True if the given elliptic curve has split multiplicative reduction.

EXAMPLES:

Sage

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

sage: eq = EllipticCurve('37a1').tate_curve(37)
sage: eq.is_split()
False

Sage Live

eq = EllipticCurve('130a1').tate_curve(5)
eq.is_split()
eq = EllipticCurve('37a1').tate_curve(37)
eq.is_split()

Python

>>> 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)

Given a point $P$ in the formal group of the elliptic curve $E$ with split multiplicative reduction, this produces an element $u$ in ${\mathbf{Q}}_p^{\times }$ mapped to the point $P$ by the Tate parametrisation. The algorithm return the unique such element in $1+p{\mathbf{Z}}_p$.

INPUT:

• P – a point on the elliptic curve.

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

EXAMPLES:

Sage

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)

Sage Live

e = EllipticCurve('130a1')
eq = e.tate_curve(5)
P = e([-6,10])
l = eq.lift(12*P, prec=10); l

Python

>>> 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

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))

Sage Live

eq.parametrisation_onto_original_curve(l)
e5 = e.change_ring(Qp(5,9))
e5(12*P)

Python

>>> 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()

Return the elliptic curve the Tate curve was constructed from.

EXAMPLES:

Sage

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 Live

eq = EllipticCurve('130a1').tate_curve(5)
eq.original_curve()

Python

>>> 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)

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

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)

Sage Live

e = EllipticCurve('130a1')
eq = e.tate_curve(5)
h = eq.padic_height(prec=10)
P = e.gens()[0]
h(P)

Python

>>> 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

sage: h(3*P)-3^2*h(P)
O(5^9)

Sage Live

h(3*P)-3^2*h(P)

Python

>>> from sage.all import *
>>> h(Integer(3)*P)-Integer(3)**Integer(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

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)

Sage Live

eq = EllipticCurve('130a1').tate_curve(5)
eq.padic_regulator()

Python

>>> 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)

Return the Tate parameter $q$ such that the curve is isomorphic over the algebraic closure of ${\mathbf{Q}}_p$ to the curve ${\mathbf{Q}}_p^{\times }/q^{\mathbf{Z}}$.

INPUT:

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

EXAMPLES:

Sage

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)

Sage Live

eq = EllipticCurve('130a1').tate_curve(5)
eq.parameter(prec=5)

Python

>>> 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)

Given an element $u$ in ${\mathbf{Q}}_p^{\times }$, 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 of u otherwise 20.

EXAMPLES:

Sage

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

Sage Live

eq = EllipticCurve('130a1').tate_curve(5)
eq.parametrisation_onto_original_curve(1+5+5^2+O(5^10))
eq.parametrisation_onto_original_curve(1+5+5^2+O(5^10), prec=20)

Python

>>> 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 ${\mathbf{Q}}_5$:

Sage

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)

Sage Live

R = Qp(5,30)
i = R(-1).sqrt()
T = eq.parametrisation_onto_original_curve(i, prec=30); T
4*T

Python

>>> 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)

Given an element $u$ in ${\mathbf{Q}}_p^{\times }$, 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 of u otherwise 20.

EXAMPLES:

Sage

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

Sage Live

eq = EllipticCurve('130a1').tate_curve(5)
eq.parametrisation_onto_tate_curve(1+5+5^2+O(5^10), prec=10)
eq.parametrisation_onto_tate_curve(1+5+5^2+O(5^10))
eq.parametrisation_onto_tate_curve(1+5+5^2+O(5^10), prec=20)

Python

>>> 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()

Return the residual characteristic $p$.

EXAMPLES:

Sage

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

Sage Live

eq = EllipticCurve('130a1').tate_curve(5)
eq.original_curve()
eq.prime()

Python

>>> 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

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