To an elliptic curve over the rational numbers and a prime
, one
can associate a
-adic L-function; at least if
does not have additive
reduction at
. This function is defined by interpolation of L-values of
at twists. Through the main conjecture of Iwasawa theory it should also be
equal to a characteristic series of a certain Selmer group.
If is ordinary, then it is an element of the Iwasawa algebra
, where
is the group
of
-st roots of unity in
, and
where
is a generator of
. (There is a slightly different
description for
.)
One can decompose this algebra as the direct product of the subalgebras
corresponding to the characters of , which are simply the powers
(
) of the Teichmueller character
. Projecting the L-function into these components gives
power series in
, each with coefficients in
.
If is supersingular, the series will have coefficients in a quadratic
extension of
, and the coefficients will be unbounded. In this case we
have only implemented the series for
. We have also implemented the
-adic L-series as formulated by Perrin-Riou [BP], which has coefficients in
the Dieudonne module
of
. There is a different
description by Pollack [Po] which is not available here.
According to the -adic version of the Birch and Swinnerton-Dyer conjecture
[MTT], the order of vanishing of the
-function at the trivial character
(i.e. of the series for
at
) is just the rank of
, or
this rank plus one if the reduction at
is split multiplicative.
See [SW] for more details.
REFERENCES:
AUTHORS:
Bases: sage.structure.sage_object.SageObject
The -adic L-series of an elliptic curve.
EXAMPLES: An ordinary example:
sage: e = EllipticCurve('389a')
sage: L = e.padic_lseries(5)
sage: L.series(0)
Traceback (most recent call last):
...
ValueError: n (=0) must be a positive integer
sage: L.series(1)
O(T^1)
sage: L.series(2)
O(5^4) + O(5)*T + (4 + O(5))*T^2 + (2 + O(5))*T^3 + (3 + O(5))*T^4 + O(T^5)
sage: L.series(3, prec=10)
O(5^5) + O(5^2)*T + (4 + 4*5 + O(5^2))*T^2 + (2 + 4*5 + O(5^2))*T^3 + (3 + O(5^2))*T^4 + (1 + O(5))*T^5 + O(5)*T^6 + (4 + O(5))*T^7 + (2 + O(5))*T^8 + O(5)*T^9 + O(T^10)
sage: L.series(2,quadratic_twist=-3)
2 + 4*5 + 4*5^2 + O(5^4) + O(5)*T + (1 + O(5))*T^2 + (4 + O(5))*T^3 + O(5)*T^4 + O(T^5)
A prime p such that E[p] is reducible:
sage: L = EllipticCurve('11a').padic_lseries(5)
sage: L.series(1)
5 + O(5^2) + O(T)
sage: L.series(2)
5 + 4*5^2 + O(5^3) + O(5^0)*T + O(5^0)*T^2 + O(5^0)*T^3 + O(5^0)*T^4 + O(T^5)
sage: L.series(3)
5 + 4*5^2 + 4*5^3 + O(5^4) + O(5)*T + O(5)*T^2 + O(5)*T^3 + O(5)*T^4 + O(T^5)
An example showing the calculation of nontrivial Teichmueller twists:
sage: E=EllipticCurve('11a1')
sage: lp=E.padic_lseries(7)
sage: lp.series(4,eta=1)
6 + 2*7^3 + 5*7^4 + O(7^6) + (4*7 + 2*7^2 + O(7^3))*T + (2 + 3*7^2 + O(7^3))*T^2 + (1 + 2*7 + 2*7^2 + O(7^3))*T^3 + (1 + 3*7^2 + O(7^3))*T^4 + O(T^5)
sage: lp.series(4,eta=2)
5 + 6*7 + 4*7^2 + 2*7^3 + 3*7^4 + 2*7^5 + O(7^6) + (6 + 4*7 + 7^2 + O(7^3))*T + (3 + 2*7^2 + O(7^3))*T^2 + (1 + 4*7 + 7^2 + O(7^3))*T^3 + (6 + 6*7 + 6*7^2 + O(7^3))*T^4 + O(T^5)
sage: lp.series(4,eta=3)
O(7^6) + (3 + 2*7 + 5*7^2 + O(7^3))*T + (5 + 4*7 + 5*7^2 + O(7^3))*T^2 + (3*7 + 7^2 + O(7^3))*T^3 + (2*7 + 7^2 + O(7^3))*T^4 + O(T^5)
(Note that the last series vanishes at , which is consistent with
sage: E.quadratic_twist(-7).rank()
1
This proves that has rank 1 over
.)
the load-dumps test:
sage: lp = EllipticCurve('11a').padic_lseries(5)
sage: lp == loads(dumps(lp))
True
Return a -adic root
of the polynomial
with
. In the ordinary case this is
just the unit root.
INPUT:
- prec - positive integer, the -adic precision of the root.
EXAMPLES: Consider the elliptic curve 37a:
sage: E = EllipticCurve('37a')
An ordinary prime:
sage: L = E.padic_lseries(5)
sage: alpha = L.alpha(10); alpha
3 + 2*5 + 4*5^2 + 2*5^3 + 5^4 + 4*5^5 + 2*5^7 + 5^8 + 5^9 + O(5^10)
sage: alpha^2 - E.ap(5)*alpha + 5
O(5^10)
A supersingular prime:
sage: L = E.padic_lseries(3)
sage: alpha = L.alpha(10); alpha
(1 + O(3^10))*alpha
sage: alpha^2 - E.ap(3)*alpha + 3
(O(3^10))*alpha^2 + (O(3^11))*alpha + (O(3^11))
A reducible prime:
sage: L = EllipticCurve('11a').padic_lseries(5)
sage: L.alpha(5)
1 + 4*5 + 3*5^2 + 2*5^3 + 4*5^4 + O(5^5)
Return the elliptic curve to which this -adic L-series is associated.
EXAMPLES:
sage: L = EllipticCurve('11a').padic_lseries(5)
sage: L.elliptic_curve()
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
Return the measure on defined by
where is the modular symbol. This is used to define
this
-adic L-function (at least when the reduction is good).
The optional argument sign allows the minus symbol to
be substituted for the plus symbol.
The optional argument quadratic_twist replaces by the twist in
the above formula, but the twisted modular symbol is computed using a
sum over modular symbols of
rather then finding the modular symbols
for the twist. Quadratic twists are only implemented if the sign is
.
Note that the normalisation is not correct at this stage: use _quotient_of periods and _quotient_of periods_to_twist to correct.
Note also that this function does not check if the condition
on the quadratic_twist=D is satisfied. So the result will only
be correct if for each prime dividing
, we have
, where
is the conductor of the curve.
INPUT:
EXAMPLES:
sage: E = EllipticCurve('37a')
sage: L = E.padic_lseries(5)
sage: L.measure(1,2, prec=9)
2 + 3*5 + 4*5^3 + 2*5^4 + 3*5^5 + 3*5^6 + 4*5^7 + 4*5^8 + O(5^9)
sage: L.measure(1,2, quadratic_twist=8,prec=15)
O(5^15)
sage: L.measure(1,2, quadratic_twist=-4,prec=15)
4 + 4*5 + 4*5^2 + 3*5^3 + 2*5^4 + 5^5 + 3*5^6 + 5^8 + 2*5^9 + 3*5^12 + 2*5^13 + 4*5^14 + O(5^15)
sage: E = EllipticCurve('11a1')
sage: a = E.quadratic_twist(-3).padic_lseries(5).measure(1,2,prec=15)
sage: b = E.padic_lseries(5).measure(1,2, quadratic_twist=-3,prec=15)
sage: a == b/E.padic_lseries(5)._quotient_of_periods_to_twist(-3)
True
Return the modular symbol evaluated at .
This is used to compute this
-adic
L-series.
Note that the normalisation is not correct at this stage: use _quotient_of periods_to_twist to correct.
Note also that this function does not check if the condition
on the quadratic_twist=D is satisfied. So the result will only
be correct if for each prime dividing
, we have
, where
is the conductor of the curve.
INPUT:
EXAMPLES:
sage: E = EllipticCurve('11a1')
sage: lp = E.padic_lseries(5)
sage: [lp.modular_symbol(r) for r in [0,1/5,oo,1/11]]
[1/5, 6/5, 0, 0]
sage: [lp.modular_symbol(r,sign=-1) for r in [0,1/3,oo,1/7]]
[0, 1, 0, -1]
sage: [lp.modular_symbol(r,quadratic_twist=-20) for r in [0,1/5,oo,1/11]]
[2, 2, 0, 1]
sage: lpt = E.quadratic_twist(-3).padic_lseries(5)
sage: et = E.padic_lseries(5)._quotient_of_periods_to_twist(-3)
sage: lpt.modular_symbol(0) == lp.modular_symbol(0,quadratic_twist=-3)/et
True
Return the order of vanishing of this -adic L-series.
The output of this function is provably correct, due to a theorem of Kato [Ka].
NOTE: currently must be a prime of good ordinary reduction.
REFERENCES:
EXAMPLES:
sage: L = EllipticCurve('11a').padic_lseries(3)
sage: L.order_of_vanishing()
0
sage: L = EllipticCurve('11a').padic_lseries(5)
sage: L.order_of_vanishing()
0
sage: L = EllipticCurve('37a').padic_lseries(5)
sage: L.order_of_vanishing()
1
sage: L = EllipticCurve('43a').padic_lseries(3)
sage: L.order_of_vanishing()
1
sage: L = EllipticCurve('37b').padic_lseries(3)
sage: L.order_of_vanishing()
0
sage: L = EllipticCurve('389a').padic_lseries(3)
sage: L.order_of_vanishing()
2
sage: L = EllipticCurve('389a').padic_lseries(5)
sage: L.order_of_vanishing()
2
sage: L = EllipticCurve('5077a').padic_lseries(5, use_eclib=True)
sage: L.order_of_vanishing()
3
Returns the prime as in ‘p-adic L-function’.
EXAMPLES:
sage: L = EllipticCurve('11a').padic_lseries(5)
sage: L.prime()
5
Return Teichmuller lifts to the given precision.
INPUT:
OUTPUT:
EXAMPLES:
sage: L = EllipticCurve('11a').padic_lseries(7)
sage: L.teichmuller(1)
[0, 1, 2, 3, 4, 5, 6]
sage: L.teichmuller(2)
[0, 1, 30, 31, 18, 19, 48]
Bases: sage.schemes.elliptic_curves.padic_lseries.pAdicLseries
INPUT:
EXAMPLES:
sage: E = EllipticCurve('11a1')
sage: Lp = E.padic_lseries(3)
sage: Lp.series(2,prec=3)
2 + 3 + 3^2 + 2*3^3 + O(3^4) + (1 + O(3))*T + (1 + O(3))*T^2 + O(T^3)
Return True if the elliptic curve that this L-function is attached to is ordinary.
EXAMPLES:
sage: L = EllipticCurve('11a').padic_lseries(5)
sage: L.is_ordinary()
True
Return True if the elliptic curve that this L function is attached to is supersingular.
EXAMPLES:
sage: L = EllipticCurve('11a').padic_lseries(5)
sage: L.is_supersingular()
False
Returns the -th approximation to the
-adic L-series, in the
component corresponding to the
-th power of the Teichmueller
character, as a power series in
(corresponding to
with
as a generator of
). Each coefficient is a
-adic number whose precision is provably correct.
Here the normalization of the -adic L-series is chosen
such that
where
is the unit root of the characteristic
polynomial of Frobenius on
and
is the
Neron period of
.
INPUT:
ALIAS: power_series is identical to series.
EXAMPLES:
We compute some -adic L-functions associated to the elliptic
curve 11a:
sage: E = EllipticCurve('11a')
sage: p = 3
sage: E.is_ordinary(p)
True
sage: L = E.padic_lseries(p)
sage: L.series(3)
2 + 3 + 3^2 + 2*3^3 + O(3^5) + (1 + 3 + O(3^2))*T + (1 + 2*3 + O(3^2))*T^2 + O(3)*T^3 + O(3)*T^4 + O(T^5)
Another example at a prime of bad reduction, where the
-adic L-function has an extra 0 (compared to the non
-adic L-function):
sage: E = EllipticCurve('11a')
sage: p = 11
sage: E.is_ordinary(p)
True
sage: L = E.padic_lseries(p)
sage: L.series(2)
O(11^4) + (10 + O(11))*T + (6 + O(11))*T^2 + (2 + O(11))*T^3 + (5 + O(11))*T^4 + O(T^5)
We compute a -adic L-function that vanishes to order 2:
sage: E = EllipticCurve('389a')
sage: p = 3
sage: E.is_ordinary(p)
True
sage: L = E.padic_lseries(p)
sage: L.series(1)
O(T^1)
sage: L.series(2)
O(3^4) + O(3)*T + (2 + O(3))*T^2 + O(T^3)
sage: L.series(3)
O(3^5) + O(3^2)*T + (2 + 2*3 + O(3^2))*T^2 + (2 + O(3))*T^3 + (1 + O(3))*T^4 + O(T^5)
Checks if the precision can be changed (:trac: ):
sage: L.series(3,prec=4)
O(3^5) + O(3^2)*T + (2 + 2*3 + O(3^2))*T^2 + (2 + O(3))*T^3 + O(T^4)
sage: L.series(3,prec=6)
O(3^5) + O(3^2)*T + (2 + 2*3 + O(3^2))*T^2 + (2 + O(3))*T^3 + (1 + O(3))*T^4 + (1 + O(3))*T^5 + O(T^6)
Rather than computing the -adic L-function for the curve ‘15523a1’, one can
compute it as a quadratic_twist:
sage: E = EllipticCurve('43a1')
sage: lp = E.padic_lseries(3)
sage: lp.series(2,quadratic_twist=-19)
2 + 2*3 + 2*3^2 + O(3^4) + (1 + O(3))*T + (1 + O(3))*T^2 + O(T^3)
sage: E.quadratic_twist(-19).label() # optional -- database_cremona_ellcurve
'15523a1'
This proves that the rank of ‘15523a1’ is zero, even if mwrank can not determine this.
We calculate the -series in the nontrivial Teichmueller components:
sage: L = EllipticCurve('110a1').padic_lseries(5)
sage: for j in [0..3]: print L.series(4, eta=j)
O(5^6) + (2 + 2*5 + 2*5^2 + O(5^3))*T + (5 + 5^2 + O(5^3))*T^2 + (4 + 4*5 + 2*5^2 + O(5^3))*T^3 + (1 + 5 + 3*5^2 + O(5^3))*T^4 + O(T^5)
3 + 2*5 + 2*5^3 + 3*5^4 + O(5^6) + (2 + 5 + 4*5^2 + O(5^3))*T + (1 + 4*5 + 2*5^2 + O(5^3))*T^2 + (1 + 5 + 5^2 + O(5^3))*T^3 + (2 + 4*5 + 4*5^2 + O(5^3))*T^4 + O(T^5)
2 + O(5^6) + (1 + 5 + O(5^3))*T + (2 + 4*5 + 3*5^2 + O(5^3))*T^2 + (4 + 5 + 2*5^2 + O(5^3))*T^3 + (4 + O(5^3))*T^4 + O(T^5)
1 + 3*5 + 4*5^2 + 2*5^3 + 5^4 + 4*5^5 + O(5^6) + (2 + 4*5 + 3*5^2 + O(5^3))*T + (2 + 3*5 + 5^2 + O(5^3))*T^2 + (1 + O(5^3))*T^3 + (2*5 + 2*5^2 + O(5^3))*T^4 + O(T^5)
Returns the -th approximation to the
-adic L-series, in the
component corresponding to the
-th power of the Teichmueller
character, as a power series in
(corresponding to
with
as a generator of
). Each coefficient is a
-adic number whose precision is provably correct.
Here the normalization of the -adic L-series is chosen
such that
where
is the unit root of the characteristic
polynomial of Frobenius on
and
is the
Neron period of
.
INPUT:
ALIAS: power_series is identical to series.
EXAMPLES:
We compute some -adic L-functions associated to the elliptic
curve 11a:
sage: E = EllipticCurve('11a')
sage: p = 3
sage: E.is_ordinary(p)
True
sage: L = E.padic_lseries(p)
sage: L.series(3)
2 + 3 + 3^2 + 2*3^3 + O(3^5) + (1 + 3 + O(3^2))*T + (1 + 2*3 + O(3^2))*T^2 + O(3)*T^3 + O(3)*T^4 + O(T^5)
Another example at a prime of bad reduction, where the
-adic L-function has an extra 0 (compared to the non
-adic L-function):
sage: E = EllipticCurve('11a')
sage: p = 11
sage: E.is_ordinary(p)
True
sage: L = E.padic_lseries(p)
sage: L.series(2)
O(11^4) + (10 + O(11))*T + (6 + O(11))*T^2 + (2 + O(11))*T^3 + (5 + O(11))*T^4 + O(T^5)
We compute a -adic L-function that vanishes to order 2:
sage: E = EllipticCurve('389a')
sage: p = 3
sage: E.is_ordinary(p)
True
sage: L = E.padic_lseries(p)
sage: L.series(1)
O(T^1)
sage: L.series(2)
O(3^4) + O(3)*T + (2 + O(3))*T^2 + O(T^3)
sage: L.series(3)
O(3^5) + O(3^2)*T + (2 + 2*3 + O(3^2))*T^2 + (2 + O(3))*T^3 + (1 + O(3))*T^4 + O(T^5)
Checks if the precision can be changed (:trac: ):
sage: L.series(3,prec=4)
O(3^5) + O(3^2)*T + (2 + 2*3 + O(3^2))*T^2 + (2 + O(3))*T^3 + O(T^4)
sage: L.series(3,prec=6)
O(3^5) + O(3^2)*T + (2 + 2*3 + O(3^2))*T^2 + (2 + O(3))*T^3 + (1 + O(3))*T^4 + (1 + O(3))*T^5 + O(T^6)
Rather than computing the -adic L-function for the curve ‘15523a1’, one can
compute it as a quadratic_twist:
sage: E = EllipticCurve('43a1')
sage: lp = E.padic_lseries(3)
sage: lp.series(2,quadratic_twist=-19)
2 + 2*3 + 2*3^2 + O(3^4) + (1 + O(3))*T + (1 + O(3))*T^2 + O(T^3)
sage: E.quadratic_twist(-19).label() # optional -- database_cremona_ellcurve
'15523a1'
This proves that the rank of ‘15523a1’ is zero, even if mwrank can not determine this.
We calculate the -series in the nontrivial Teichmueller components:
sage: L = EllipticCurve('110a1').padic_lseries(5)
sage: for j in [0..3]: print L.series(4, eta=j)
O(5^6) + (2 + 2*5 + 2*5^2 + O(5^3))*T + (5 + 5^2 + O(5^3))*T^2 + (4 + 4*5 + 2*5^2 + O(5^3))*T^3 + (1 + 5 + 3*5^2 + O(5^3))*T^4 + O(T^5)
3 + 2*5 + 2*5^3 + 3*5^4 + O(5^6) + (2 + 5 + 4*5^2 + O(5^3))*T + (1 + 4*5 + 2*5^2 + O(5^3))*T^2 + (1 + 5 + 5^2 + O(5^3))*T^3 + (2 + 4*5 + 4*5^2 + O(5^3))*T^4 + O(T^5)
2 + O(5^6) + (1 + 5 + O(5^3))*T + (2 + 4*5 + 3*5^2 + O(5^3))*T^2 + (4 + 5 + 2*5^2 + O(5^3))*T^3 + (4 + O(5^3))*T^4 + O(T^5)
1 + 3*5 + 4*5^2 + 2*5^3 + 5^4 + 4*5^5 + O(5^6) + (2 + 4*5 + 3*5^2 + O(5^3))*T + (2 + 3*5 + 5^2 + O(5^3))*T^2 + (1 + O(5^3))*T^3 + (2*5 + 2*5^2 + O(5^3))*T^4 + O(T^5)
Bases: sage.schemes.elliptic_curves.padic_lseries.pAdicLseries
INPUT:
EXAMPLES:
sage: E = EllipticCurve('11a1')
sage: Lp = E.padic_lseries(3)
sage: Lp.series(2,prec=3)
2 + 3 + 3^2 + 2*3^3 + O(3^4) + (1 + O(3))*T + (1 + O(3))*T^2 + O(T^3)
Returns the canonical -adic height with values in the Dieudonne module
.
It is defined to be
where is made out of the sigma function of Bernardi and
is
.
The answer v is given as v[1]*omega + v[2]*eta.
The coordinates of v are dependent of the
Weierstrass equation.
EXAMPLES:
sage: E = EllipticCurve('53a')
sage: L = E.padic_lseries(5)
sage: h = L.Dp_valued_height(7)
sage: h(E.gens()[0])
(3*5 + 5^2 + 2*5^3 + 3*5^4 + 4*5^5 + 5^6 + 5^7 + O(5^8), 5^2 + 4*5^4 + 2*5^7 + 3*5^8 + O(5^9))
Returns the canonical -adic regulator with values in the Dieudonne module
as defined by Perrin-Riou using the
-adic height with values in
.
The result is written in the basis
,
, and hence the
coordinates of the result are independent of the chosen Weierstrass equation.
NOTE: The definition here is corrected with respect to Perrin-Riou’s article [PR]. See [SW].
REFERENCES:
EXAMPLES:
sage: E = EllipticCurve('43a')
sage: L = E.padic_lseries(7)
sage: L.Dp_valued_regulator(7)
(5*7 + 6*7^2 + 4*7^3 + 4*7^4 + 7^5 + 4*7^7 + O(7^8), 4*7^2 + 2*7^3 + 3*7^4 + 7^5 + 6*7^6 + 4*7^7 + O(7^8))
Returns a vector of two components which are p-adic power series. The answer v is such that
v[1]
v[2]
as an element of the Dieudonne module where
is the invariant differential and
is the Frobenius on
.
According to the
-adic Birch and Swinnerton-Dyer
conjecture [BP] this function has a zero of order
rank of
and it’s leading term is contains the order of
the Tate-Shafarevich group, the Tamagawa numbers, the order of the
torsion subgroup and the
-valued
-adic regulator.
INPUT:
REFERENCE:
EXAMPLES:
sage: E = EllipticCurve('14a')
sage: L = E.padic_lseries(5)
sage: L.Dp_valued_series(4) # long time (9s on sage.math, 2011)
(1 + 4*5 + 4*5^3 + O(5^4) + (4 + O(5))*T + (1 + O(5))*T^2 + (4 + O(5))*T^3 + (2 + O(5))*T^4 + O(T^5), O(5^4) + O(5)*T + O(5)*T^2 + O(5)*T^3 + (2 + O(5))*T^4 + O(T^5))
Return the -adic sigma function of Bernardi in terms of
.
This is the same as padic_sigma with E2 = 0.
EXAMPLES:
sage: E = EllipticCurve('14a')
sage: L = E.padic_lseries(5)
sage: L.bernardi_sigma_function(prec=5) # Todo: some sort of consistency check!?
z + 1/24*z^3 + 29/384*z^5 - 8399/322560*z^7 - 291743/92897280*z^9 + O(z^10)
This returns a geometric Frobenius on the Diedonne module
with respect to the basis
, the invariant differential, and
.
It satisfies
.
INPUT:
EXAMPLES:
sage: E = EllipticCurve('14a')
sage: L = E.padic_lseries(5)
sage: phi = L.frobenius(5)
sage: phi
[ 2 + 5^2 + 5^4 + O(5^5) 3*5^-1 + 3 + 5 + 4*5^2 + 5^3 + O(5^4)]
[ 3 + 3*5^2 + 4*5^3 + 3*5^4 + O(5^5) 3 + 4*5 + 3*5^2 + 4*5^3 + 3*5^4 + O(5^5)]
sage: -phi^2
[5^-1 + O(5^4) O(5^4)]
[ O(5^5) 5^-1 + O(5^4)]
Return True if the elliptic curve that this L-function is attached to is ordinary.
EXAMPLES:
sage: L = EllipticCurve('11a').padic_lseries(19)
sage: L.is_ordinary()
False
Return True if the elliptic curve that this L function is attached to is supersingular.
EXAMPLES:
sage: L = EllipticCurve('11a').padic_lseries(19)
sage: L.is_supersingular()
True
Return the -th approximation to the
-adic L-series as a
power series in
(corresponding to
with
as a generator of
). Each
coefficient is an element of a quadratic extension of the
-adic
number whose precision is probably correct.
Here the normalization of the -adic L-series is chosen
such that
where
is the unit root of the characteristic
polynomial of Frobenius on
and
is the
Neron period of
.
INPUT:
ALIAS: power_series is identical to series.
EXAMPLES: A superingular example, where we must compute to higher precision to see anything:
sage: e = EllipticCurve('37a')
sage: L = e.padic_lseries(3); L
3-adic L-series of Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: L.series(2)
O(T^3)
sage: L.series(4) # takes a long time (several seconds)
(O(3))*alpha + (O(3^2)) + ((O(3^-1))*alpha + (2*3^-1 + O(3^0)))*T + ((O(3^-1))*alpha + (2*3^-1 + O(3^0)))*T^2 + O(T^5)
sage: L.alpha(2).parent()
Univariate Quotient Polynomial Ring in alpha over 3-adic Field with capped
relative precision 2 with modulus (1 + O(3^2))*x^2 + (3 + O(3^3))*x + (3 + O(3^3))
An example where we only compute the leading term (:trac: ):
sage: E = EllipticCurve("17a1")
sage: L = E.padic_lseries(3)
sage: L.series(4,prec=1)
(O(3^18))*alpha^2 + (2*3^-1 + 1 + 3 + 3^2 + 3^3 + ... + 3^18 + O(3^19))*alpha + (2*3^-1 + 1 + 3 + 3^2 + 3^3 + 3^4 + ... + 3^18 + O(3^19)) + O(T)
Return the -th approximation to the
-adic L-series as a
power series in
(corresponding to
with
as a generator of
). Each
coefficient is an element of a quadratic extension of the
-adic
number whose precision is probably correct.
Here the normalization of the -adic L-series is chosen
such that
where
is the unit root of the characteristic
polynomial of Frobenius on
and
is the
Neron period of
.
INPUT:
ALIAS: power_series is identical to series.
EXAMPLES: A superingular example, where we must compute to higher precision to see anything:
sage: e = EllipticCurve('37a')
sage: L = e.padic_lseries(3); L
3-adic L-series of Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: L.series(2)
O(T^3)
sage: L.series(4) # takes a long time (several seconds)
(O(3))*alpha + (O(3^2)) + ((O(3^-1))*alpha + (2*3^-1 + O(3^0)))*T + ((O(3^-1))*alpha + (2*3^-1 + O(3^0)))*T^2 + O(T^5)
sage: L.alpha(2).parent()
Univariate Quotient Polynomial Ring in alpha over 3-adic Field with capped
relative precision 2 with modulus (1 + O(3^2))*x^2 + (3 + O(3^3))*x + (3 + O(3^3))
An example where we only compute the leading term (:trac: ):
sage: E = EllipticCurve("17a1")
sage: L = E.padic_lseries(3)
sage: L.series(4,prec=1)
(O(3^18))*alpha^2 + (2*3^-1 + 1 + 3 + 3^2 + 3^3 + ... + 3^18 + O(3^19))*alpha + (2*3^-1 + 1 + 3 + 3^2 + 3^3 + 3^4 + ... + 3^18 + O(3^19)) + O(T)