# $$L$$-series for elliptic curves¶

AUTHORS:

• Simon Spicer (2014-08-15) - Added LFunctionZeroSum class interface method

• Jeroen Demeyer (2013-10-17) - Compute L series with arbitrary precision instead of floats.

• William Stein et al. (2005 and later)

class sage.schemes.elliptic_curves.lseries_ell.Lseries_ell(E)

An elliptic curve $$L$$-series.

L1_vanishes()

Returns whether or not $$L(E,1) = 0$$. The result is provably correct if the Manin constant of the associated optimal quotient is <= 2. This hypothesis on the Manin constant is true for all curves of conductor <= 40000 (by Cremona) and all semistable curves (i.e., squarefree conductor).

ALGORITHM: see L_ratio().

EXAMPLES:

sage: E = EllipticCurve([0, -1, 1, -10, -20])   # 11A  = X_0(11)
sage: E.lseries().L1_vanishes()
False
sage: E = EllipticCurve([0, -1, 1, 0, 0])       # X_1(11)
sage: E.lseries().L1_vanishes()
False
sage: E = EllipticCurve([0, 0, 1, -1, 0])       # 37A  (rank 1)
sage: E.lseries().L1_vanishes()
True
sage: E = EllipticCurve([0, 1, 1, -2, 0])       # 389A (rank 2)
sage: E.lseries().L1_vanishes()
True
sage: E = EllipticCurve([0, 0, 1, -38, 90])     # 361A (CM curve))
sage: E.lseries().L1_vanishes()
True
sage: E = EllipticCurve([0,-1,1,-2,-1])         # 141C (13-isogeny)
sage: E.lseries().L1_vanishes()
False


AUTHOR: William Stein, 2005-04-20.

L_ratio()

Return the ratio $$L(E,1) / \Omega$$ as an exact rational number.

The result is provably correct if the Manin constant of the associated optimal quotient is $$\leq 2$$. This hypothesis on the Manin constant is true for all semistable curves (i.e., squarefree conductor), by a theorem of Mazur from his Rational Isogenies of Prime Degree paper.

EXAMPLES:

sage: E = EllipticCurve([0, -1, 1, -10, -20])   # 11A  = X_0(11)
sage: E.lseries().L_ratio()
1/5
sage: E = EllipticCurve([0, -1, 1, 0, 0])       # X_1(11)
sage: E.lseries().L_ratio()
1/25
sage: E = EllipticCurve([0, 0, 1, -1, 0])       # 37A  (rank 1)
sage: E.lseries().L_ratio()
0
sage: E = EllipticCurve([0, 1, 1, -2, 0])       # 389A (rank 2)
sage: E.lseries().L_ratio()
0
sage: E = EllipticCurve([0, 0, 1, -38, 90])     # 361A (CM curve))
sage: E.lseries().L_ratio()
0
sage: E = EllipticCurve([0,-1,1,-2,-1])         # 141C (13-isogeny)
sage: E.lseries().L_ratio()
1
sage: E = EllipticCurve(RationalField(), [1, 0, 0, 1/24624, 1/886464])
sage: E.lseries().L_ratio()
2

sage: EllipticCurve([0,0,0,-193^2,0]).sha().an()
4
sage: EllipticCurve([1, 0, 1, -131, 558]).sha().an()  # long time
1.00000000000000


ALGORITHM: Compute the root number. If it is -1 then $$L(E,s)$$ vanishes to odd order at 1, hence vanishes. If it is +1, use a result about modular symbols and Mazur’s Rational Isogenies paper to determine a provably correct bound (assuming Manin constant is <= 2) so that we can determine whether $$L(E,1) = 0$$.

AUTHOR: William Stein, 2005-04-20.

at1(k=None, prec=None)

Compute $$L(E,1)$$ using $$k$$ terms of the series for $$L(E,1)$$ as explained in Section 7.5.3 of Henri Cohen’s book A Course in Computational Algebraic Number Theory. If the argument $$k$$ is not specified, then it defaults to $$\sqrt{N}$$, where $$N$$ is the conductor.

INPUT:

• k – number of terms of the series. If zero or None, use $$k = \sqrt{N}$$, where $$N$$ is the conductor.

• prec – numerical precision in bits. If zero or None, use a reasonable automatic default.

OUTPUT:

A tuple of real numbers (L, err) where L is an approximation for $$L(E,1)$$ and err is a bound on the error in the approximation.

This function is disjoint from the PARI elllseries command, which is for a similar purpose. To use that command (via the PARI C library), simply type E.pari_mincurve().elllseries(1).

ALGORITHM:

• Compute the root number eps. If it is -1, return 0.

• Compute the Fourier coefficients $$a_n$$, for $$n$$ up to and including $$k$$.

• Compute the sum

$2 \cdot \sum_{n=1}^{k} \frac{a_n}{n} \cdot \exp(-2*pi*n/\sqrt{N}),$

where $$N$$ is the conductor of $$E$$.

• Compute a bound on the tail end of the series, which is

$2 e^{-2 \pi (k+1) / \sqrt{N}} / (1 - e^{-2 \pi/\sqrt{N}}).$

For a proof see [Grigov-Jorza-Patrascu-Patrikis-Stein].

EXAMPLES:

sage: L, err = EllipticCurve('11a1').lseries().at1()
sage: L, err
(0.253804, 0.000181444)
sage: parent(L)
Real Field with 24 bits of precision
sage: E = EllipticCurve('37b')
sage: E.lseries().at1()
(0.7257177, 0.000800697)
sage: E.lseries().at1(100)
(0.7256810619361527823362055410263965487367603361763, 1.52469e-45)
sage: L,err = E.lseries().at1(100, prec=128)
sage: L
0.72568106193615278233620554102639654873
sage: parent(L)
Real Field with 128 bits of precision
sage: err
1.70693e-37
sage: parent(err)
Real Field with 24 bits of precision and rounding RNDU


Rank 1 through 3 elliptic curves:

sage: E = EllipticCurve('37a1')
sage: E.lseries().at1()
(0.0000000, 0.000000)
sage: E = EllipticCurve('389a1')
sage: E.lseries().at1()
(-0.001769566, 0.00911776)
sage: E = EllipticCurve('5077a1')
sage: E.lseries().at1()
(0.0000000, 0.000000)

deriv_at1(k=None, prec=None)

Compute $$L'(E,1)$$ using $$k$$ terms of the series for $$L'(E,1)$$, under the assumption that $$L(E,1) = 0$$.

The algorithm used is from Section 7.5.3 of Henri Cohen’s book A Course in Computational Algebraic Number Theory.

INPUT:

• k – number of terms of the series. If zero or None, use $$k = \sqrt{N}$$, where $$N$$ is the conductor.

• prec – numerical precision in bits. If zero or None, use a reasonable automatic default.

OUTPUT:

A tuple of real numbers (L1, err) where L1 is an approximation for $$L'(E,1)$$ and err is a bound on the error in the approximation.

Warning

This function only makes sense if $$L(E)$$ has positive order of vanishing at 1, or equivalently if $$L(E,1) = 0$$.

ALGORITHM:

• Compute the root number eps. If it is 1, return 0.

• Compute the Fourier coefficients $$a_n$$, for $$n$$ up to and including $$k$$.

• Compute the sum

$2 \cdot \sum_{n=1}^{k} (a_n / n) \cdot E_1(2 \pi n/\sqrt{N}),$

where $$N$$ is the conductor of $$E$$, and $$E_1$$ is the exponential integral function.

• Compute a bound on the tail end of the series, which is

$2 e^{-2 \pi (k+1) / \sqrt{N}} / (1 - e^{-2 \pi/\sqrt{N}}).$

For a proof see [Grigorov-Jorza-Patrascu-Patrikis-Stein]. This is exactly the same as the bound for the approximation to $$L(E,1)$$ produced by at1().

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E.lseries().deriv_at1()
(0.3059866, 0.000801045)
sage: E.lseries().deriv_at1(100)
(0.3059997738340523018204836833216764744526377745903, 1.52493e-45)
sage: E.lseries().deriv_at1(1000)
(0.305999773834052301820483683321676474452637774590771998..., 2.75031e-449)


With less numerical precision, the error is bounded by numerical accuracy:

sage: L,err = E.lseries().deriv_at1(100, prec=64)
sage: L,err
(0.305999773834052302, 5.55318e-18)
sage: parent(L)
Real Field with 64 bits of precision
sage: parent(err)
Real Field with 24 bits of precision and rounding RNDU


Rank 2 and rank 3 elliptic curves:

sage: E = EllipticCurve('389a1')
sage: E.lseries().deriv_at1()
(0.0000000, 0.000000)
sage: E = EllipticCurve((1, 0, 1, -131, 558))  # curve 59450i1
sage: E.lseries().deriv_at1()
(-0.00010911444, 0.142428)
sage: E.lseries().deriv_at1(4000)
(6.990...e-50, 1.31318e-43)

dokchitser(prec=53, max_imaginary_part=0, max_asymp_coeffs=40, algorithm=None)

Return an interface for computing with the $$L$$-series of this elliptic curve.

This provides a way to compute Taylor expansions and higher derivatives of $$L$$-series.

INPUT:

• prec – optional integer (default 53) bits precision

• max_imaginary_part – optional real number (default 0)

• max_asymp_coeffs – optional integer (default 40)

• algorithm – optional string: ‘gp’ (default), ‘pari’ or ‘magma’

If algorithm is “gp”, this returns an interface to Tim Dokchitser’s program for computing with the L-functions.

If algorithm is “pari”, this returns instead an interface to Pari’s own general implementation of L-functions.

Note

If algorithm=’magma’, then the precision is in digits rather than bits and the object returned is a Magma L-series, which has different functionality from the Sage L-series.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser()
sage: L(2)
0.381575408260711
sage: L = E.lseries().dokchitser(algorithm='magma')         # optional - magma
sage: L.Evaluate(2)                                         # optional - magma
0.38157540826071121129371040958008663667709753398892116


If the curve has too large a conductor, it is not possible to compute with the $$L$$-series using this command. Instead a RuntimeError is raised:

sage: e = EllipticCurve([1,1,0,-63900,-1964465932632])
sage: L = e.lseries().dokchitser(15, algorithm='gp')
Traceback (most recent call last):
...
RuntimeError: Unable to create L-series, due to precision or other limits in PARI.


Using the “pari” algorithm:

sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser(algorithm="pari")
sage: L(2)
0.381575408260711

elliptic_curve()

Return the elliptic curve that this $$L$$-series is attached to.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: L = E.lseries()
sage: L.elliptic_curve ()
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field

sympow(n, prec)

Return $$L( Sym^{(n)}(E, \text{edge}))$$ to prec digits of precision.

INPUT:

• n – integer

• prec – integer

OUTPUT:

• string – real number to prec digits of precision as a string.

Note

Before using this function for the first time for a given n, you may have to type sympow('-new_data <n>'), where <n> is replaced by your value of n. This command takes a long time to run.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: a = E.lseries().sympow(2,16)   # not tested - requires precomputing "sympow('-new_data 2')"
sage: a                              # not tested
'2.492262044273650E+00'
sage: RR(a)                          # not tested
2.49226204427365

sympow_derivs(n, prec, d)

Return 0-th to $$d$$-th derivatives of $$L( Sym^{(n)}(E, \text{edge}))$$ to prec digits of precision.

INPUT:

• n – integer

• prec – integer

• d – integer

OUTPUT:

• a string, exactly as output by sympow

Note

To use this function you may have to run a few commands like sympow('-new_data 1d2'), each which takes a few minutes. If this function fails it will indicate what commands have to be run.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: print(E.lseries().sympow_derivs(1,16,2))    # not tested -- requires precomputing "sympow('-new_data 2')"
sympow 1.018 RELEASE  (c) Mark Watkins --- see README and COPYING for details
Minimal model of curve  is [0,0,1,-1,0]
At 37: Inertia Group is  C1 MULTIPLICATIVE REDUCTION
Conductor is 37
sp 1: Conductor at 37 is 1+0, root number is 1
sp 1: Euler factor at 37 is 1+1*x
1st sym power conductor is 37, global root number is -1
NT 1d0: 35
NT 1d1: 32
NT 1d2: 28
Maximal number of terms is 35
Done with small primes 1049
Computed:  1d0  1d1  1d2
Checked out:  1d1
1n0: 3.837774351482055E-01
1w0: 3.777214305638848E-01
1n1: 3.059997738340522E-01
1w1: 3.059997738340524E-01
1n2: 1.519054910249753E-01
1w2: 1.545605024269432E-01

taylor_series(a=1, prec=53, series_prec=6, var='z')

Return the Taylor series of this $$L$$-series about $$a$$ to the given precision (in bits) and the number of terms.

The output is a series in var, where you should view var as equal to $$s-a$$. Thus this function returns the formal power series whose coefficients are $$L^{(n)}(a)/n!$$.

INPUT:

• a – complex number

• prec – integer, precision in bits (default 53)

• series_prec – integer (default 6)

• var – variable (default ‘z’)

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: L = E.lseries()
sage: L.taylor_series(series_prec=3)   # abs tol 1e-14
-1.27685190980159e-23 + (7.23588070754027e-24)*z + 0.759316500288427*z^2 + O(z^3)  # 32-bit
1.34667664606157e-19 + (-7.63157535163667e-20)*z + 0.759316500288427*z^2 + O(z^3)  # 64-bit

twist_values(s, dmin, dmax)

Return values of $$L(E, s, \chi_d)$$ for each quadratic character $$\chi_d$$ for $$d_{\min} \leq d \leq d_{\max}$$.

Note

The L-series is normalized so that the center of the critical strip is 1.

INPUT:

• s – complex numbers

• dmin – integer

• dmax – integer

OUTPUT:

• list of pairs $$(d, L(E, s, \chi_d))$$

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: vals = E.lseries().twist_values(1, -12, -4)
sage: vals  # abs tol 1e-15
[(-11, 1.47824342), (-8, 8.9590946e-18), (-7, 1.85307619), (-4, 2.45138938)]
sage: F.rank()
1
sage: F.rank()
0

twist_zeros(n, dmin, dmax)

Return first $$n$$ real parts of nontrivial zeros of $$L(E,s,\chi_d)$$ for each quadratic character $$\chi_d$$ with $$d_{\min} \leq d \leq d_{\max}$$.

Note

The L-series is normalized so that the center of the critical strip is 1.

INPUT:

• n – integer

• dmin – integer

• dmax – integer

OUTPUT:

• dict – keys are the discriminants $$d$$, and

values are list of corresponding zeros.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E.lseries().twist_zeros(3, -4, -3)         # long time
{-4: [1.60813783, 2.96144840, 3.89751747], -3: [2.06170900, 3.48216881, 4.45853219]}

values_along_line(s0, s1, number_samples)

Return values of $$L(E, s)$$ at number_samples equally-spaced sample points along the line from $$s_0$$ to $$s_1$$ in the complex plane.

Note

The $$L$$-series is normalized so that the center of the critical strip is 1.

INPUT:

• s0, s1 – complex numbers

• number_samples – integer

OUTPUT:

list – list of pairs ($$s$$, $$L(E,s)$$), where the $$s$$ are

equally spaced sampled points on the line from s0 to s1.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E.lseries().values_along_line(1, 0.5 + 20*I, 5)
[(0.500000000, ...),
(0.400000000 + 4.00000000*I, 3.31920245 - 2.60028054*I),
(0.300000000 + 8.00000000*I, -0.886341185 - 0.422640337*I),
(0.200000000 + 12.0000000*I, -3.50558936 - 0.108531690*I),
(0.100000000 + 16.0000000*I, -3.87043288 - 1.88049411*I)]

zero_sums(N=None)

Return an LFunctionZeroSum class object for efficient computation of sums over the zeros of self.

This can be used to bound analytic rank from above without having to compute with the $$L$$-series directly.

INPUT:

• N – (default: None) If not None, the conductor of the elliptic curve attached to self. This is passable so that zero sum computations can be done on curves for which the conductor has been precomputed.

OUTPUT:

A LFunctionZeroSum_EllipticCurve instance.

EXAMPLES:

sage: E = EllipticCurve("5077a")
sage: E.lseries().zero_sums()
Zero sum estimator for L-function attached to Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field

zeros(n)

Return the imaginary parts of the first $$n$$ nontrivial zeros on the critical line of the $$L$$-function in the upper half plane, as 32-bit reals.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E.lseries().zeros(2)
[0.000000000, 5.00317001]

sage: a = E.lseries().zeros(20)             # long time
sage: point([(1,x) for x in a])             # graph  (long time)
Graphics object consisting of 1 graphics primitive

AUTHOR:

– Uses Rubinstein’s L-functions calculator.

zeros_in_interval(x, y, stepsize)

Return the imaginary parts of (most of) the nontrivial zeros on the critical line $$\Re(s)=1$$ with positive imaginary part between x and y, along with a technical quantity for each.

INPUT:

• x– positive floating point number

• y– positive floating point number

• stepsize – positive floating point number

OUTPUT:

• list of pairs (zero, S(T)).

Rubinstein writes: The first column outputs the imaginary part of the zero, the second column a quantity related to S(T) (it increases roughly by 2 whenever a sign change, i.e. pair of zeros, is missed). Higher up the critical strip you should use a smaller stepsize so as not to miss zeros.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E.lseries().zeros_in_interval(6, 10, 0.1)      # long time
[(6.87039122, 0.248922780), (8.01433081, -0.140168533), (9.93309835, -0.129943029)]