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 [BP], which has coefficients in the Dieudonne module $D_pE = H^1_{dR}(E/\QQ_p)$ of $E$ . There is a different description by Pollack [Po] which is not available here.

According to the $p$ -adic version of the Birch and Swinnerton-Dyer conjecture [MTT], 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 [SW] for more details.

REFERENCES:

[MTT] B. Mazur, J. Tate, and J. Teitelbaum, On $p$ -adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones mathematicae 84, (1986), 1-48.
[BP] Dominique Bernardi and Bernadette Perrin-Riou, Variante $p$ -adique de la conjecture de Birch et Swinnerton-Dyer (le cas supersingulier), C. R. Acad. Sci. Paris, Sér I. Math., 317 (1993), no. 3, 227-232.
[Po] Robert Pollack, On the $p$ -adic $L$ -function of a modular form at a supersingular prime, Duke Math. J. 118 (2003), no. 3, 523-558.
[SW] William Stein and Christian Wuthrich, Algorithms for the Arithmetic of Elliptic Curves using Iwasawa Theory, Mathematics of Computation 82 (2013), 1757-1792.

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: sage.structure.sage_object.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 $\QQ(\zeta_7)$ .)

the load-dumps test:

sage: lp = EllipticCurve('11a').padic_lseries(5)
sage: lp == loads(dumps(lp))
True

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 then finding the modular symbols for the twist. Quadratic twists are only implemented if the sign is $+1$ .

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 $\ell$ 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 = 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 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 $\ell$ 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 = 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 [Ka].

NOTE: currently $p$ must be a prime of good ordinary reduction.

REFERENCES:

[MTT] B. Mazur, J. Tate, and J. Teitelbaum, On $p$ -adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones mathematicae 84, (1986), 1-48.
[Ka] Kayuza Kato, $p$ -adic Hodge theory and values of zeta functions of modular forms, Cohomologies $p$ -adiques et applications arithmetiques III, Asterisque vol 295, SMF, Paris, 2004.

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: sage.schemes.elliptic_curves.padic_lseries.pAdicLseries

INPUT:

E - an elliptic curve
p - a prime of good reduction
implementation - string (default:’eclib’); either ‘eclib’ to use John Cremona’s eclib for the computation of modular symbols or ‘sage’ to use Sage’s own implementation
normalize - 'L_ratio' (default), 'period' or 'none'; this is describes the way the modular symbols are normalized. See modular_symbol of an elliptic curve over Q for more details.

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)

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 Neron 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 $\ZZ_p^\times$ )

ALIAS: 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 = 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 (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')
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 $L$ -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)
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 = 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)

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 Neron 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 $\ZZ_p^\times$ )

ALIAS: 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 = 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 (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')
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 $L$ -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)
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 = 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)

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

Bases: sage.schemes.elliptic_curves.padic_lseries.pAdicLseries

INPUT:

E - an elliptic curve
p - a prime of good reduction
implementation - string (default:’eclib’); either ‘eclib’ to use John Cremona’s eclib for the computation of modular symbols or ‘sage’ to use Sage’s own implementation
normalize - 'L_ratio' (default), 'period' or 'none'; this is describes the way the modular symbols are normalized. See modular_symbol of an elliptic curve over Q for more details.

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)

Dp_valued_height(prec=20)¶

Return the canonical $p$ -adic height with values in the Dieudonne 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[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))

Dp_valued_regulator(prec=20, v1=0, v2=0)¶

Return the canonical $p$ -adic regulator with values in the Dieudonne 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 [PR]. See [SW].

REFERENCES:

[PR] Perrin Riou, Arithmetique des courbes elliptiques a reduction supersinguliere en $p$ , Experiment. Math. 12 (2003), no. 2, 155-186.
[SW] William Stein and Christian Wuthrich, Computations About Tate-Shafarevich Groups using Iwasawa theory, preprint 2009.

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 Dieudonne 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 [BP] 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

REFERENCE:

[BP] Dominique Bernardi and Bernadette Perrin-Riou, Variante $p$ -adique de la conjecture de Birch et Swinnerton-Dyer (le cas supersingulier), C. R. Acad. Sci. Paris, Ser I. Math, 317 (1993), no 3, 227-232.

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 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)

frobenius(prec=20, algorithm='mw')¶

This returns a geometric Frobenius $\varphi$ on the Diedonne 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-Washintzer 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 probably 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 Neron 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 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(alpha) + (alpha^-2 + O(alpha^0))*T + (alpha^-2 + O(alpha^0))*T^2 + O(T^5)
sage: L.alpha(2).parent()
Eisenstein Extension of 3-adic Field with capped relative precision 2 in alpha defined by (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 ticket #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 probably 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 Neron 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 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(alpha) + (alpha^-2 + O(alpha^0))*T + (alpha^-2 + O(alpha^0))*T^2 + O(T^5)
sage: L.alpha(2).parent()
Eisenstein Extension of 3-adic Field with capped relative precision 2 in alpha defined by (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 ticket #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)

p-adic L-functions of elliptic curves¶

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