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# -*- coding: utf-8 -*- """ 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)
""" #***************************************************************************** # Copyright (C) 2005 William Stein # Copyright (C) 2013 Jeroen Demeyer # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # http://www.gnu.org/licenses/ #***************************************************************************** from six.moves import range
from sage.structure.sage_object import SageObject from sage.rings.all import RealField, RationalField from math import sqrt, exp, log, ceil import sage.functions.exp_integral as exp_integral from sage.misc.all import verbose
class Lseries_ell(SageObject): """ An elliptic curve `L`-series. """ def __init__(self, E): r""" Create an elliptic curve `L`-series.
EXAMPLES::
sage: EllipticCurve([1..5]).lseries() Complex L-series of the Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field """
def elliptic_curve(self): r""" 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 """
def taylor_series(self, a=1, prec=53, series_prec=6, var='z'): r""" 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) -1.27685190980159e-23 + (7.23588070754027e-24)*z + 0.759316500288427*z^2 + O(z^3) # 32-bit -2.72911738151096e-23 + (1.54658247036311e-23)*z + 0.759316500288427*z^2 + O(z^3) # 64-bit """
def _repr_(self): r""" Return string representation of this `L`-series.
EXAMPLES::
sage: E = EllipticCurve('37a') sage: L = E.lseries() sage: L._repr_() 'Complex L-series of the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field' """
def dokchitser(self, prec=53, max_imaginary_part=0, max_asymp_coeffs=40, algorithm='gp'): r""" Return interface to Tim Dokchitser's program 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`` -- integer (bits precision)
- ``max_imaginary_part`` -- real number
- ``max_asymp_coeffs`` -- integer
- ``algorithm`` -- string: 'gp' or 'magma'
.. 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 isn't 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) Traceback (most recent call last): ... RuntimeError: Unable to create L-series, due to precision or other limits in PARI. """ from sage.interfaces.all import magma return magma(self.__E).LSeries(Precision = prec)
gammaV = [0,1], weight = 2, eps = self.__E.root_number(), poles = [], prec = prec) max_imaginary_part=max_imaginary_part, max_asymp_coeffs=max_asymp_coeffs)
def sympow(self, n, prec): r""" 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 """ from sage.lfunctions.sympow import sympow return sympow.L(self.__E, n, prec)
def sympow_derivs(self, n, prec, d): r""" 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 """ from sage.lfunctions.sympow import sympow return sympow.Lderivs(self.__E, n, prec, d)
def zeros(self, n): r""" 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. """
def zeros_in_interval(self, x, y, stepsize): r""" 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)] """ from sage.lfunctions.lcalc import lcalc return lcalc.zeros_in_interval(x, y, stepsize, L=self.__E)
def values_along_line(self, s0, s1, number_samples): r""" 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)]
""" s1-RationalField()('1/2'), number_samples, L=self.__E)
def twist_values(self, s, dmin, dmax): r""" 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 = E.quadratic_twist(-8) sage: F.rank() 1 sage: F = E.quadratic_twist(-7) sage: F.rank() 0 """
def twist_zeros(self, n, dmin, dmax): r""" 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]} """ from sage.lfunctions.lcalc import lcalc return lcalc.twist_zeros(n, dmin, dmax, L=self.__E)
def at1(self, k=None, prec=None): r""" 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
.. MATH::
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
.. MATH::
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) """ else:
else: # Use the same precision as deriv_at1() below for # consistency # Compute error term with bounded precision of 24 bits and # round towards +infinity
# Compute series sum and accumulate floating point errors
# We express relative error in units of epsilon, where # epsilon is a number divided by 2^precision. # Instead of multiplying the error by 2 after the loop # (to account for L *= 2), we already multiply it now. # # For multiplication and division, the relative error # in epsilons is bounded by (1+e)^n - 1, where n is the # number of operations (assuming exact inputs). # exp(x) additionally multiplies this error by abs(x) and # adds one epsilon. The inputs pi and sqrtN each contribute # another epsilon. # Assuming that 2*pi/sqrtN <= 2, the relative error for z is # 7 epsilon. This implies a relative error of (8n-1) epsilon # for zpow. We add 2 for the computation of term and 1/2 to # compensate for the approximation (1+e)^n = 1+ne. # # The error of the addition is at most half an ulp of the # result. # # Multiplying everything by two gives:
# Add series error (we use (-2)/(z-1) instead of 2/(1-z) # because this causes 1/(1-z) to be rounded up)
def deriv_at1(self, k=None, prec=None): r""" 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
.. MATH::
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
.. MATH::
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 :meth:`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) """ else:
else: # Estimate number of bits for the computation, based on error # estimate below (the denominator of that error is close enough # to 1 that we can ignore it). # 9.065 = 2*Pi/log(2) # 1.443 = 1/log(2) # 12 is an arbitrary extra number of bits (it is chosen # such that the precision is 24 bits when the conductor # equals 11 and k is the default value 4) # Compute error term with bounded precision of 24 bits and # round towards +infinity
# Order of vanishing at 1 of L(E) is even and assumed to be # positive, so L'(E,1) = 0.
# Compute series sum and accumulate floating point errors # Sum of |an[n]|/n
# Add error term for exponential_integral_1() errors. # Absolute error for 2*v[i] is 4*max(1, v[0])*2^-prec
# Add series error (we use (-2)/(z-1) instead of 2/(1-z) # because this causes 1/(1-z) to be rounded up)
def __call__(self, s): r""" Returns the value of the L-series of the elliptic curve E at s, where s must be a real number.
.. NOTE::
If the conductor of the curve is large, say `>10^{12}`, then this function will take a very long time, since it uses an `O(\sqrt{N})` algorithm.
EXAMPLES::
sage: E = EllipticCurve([1,2,3,4,5]) sage: L = E.lseries() sage: L(1) 0.000000000000000 sage: L(1.1) 0.285491007678148 sage: L(1.1 + I) 0.174851377216615 + 0.816965038124457*I """
def L1_vanishes(self): r""" 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 :meth:`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. """
def L_ratio(self): r""" Returns 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
See :trac:`3651` and :trac:`15299`::
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. """
# Even root number. Decide if L(E,1) = 0. If E is a modular # *OPTIMAL* quotient of J_0(N) elliptic curve, we know that T * # L(E,1)/omega is an integer n, where T is the order of the # image of the rational torsion point (0)-(oo) in E(Q), and # omega is the least real Neron period. (This is proved in my # Ph.D. thesis, but is probably well known.) We can easily # compute omega to very high precision using AGM. So to prove # that L(E,1) = 0 we compute T/omega * L(E,1) to sufficient # precision to determine it as an integer. If eps is the # error in computation of L(E,1), then the error in computing # the product is (2T/Omega_E) * eps, and we need this to be # less than 0.5, i.e., # (2T/Omega_E) * eps < 0.5, # so # eps < 0.5 * Omega_E / (2T) = Omega_E / (4*T). # # Since in general E need not be optimal, we have to choose # eps = Omega_E/(8*t*B), where t is the exponent of E(Q)_tor, # and is a multiple of the degree of an isogeny between E # and the optimal curve. # # NOTES: We *do* have to worry about the Manin constant, since # we are using the Neron model to compute omega, not the # newform. My theorem replaces the omega above by omega/c, # where c is the Manin constant, and the bound must be # correspondingly smaller. If the level is square free, then # the Manin constant is 1 or 2, so there's no problem (since # we took 8 instead of 4 in the denominator). If the level # is divisible by a square, then the Manin constant could # be a divisible by an arbitrary power of that prime, except # that Edixhoven claims the primes that appear are <= 7.
k += sqrtN verbose("Increasing precision to %s terms." % k)
def zero_sums(self, N=None): r""" 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 """ |