Tate’s parametrisation of
-adic curves with multiplicative reduction¶
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 :
[Sil2] | (1, 2) 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
-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: [Sil2]
-
E2
(prec=20)¶ Return the value of the
-adic Eisenstein series of weight 2 evaluated on the elliptic curve having split multiplicative reduction.
INPUT:
prec
- the-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
-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:
prec
- the-adic precision, default is 20.
REFERENCES:
[Col] Pierre Colmez, Invariant et derivees de valeurs propres 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)
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curve
(prec=20)¶ Return 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:
prec
- the-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
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lift
(P, prec=20)¶ 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:
P
- a point on the elliptic curve.prec
- the-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))
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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
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padic_height
(prec=20)¶ Return the canonical
-adic height function on the original curve.
INPUT:
prec
- the-adic precision, default is 20.
OUTPUT:
- A function that can be evaluated on rational points of
.
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)
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padic_regulator
(prec=20)¶ Compute 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:
prec
- the-adic precision, default is 20.
REFERENCES:
[Wer] Annette Werner, Local heights on abelian varieties and rigid analytic unifomization, Doc. Math. 3 (1998), 301-319. 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)
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parameter
(prec=20)¶ Return the Tate parameter
such that the curve is isomorphic over the algebraic closure of
to the curve
.
INPUT:
prec
- the-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)
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parametrisation_onto_original_curve
(u, prec=20)¶ Given an element
in
, this computes its image on the original curve under the
-adic uniformisation of
.
INPUT:
u
- a non-zero-adic number.
prec
- the-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
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)
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parametrisation_onto_tate_curve
(u, prec=20)¶ Given an element
in
, this computes its image on the Tate curve under the
-adic uniformisation of
.
INPUT:
u
- a non-zero-adic number.
prec
- the-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))
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prime
()¶ Return 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
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