# $$p$$-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

• David Loeffler (01/2011): added nontrivial Teichmueller components

class sage.schemes.elliptic_curves.padic_lseries.pAdicLseries(E, p, implementation='eclib', normalize='L_ratio')

The $$p$$-adic L-series of an elliptic curve.

EXAMPLES:

An ordinary example:

sage: e = EllipticCurve('389a')
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)
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.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 integer

• n – a non-negative integer

• prec – an integer

• quadratic_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.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)
O(5^15)
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 == 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 oo

• sign – +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.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: 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.order_of_vanishing()
0
sage: L.order_of_vanishing()
1
sage: L.order_of_vanishing()
1
sage: L.order_of_vanishing()
0
sage: L.order_of_vanishing()
2
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')
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 integer

• quadratic_twist - (default: +1) a fundamental discriminant of a quadratic field, coprime to the conductor of the curve

• prec - (default: 5) maximal number of terms of the series to compute; to compute as many as possible just give a very large number for prec; 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 to series.

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.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.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.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 ticket #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')
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 $$L$$-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 $$p=2$$ (trac ticket #20798):

sage: E = EllipticCurve("53a1")
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.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)

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 integer

• quadratic_twist - (default: +1) a fundamental discriminant of a quadratic field, coprime to the conductor of the curve

• prec - (default: 5) maximal number of terms of the series to compute; to compute as many as possible just give a very large number for prec; 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 to series.

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.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.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.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 ticket #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')
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 $$L$$-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 $$p=2$$ (trac ticket #20798):

sage: E = EllipticCurve("53a1")
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.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)

class sage.schemes.elliptic_curves.padic_lseries.pAdicLseriesSupersingular(E, p, implementation='eclib', normalize='L_ratio')
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 as v*omega + v*eta. The coordinates of v are dependent of the Weierstrass equation.

EXAMPLES:

sage: E = EllipticCurve('53a')
sage: h = L.Dp_valued_height(7)
sage: h(E.gens())
(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.

Note

The definition here is corrected with respect to Perrin-Riou’s article [PR2003]. See [SW2013].

EXAMPLES:

sage: E = EllipticCurve('43a')
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 $$\cdot \omega +$$ v $$\cdot \varphi(\omega)$$

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 integer

• prec – (default: 5) a positive integer

EXAMPLES:

sage: E = EllipticCurve('14a')
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 with E2 = 0.

EXAMPLES:

sage: E = EllipticCurve('14a')
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 integer

• algorithm - 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: 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 integer

• quadratic_twist - (default: +1) a fundamental discriminant of a quadratic field, coprime to the conductor of the curve

• prec - (default: 5) maximal number of terms of the series to compute; to compute as many as possible just give a very large number for prec; 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')
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 (trac ticket #15737):

sage: E = EllipticCurve("17a1")
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.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 integer

• quadratic_twist - (default: +1) a fundamental discriminant of a quadratic field, coprime to the conductor of the curve

• prec - (default: 5) maximal number of terms of the series to compute; to compute as many as possible just give a very large number for prec; 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')
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 (trac ticket #15737):

sage: E = EllipticCurve("17a1")

It works also for $$p=2$$:
sage: E = EllipticCurve("11a1")