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""" Dokchitser's L-functions Calculator
AUTHORS:
- Tim Dokchitser (2002): original PARI code and algorithm (and the documentation below is based on Dokchitser's docs).
- William Stein (2006-03-08): Sage interface
.. TODO::
- add more examples from SAGE_EXTCODE/pari/dokchitser that illustrate use with Eisenstein series, number fields, etc.
- plug this code into number fields and modular forms code (elliptic curves are done). """
#***************************************************************************** # Copyright (C) 2006 William Stein <wstein@gmail.com> # # 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/ #*****************************************************************************
r""" Dokchitser's `L`-functions Calculator
Create a Dokchitser `L`-series with
Dokchitser(conductor, gammaV, weight, eps, poles, residues, init, prec)
where
- ``conductor`` -- integer, the conductor
- ``gammaV`` -- list of Gamma-factor parameters, e.g. [0] for Riemann zeta, [0,1] for ell.curves, (see examples).
- ``weight`` -- positive real number, usually an integer e.g. 1 for Riemann zeta, 2 for `H^1` of curves/`\QQ`
- ``eps`` -- complex number; sign in functional equation
- ``poles`` -- (default: []) list of points where `L^*(s)` has (simple) poles; only poles with `Re(s)>weight/2` should be included
- ``residues`` -- vector of residues of `L^*(s)` in those poles or set residues='automatic' (default value)
- ``prec`` -- integer (default: 53) number of *bits* of precision
RIEMANN ZETA FUNCTION:
We compute with the Riemann Zeta function. ::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L Dokchitser L-series of conductor 1 and weight 1 sage: L(1) Traceback (most recent call last): ... ArithmeticError sage: L(2) 1.64493406684823 sage: L(2, 1.1) 1.64493406684823 sage: L.derivative(2) -0.937548254315844 sage: h = RR('0.0000000000001') sage: (zeta(2+h) - zeta(2.))/h -0.937028232783632 sage: L.taylor_series(2, k=5) 1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 - 1.00002430047384*z^3 + 1.00006193307...*z^4 + O(z^5)
RANK 1 ELLIPTIC CURVE:
We compute with the `L`-series of a rank `1` curve. ::
sage: E = EllipticCurve('37a') sage: L = E.lseries().dokchitser(); L Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: L(1) 0.000000000000000 sage: L.derivative(1) 0.305999773834052 sage: L.derivative(1,2) 0.373095594536324 sage: L.num_coeffs() 48 sage: L.taylor_series(1,4) 0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + O(z^4) sage: L.check_functional_equation() 6.11218974700000e-18 # 32-bit 6.04442711160669e-18 # 64-bit
RANK 2 ELLIPTIC CURVE:
We compute the leading coefficient and Taylor expansion of the `L`-series of a rank `2` elliptic curve. ::
sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser() sage: L.num_coeffs() 156 sage: L.derivative(1,E.rank()) 1.51863300057685 sage: L.taylor_series(1,4) -1.27685190980159e-23 + (7.23588070754027e-24)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 32-bit -2.72911738151096e-23 + (1.54658247036311e-23)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 64-bit
NUMBER FIELD:
We compute with the Dedekind zeta function of a number field. ::
sage: x = var('x') sage: K = NumberField(x**4 - x**2 - 1,'a') sage: L = K.zeta_function() sage: L.conductor 400 sage: L.num_coeffs() 264 sage: L(2) 1.10398438736918 sage: L.taylor_series(2,3) 1.10398438736918 - 0.215822638498759*z + 0.279836437522536*z^2 + O(z^3)
RAMANUJAN DELTA L-FUNCTION:
The coefficients are given by Ramanujan's tau function::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
We redefine the default bound on the coefficients: Deligne's estimate on tau(n) is better than the default coefgrow(n)=`(4n)^{11/2}` (by a factor 1024), so re-defining coefgrow() improves efficiency (slightly faster). ::
sage: L.num_coeffs() 12 sage: L.set_coeff_growth('2*n^(11/2)') sage: L.num_coeffs() 11
Now we're ready to evaluate, etc. ::
sage: L(1) 0.0374412812685155 sage: L(1, 1.1) 0.0374412812685155 sage: L.taylor_series(1,3) 0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3) """
# computel.gp script that are replaced by indexed copies # in the computel.gp.template 'computel.gp.template')
poles=[], residues='automatic', prec=53, init=None): """ Initialization of Dokchitser calculator EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.num_coeffs() 4 """
D = copy.copy(self.__dict__) if '_Dokchitser__gp' in D: del D['_Dokchitser__gp'] return reduce_load_dokchitser, (D, )
self.conductor, self.weight)
""" Return the gp interpreter that is used to implement this Dokchitser L-function.
EXAMPLES::
sage: E = EllipticCurve('11a') sage: L = E.lseries().dokchitser() sage: L(2) 0.546048036215014 sage: L.gp() PARI/GP interpreter """
finally:
def _instantiate_gp(cls): logfile=logfile) # Read the script template and parse out all indexed global variables # (easy because they all end in "_$i" and there's nothing else in the # script that uses $)
'([^a-zA-Z0-9_]|^)(%s)([^a-zA-Z0-9_]|$)' % '|'.join(cls.__globals))
# Clean up all global variables created by this instance
""" Call the specified PARI function in the GP interpreter, with the instance number of this `Dokchitser` instance automatically appended.
For example, ``self._gp_call_inst('L', 1)`` is equivalent to ``self.gp().eval('L_N(1)')`` where ``N`` is ``self.__instance``. """
','.join(str(a) for a in args))
""" Like ``_gp_call_inst`` but sets the variable given by ``varname`` to the given value, appending ``_N`` to the variable name.
If ``varname`` is a function (e.g. ``'func(n)'``) then this sets ``func_N(n)``). """
else:
# After init_coeffs is called, future calls to this method should # return the full output for futher parsing raise RuntimeError("Unable to create L-series, due to precision or other limits in PARI.")
raise ValueError("you must call init_coeffs on the L-function first")
""" Return number of coefficients `a_n` that are needed in order to perform most relevant `L`-function computations to the desired precision.
EXAMPLES::
sage: E = EllipticCurve('11a') sage: L = E.lseries().dokchitser() sage: L.num_coeffs() 26 sage: E = EllipticCurve('5077a') sage: L = E.lseries().dokchitser() sage: L.num_coeffs() 568 sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.num_coeffs() 4
Verify that ``num_coeffs`` works with non-real spectral parameters, e.g. for the L-function of the level 10 Maass form with eigenvalue 2.7341055592527126::
sage: ev = 2.7341055592527126 sage: L = Dokchitser(conductor=10, gammaV=[ev*i, -ev*i],weight=2,eps=1) sage: L.num_coeffs() 26 """
w=None, pari_precode='', max_imaginary_part=0, max_asymp_coeffs=40): """ Set the coefficients `a_n` of the `L`-series.
If `L(s)` is not equal to its dual, pass the coefficients of the dual as the second optional argument.
INPUT:
- ``v`` -- list of complex numbers or string (pari function of k)
- ``cutoff`` -- real number = 1 (default: 1)
- ``w`` -- list of complex numbers or string (pari function of k)
- ``pari_precode`` -- some code to execute in pari before calling initLdata
- ``max_imaginary_part`` -- (default: 0): redefine if you want to compute L(s) for s having large imaginary part,
- ``max_asymp_coeffs`` -- (default: 40): at most this many terms are generated in asymptotic series for phi(t) and G(s,t).
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode)
Evaluate the resulting L-function at a point, and compare with the answer that one gets "by definition" (of L-function attached to a modular form)::
sage: L(14) 0.998583063162746 sage: a = delta_qexp(1000) sage: sum(a[n]/float(n)^14 for n in range(1,1000)) 0.9985830631627459
Illustrate that one can give a list of complex numbers for v (see :trac:`10937`)::
sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: L2.init_coeffs(list(delta_qexp(1000))[1:]) sage: L2(14) 0.998583063162746
TESTS:
Verify that setting the `w` parameter does not raise an error (see :trac:`10937`). Note that the meaning of `w` does not seem to be documented anywhere in Dokchitser's package yet, so there is no claim that the example below is meaningful! ::
sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: L2.init_coeffs(list(delta_qexp(1000))[1:], w=[1..1000]) """ v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs = v
# Must have __globals_re instantiated
m.group(3))
# If any of the pre-code contains references to some of the # templated global variables we must replace those as well
else: raise TypeError("v (=%s) must be a list, tuple, or string" % v) else: else: '"Bvec[k]"') max_asymp_coeffs)
del self.__values
r""" INPUT:
- ``s`` -- complex number
.. NOTE::
Evaluation of the function takes a long time, so each evaluation is cached. Call ``self._clear_value_cache()`` to clear the evaluation cache.
EXAMPLES::
sage: E = EllipticCurve('5077a') sage: L = E.lseries().dokchitser(100) sage: L(1) 0.00000000000000000000000000000 sage: L(1+I) -1.3085436607849493358323930438 + 0.81298000036784359634835412129*I """ print(z) raise RuntimeError
r""" Return the `k`-th derivative of the `L`-series at `s`.
.. WARNING::
If `k` is greater than the order of vanishing of `L` at `s` you may get nonsense.
EXAMPLES::
sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser() sage: L.derivative(1,E.rank()) 1.51863300057685 """ raise ArithmeticError(z) i = z.rfind('\n') msg = z[:i].replace('digits', 'decimal digits') verbose(msg, level=-1) return self.__CC(z[i:])
r""" Return the first `k` terms of the Taylor series expansion of the `L`-series about `a`.
This is returned as 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 (default: 0); point about which to expand
- ``k`` -- integer (default: 6), series is `O(``var``^k)`
- ``var`` -- string (default: 'z'), variable of power series
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.taylor_series(2, 3) 1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 + O(z^3) sage: E = EllipticCurve('37a') sage: L = E.lseries().dokchitser() sage: L.taylor_series(1) 0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6)
We compute a Taylor series where each coefficient is to high precision. ::
sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser(200) sage: L.taylor_series(1,3) -9.094...e-82 + (5.1538...e-82)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3) """ except TypeError as msg: raise RuntimeError("%s\nUnable to compute Taylor expansion (try lowering the number of terms)" % msg) raise ArithmeticError(r) i = r.rfind('\n') msg = r[:i].replace('digits', 'decimal digits') verbose(msg, level=-1)
r""" Verifies how well numerically the functional equation is satisfied, and also determines the residues if ``self.poles != []`` and residues='automatic'.
More specifically: for `T>1` (default 1.2), ``self.check_functional_equation(T)`` should ideally return 0 (to the current precision).
- if what this function returns does not look like 0 at all, probably the functional equation is wrong (i.e. some of the parameters gammaV, conductor etc., or the coefficients are wrong),
- if checkfeq(T) is to be used, more coefficients have to be generated (approximately T times more), e.g. call cflength(1.3), initLdata("a(k)",1.3), checkfeq(1.3)
- T=1 always (!) returns 0, so T has to be away from 1
- default value `T=1.2` seems to give a reasonable balance
- if you don't have to verify the functional equation or the L-values, call num_coeffs(1) and initLdata("a(k)",1), you need slightly less coefficients.
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.check_functional_equation() -1.35525271600000e-20 # 32-bit -2.71050543121376e-20 # 64-bit
If we choose the sign in functional equation for the `\zeta` function incorrectly, the functional equation doesn't check out. ::
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=-11, poles=[1], residues=[-1], init='1') sage: L.check_functional_equation() -9.73967861488124 """
r""" You might have to redefine the coefficient growth function if the `a_n` of the `L`-series are not given by the following PARI function::
coefgrow(n) = if(length(Lpoles), 1.5*n^(vecmax(real(Lpoles))-1), sqrt(4*n)^(weight-1));
INPUT:
- ``coefgrow`` -- string that evaluates to a PARI function of n that defines a coefgrow function.
EXAMPLES::
sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode) sage: L.set_coeff_growth('2*n^(11/2)') sage: L(1) 0.0374412812685155 """ raise TypeError("coefgrow must be a string")
X = Dokchitser(1, 1, 1, 1) X.__dict__ = D X.init_coeffs(X._Dokchitser__init) return X |