\(p\)-adic \(L\)-functions of elliptic curves#
To an elliptic curve \(E\) over the rational numbers and a prime \(p\), one can associate a \(p\)-adic L-function; at least if \(E\) does not have additive reduction at \(p\). This function is defined by interpolation of L-values of \(E\) at twists. Through the main conjecture of Iwasawa theory it should also be equal to a characteristic series of a certain Selmer group.
If \(E\) is ordinary, then it is an element of the Iwasawa algebra \(\Lambda(\ZZ_p^\times) = \ZZ_p[\Delta][\![T]\!]\), where \(\Delta\) is the group of \((p-1)\)-st roots of unity in \(\ZZ_p^\times\), and \(T = [\gamma] - 1\) where \(\gamma = 1 + p\) is a generator of \(1 + p\ZZ_p\). (There is a slightly different description for \(p = 2\).)
One can decompose this algebra as the direct product of the subalgebras corresponding to the characters of \(\Delta\), which are simply the powers \(\tau^\eta\) (\(0 \le \eta \le p-2\)) of the Teichmueller character \(\tau: \Delta \to \ZZ_p^\times\). Projecting the L-function into these components gives \(p-1\) power series in \(T\), each with coefficients in \(\ZZ_p\).
If \(E\) is supersingular, the series will have coefficients in a quadratic extension of \(\QQ_p\), and the coefficients will be unbounded. In this case we have only implemented the series for \(\eta = 0\). We have also implemented the \(p\)-adic L-series as formulated by Perrin-Riou [BP1993], which has coefficients in the Dieudonné module \(D_pE = H^1_{dR}(E/\QQ_p)\) of \(E\). There is a different description by Pollack [Pol2003] which is not available here.
According to the \(p\)-adic version of the Birch and Swinnerton-Dyer conjecture [MTT1986], the order of vanishing of the \(L\)-function at the trivial character (i.e. of the series for \(\eta = 0\) at \(T = 0\)) is just the rank of \(E(\QQ)\), or this rank plus one if the reduction at \(p\) is split multiplicative.
See [SW2013] for more details.
AUTHORS:
William Stein (2007-01-01): first version
Chris Wuthrich (22/05/2007): changed minor issues and added supersingular things
Chris Wuthrich (11/2008): added quadratic_twists
David Loeffler (01/2011): added nontrivial Teichmueller components
- class sage.schemes.elliptic_curves.padic_lseries.pAdicLseries(E, p, implementation='eclib', normalize='L_ratio')#
Bases:
SageObject
The \(p\)-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) 3 + 7^3 + 6*7^4 + 3*7^5 + O(7^6) + (2*7 + 7^2 + O(7^3))*T + (1 + 5*7^2 + O(7^3))*T^2 + (4 + 4*7 + 4*7^2 + O(7^3))*T^3 + (4 + 3*7 + 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) + (5 + 4*7 + 2*7^2 + O(7^3))*T + (6 + 5*7 + 2*7^2 + O(7^3))*T^2 + (5*7 + O(7^3))*T^3 + (7 + 4*7^2 + O(7^3))*T^4 + O(T^5)
(Note that the last series vanishes at \(T = 0\), which is consistent with
sage: E.quadratic_twist(-7).rank() 1
This proves that \(E\) has rank 1 over \(\QQ(\zeta_7)\).)
- alpha(prec=20)#
Return a \(p\)-adic root \(\alpha\) of the polynomial \(x^2 - a_p x + p\) with \(ord_p(\alpha) < 1\). In the ordinary case this is just the unit root.
INPUT:
prec
– positive integer, the \(p\)-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 alpha + O(alpha^21) sage: alpha^2 - E.ap(3)*alpha + 3 O(alpha^22)
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)
- elliptic_curve()#
Return the elliptic curve to which this \(p\)-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
- measure(a, n, prec, quadratic_twist=1, sign=1)#
Return the measure on \(\ZZ_p^{\times}\) defined by
\(\mu_{E,\alpha}^+ ( a + p^n \ZZ_p ) = \frac{1}{\alpha^n} \left [\frac{a}{p^n}\right]^{+} - \frac{1}{\alpha^{n+1}} \left[\frac{a}{p^{n-1}}\right]^{+}\)
where \([\cdot]^{+}\) is the modular symbol. This is used to define this \(p\)-adic L-function (at least when the reduction is good).
The optional argument
sign
allows the minus symbol \([\cdot]^{-}\) to be substituted for the plus symbol.The optional argument
quadratic_twist
replaces \(E\) by the twist in the above formula, but the twisted modular symbol is computed using a sum over modular symbols of \(E\) rather than finding the modular symbols for the twist. Quadratic twists are only implemented if the sign is \(+1\).Note that the normalization 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 \(\ell\) dividing \(D\), we have \(ord_{\ell}(N)<= ord_{\ell}(D)\), where \(N\) is the conductor of the curve.INPUT:
a
– an integern
– a non-negative integerprec
– an integerquadratic_twist
(default = 1) – a fundamental discriminant of a quadratic field, should be coprime to the conductor of \(E\)sign
(default = 1) – an integer, which should be \(\pm 1\).
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
- modular_symbol(r, sign=1, quadratic_twist=1)#
Return the modular symbol evaluated at \(r\).
This is used to compute this \(p\)-adic L-series.
Note that the normalization 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 \(\ell\) dividing \(D\), we have \(ord_{\ell}(N)<= ord_{\ell}(D)\), where \(N\) is the conductor of the curve.
INPUT:
r
– a cusp given as either a rational number or oosign
– +1 (default) or -1 (only implemented without twists)quadratic_twist
– a fundamental discriminant of a quadratic field or +1 (default)
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, 1/2, 0, -1/2] sage: [lp.modular_symbol(r,quadratic_twist=-20) for r in [0,1/5,oo,1/11]] [1, 1, 0, 1/2] sage: E = EllipticCurve('20a1') sage: Et = E.quadratic_twist(-4) sage: lpt = Et.padic_lseries(5) sage: eta = lpt._quotient_of_periods_to_twist(-4) sage: lpt.modular_symbol(0) == lp.modular_symbol(0,quadratic_twist=-4) / eta True
- order_of_vanishing()#
Return the order of vanishing of this \(p\)-adic L-series.
The output of this function is provably correct, due to a theorem of Kato [Kat2004].
Note
currently \(p\) 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, implementation = 'eclib') sage: L.order_of_vanishing() 3
- prime()#
Return the prime \(p\) as in ‘p-adic L-function’.
EXAMPLES:
sage: L = EllipticCurve('11a').padic_lseries(5) sage: L.prime() 5
- teichmuller(prec)#
Return Teichmuller lifts to the given precision.
INPUT:
prec
– a positive integer.
OUTPUT:
a list of \(p\)-adic numbers, the cached Teichmuller lifts
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]
- class sage.schemes.elliptic_curves.padic_lseries.pAdicLseriesOrdinary(E, p, implementation='eclib', normalize='L_ratio')#
Bases:
pAdicLseries
- is_ordinary()#
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
- is_supersingular()#
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
- power_series(n=2, quadratic_twist=1, prec=5, eta=0)#
Return the \(n\)-th approximation to the \(p\)-adic L-series, in the component corresponding to the \(\eta\)-th power of the Teichmueller character, as a power series in \(T\) (corresponding to \(\gamma-1\) with \(\gamma=1+p\) as a generator of \(1+p\ZZ_p\)). Each coefficient is a \(p\)-adic number whose precision is provably correct.
Here the normalization of the \(p\)-adic L-series is chosen such that \(L_p(E,1) = (1-1/\alpha)^2 L(E,1)/\Omega_E\) where \(\alpha\) is the unit root of the characteristic polynomial of Frobenius on \(T_pE\) and \(\Omega_E\) is the Néron period of \(E\).
INPUT:
n
– (default: 2) a positive integerquadratic_twist
– (default: +1) a fundamental discriminant of a quadratic field, coprime to the conductor of the curveprec
– (default: 5) maximal number of terms of the series to compute; to compute as many as possible just give a very large number forprec
; the result will still be correct.eta
(default: 0) an integer (specifying the power of the Teichmueller character on the group of roots of unity in \(\ZZ_p^\times\))
power_series()
is identical toseries
.EXAMPLES:
We compute some \(p\)-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 \(p\)-adic L-function has an extra 0 (compared to the non \(p\)-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 \(p\)-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 (github issue #5846):
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 \(p\)-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
cannot determine this.We calculate the \(L\)-series in the nontrivial Teichmueller components:
sage: L = EllipticCurve('110a1').padic_lseries(5, implementation="sage") 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) 4 + 3*5 + 2*5^2 + 3*5^3 + 5^4 + O(5^6) + (1 + 3*5 + 4*5^2 + O(5^3))*T + (3 + 4*5 + 3*5^2 + O(5^3))*T^2 + (3 + 3*5^2 + O(5^3))*T^3 + (1 + 2*5 + 2*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) 3 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + O(5^6) + (1 + 2*5 + 4*5^2 + O(5^3))*T + (1 + 4*5 + O(5^3))*T^2 + (3 + 2*5 + 2*5^2 + O(5^3))*T^3 + (5 + 5^2 + O(5^3))*T^4 + O(T^5)
It should now also work with \(p=2\) (github issue #20798):
sage: E = EllipticCurve("53a1") sage: lp = E.padic_lseries(2) sage: lp.series(7) O(2^8) + (1 + 2^2 + 2^3 + O(2^5))*T + (1 + 2^3 + O(2^4))*T^2 + (2^2 + 2^3 + O(2^4))*T^3 + (2 + 2^2 + O(2^3))*T^4 + O(T^5) sage: E = EllipticCurve("109a1") sage: lp = E.padic_lseries(2) sage: lp.series(6) 2^2 + 2^6 + O(2^7) + (2 + O(2^4))*T + O(2^3)*T^2 + (2^2 + O(2^3))*T^3 + (2 + O(2^2))*T^4 + O(T^5)
Check that twists by odd Teichmuller characters are ok (github issue #32258):
sage: E = EllipticCurve("443c1") sage: lp = E.padic_lseries(17, implementation="num") sage: l8 = lp.series(2,eta=8,prec=3) sage: l8.list()[0] - 1/lp.alpha() O(17^4) sage: lp = E.padic_lseries(2, implementation="num") sage: l1 = lp.series(8,eta=1,prec=3) sage: l1.list()[0] - 4/lp.alpha()^2 O(2^9)
- series(n=2, quadratic_twist=1, prec=5, eta=0)#
Return the \(n\)-th approximation to the \(p\)-adic L-series, in the component corresponding to the \(\eta\)-th power of the Teichmueller character, as a power series in \(T\) (corresponding to \(\gamma-1\) with \(\gamma=1+p\) as a generator of \(1+p\ZZ_p\)). Each coefficient is a \(p\)-adic number whose precision is provably correct.
Here the normalization of the \(p\)-adic L-series is chosen such that \(L_p(E,1) = (1-1/\alpha)^2 L(E,1)/\Omega_E\) where \(\alpha\) is the unit root of the characteristic polynomial of Frobenius on \(T_pE\) and \(\Omega_E\) is the Néron period of \(E\).
INPUT:
n
– (default: 2) a positive integerquadratic_twist
– (default: +1) a fundamental discriminant of a quadratic field, coprime to the conductor of the curveprec
– (default: 5) maximal number of terms of the series to compute; to compute as many as possible just give a very large number forprec
; the result will still be correct.eta
(default: 0) an integer (specifying the power of the Teichmueller character on the group of roots of unity in \(\ZZ_p^\times\))
power_series()
is identical toseries
.EXAMPLES:
We compute some \(p\)-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 \(p\)-adic L-function has an extra 0 (compared to the non \(p\)-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 \(p\)-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 (github issue #5846):
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 \(p\)-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
cannot determine this.We calculate the \(L\)-series in the nontrivial Teichmueller components:
sage: L = EllipticCurve('110a1').padic_lseries(5, implementation="sage") 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) 4 + 3*5 + 2*5^2 + 3*5^3 + 5^4 + O(5^6) + (1 + 3*5 + 4*5^2 + O(5^3))*T + (3 + 4*5 + 3*5^2 + O(5^3))*T^2 + (3 + 3*5^2 + O(5^3))*T^3 + (1 + 2*5 + 2*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) 3 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + O(5^6) + (1 + 2*5 + 4*5^2 + O(5^3))*T + (1 + 4*5 + O(5^3))*T^2 + (3 + 2*5 + 2*5^2 + O(5^3))*T^3 + (5 + 5^2 + O(5^3))*T^4 + O(T^5)
It should now also work with \(p=2\) (github issue #20798):
sage: E = EllipticCurve("53a1") sage: lp = E.padic_lseries(2) sage: lp.series(7) O(2^8) + (1 + 2^2 + 2^3 + O(2^5))*T + (1 + 2^3 + O(2^4))*T^2 + (2^2 + 2^3 + O(2^4))*T^3 + (2 + 2^2 + O(2^3))*T^4 + O(T^5) sage: E = EllipticCurve("109a1") sage: lp = E.padic_lseries(2) sage: lp.series(6) 2^2 + 2^6 + O(2^7) + (2 + O(2^4))*T + O(2^3)*T^2 + (2^2 + O(2^3))*T^3 + (2 + O(2^2))*T^4 + O(T^5)
Check that twists by odd Teichmuller characters are ok (github issue #32258):
sage: E = EllipticCurve("443c1") sage: lp = E.padic_lseries(17, implementation="num") sage: l8 = lp.series(2,eta=8,prec=3) sage: l8.list()[0] - 1/lp.alpha() O(17^4) sage: lp = E.padic_lseries(2, implementation="num") sage: l1 = lp.series(8,eta=1,prec=3) sage: l1.list()[0] - 4/lp.alpha()^2 O(2^9)
- class sage.schemes.elliptic_curves.padic_lseries.pAdicLseriesSupersingular(E, p, implementation='eclib', normalize='L_ratio')#
Bases:
pAdicLseries
- Dp_valued_height(prec=20)#
Return the canonical \(p\)-adic height with values in the Dieudonné module \(D_p(E)\).
It is defined to be
\(h_{\eta} \cdot \omega - h_{\omega} \cdot \eta\)
where \(h_{\eta}\) is made out of the sigma function of Bernardi and \(h_{\omega}\) is \(log_E^2\).
The answer
v
is given asv[1]*omega + v[2]*eta
. The coordinates ofv
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))
- Dp_valued_regulator(prec=20, v1=0, v2=0)#
Return the canonical \(p\)-adic regulator with values in the Dieudonné module \(D_p(E)\) as defined by Perrin-Riou using the \(p\)-adic height with values in \(D_p(E)\).
The result is written in the basis \(\omega\), \(\varphi(\omega)\), and hence the coordinates of the result are independent of the chosen Weierstrass equation.
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))
- Dp_valued_series(n=3, quadratic_twist=1, prec=5)#
Return a vector of two components which are p-adic power series.
The answer v is such that
\((1-\varphi)^{-2}\cdot L_p(E,T) =\)
v[1]
\(\cdot \omega +\)v[2]
\(\cdot \varphi(\omega)\)as an element of the Dieudonné module \(D_p(E) = H^1_{dR}(E/\QQ_p)\) where \(\omega\) is the invariant differential and \(\varphi\) is the Frobenius on \(D_p(E)\).
According to the \(p\)-adic Birch and Swinnerton-Dyer conjecture [BP1993] this function has a zero of order rank of \(E(\QQ)\) 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 \(D_p\)-valued \(p\)-adic regulator.
INPUT:
n
– (default: 3) a positive integerprec
– (default: 5) a positive integer
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 + O(5^2) + (4 + O(5))*T + (1 + O(5))*T^2 + (4 + O(5))*T^3 + (2 + O(5))*T^4 + O(T^5), 5^2 + O(5^3) + O(5^2)*T + (4*5 + O(5^2))*T^2 + (2*5 + O(5^2))*T^3 + (2 + 2*5 + O(5^2))*T^4 + O(T^5))
- bernardi_sigma_function(prec=20)#
Return the \(p\)-adic sigma function of Bernardi in terms of \(z = log(t)\).
This is the same as
padic_sigma
withE2 = 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)
- frobenius(prec=20, algorithm='mw')#
Return a geometric Frobenius \(\varphi\) on the Dieudonné module \(D_p(E)\) with respect to the basis \(\omega\), the invariant differential, and \(\eta=x\omega\).
It satisfies \(\varphi^2 - a_p/p\, \varphi + 1/p = 0\).
INPUT:
prec
– (default: 20) a positive integeralgorithm
– either ‘mw’ (default) for Monsky-Washnitzer or ‘approx’ for the algorithm described by Bernardi and Perrin-Riou (much slower and not fully tested)
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)]
- is_ordinary()#
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
- is_supersingular()#
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
- power_series(n=3, quadratic_twist=1, prec=5, eta=0)#
Return the \(n\)-th approximation to the \(p\)-adic L-series as a power series in \(T\) (corresponding to \(\gamma-1\) with \(\gamma=1+p\) as a generator of \(1+p\ZZ_p\)). Each coefficient is an element of a quadratic extension of the \(p\)-adic number whose precision is provably correct.
Here the normalization of the \(p\)-adic L-series is chosen such that \(L_p(E,1) = (1-1/\alpha)^2 L(E,1)/\Omega_E\) where \(\alpha\) is a root of the characteristic polynomial of Frobenius on \(T_pE\) and \(\Omega_E\) is the Néron period of \(E\).
INPUT:
n
– (default: 2) a positive integerquadratic_twist
– (default: +1) a fundamental discriminant of a quadratic field, coprime to the conductor of the curveprec
– (default: 5) maximal number of terms of the series to compute; to compute as many as possible just give a very large number forprec
; the result will still be correct.eta
(default: 0) an integer (specifying the power of the Teichmueller character on the group of roots of unity in \(\ZZ_p^\times\))
OUTPUT:
a power series with coefficients in a quadratic ramified extension of the \(p\)-adic numbers generated by a root \(alpha\) of the characteristic polynomial of Frobenius on \(T_pE\).
ALIAS: power_series is identical to series.
EXAMPLES:
A supersingular 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(alpha) + (alpha^-2 + O(alpha^0))*T + (alpha^-2 + O(alpha^0))*T^2 + O(T^5) sage: L.alpha(2).parent() 3-adic Eisenstein Extension Field in alpha defined by x^2 + 3*x + 3
An example where we only compute the leading term (github issue #15737):
sage: E = EllipticCurve("17a1") sage: L = E.padic_lseries(3) sage: L.series(4,prec=1) alpha^-2 + alpha^-1 + 2 + 2*alpha + ... + O(alpha^38) + O(T)
It works also for \(p=2\):
sage: E = EllipticCurve("11a1") sage: lp = E.padic_lseries(2) sage: lp.series(10) O(alpha^-3) + (alpha^-4 + O(alpha^-3))*T + (alpha^-4 + O(alpha^-3))*T^2 + (alpha^-5 + alpha^-4 + O(alpha^-3))*T^3 + (alpha^-4 + O(alpha^-3))*T^4 + O(T^5)
- series(n=3, quadratic_twist=1, prec=5, eta=0)#
Return the \(n\)-th approximation to the \(p\)-adic L-series as a power series in \(T\) (corresponding to \(\gamma-1\) with \(\gamma=1+p\) as a generator of \(1+p\ZZ_p\)). Each coefficient is an element of a quadratic extension of the \(p\)-adic number whose precision is provably correct.
Here the normalization of the \(p\)-adic L-series is chosen such that \(L_p(E,1) = (1-1/\alpha)^2 L(E,1)/\Omega_E\) where \(\alpha\) is a root of the characteristic polynomial of Frobenius on \(T_pE\) and \(\Omega_E\) is the Néron period of \(E\).
INPUT:
n
– (default: 2) a positive integerquadratic_twist
– (default: +1) a fundamental discriminant of a quadratic field, coprime to the conductor of the curveprec
– (default: 5) maximal number of terms of the series to compute; to compute as many as possible just give a very large number forprec
; the result will still be correct.eta
(default: 0) an integer (specifying the power of the Teichmueller character on the group of roots of unity in \(\ZZ_p^\times\))
OUTPUT:
a power series with coefficients in a quadratic ramified extension of the \(p\)-adic numbers generated by a root \(alpha\) of the characteristic polynomial of Frobenius on \(T_pE\).
ALIAS: power_series is identical to series.
EXAMPLES:
A supersingular 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(alpha) + (alpha^-2 + O(alpha^0))*T + (alpha^-2 + O(alpha^0))*T^2 + O(T^5) sage: L.alpha(2).parent() 3-adic Eisenstein Extension Field in alpha defined by x^2 + 3*x + 3
An example where we only compute the leading term (github issue #15737):
sage: E = EllipticCurve("17a1") sage: L = E.padic_lseries(3) sage: L.series(4,prec=1) alpha^-2 + alpha^-1 + 2 + 2*alpha + ... + O(alpha^38) + O(T)
It works also for \(p=2\):
sage: E = EllipticCurve("11a1") sage: lp = E.padic_lseries(2) sage: lp.series(10) O(alpha^-3) + (alpha^-4 + O(alpha^-3))*T + (alpha^-4 + O(alpha^-3))*T^2 + (alpha^-5 + alpha^-4 + O(alpha^-3))*T^3 + (alpha^-4 + O(alpha^-3))*T^4 + O(T^5)