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).
-
class
sage.lfunctions.dokchitser.
Dokchitser
(conductor, gammaV, weight, eps, poles=[], residues='automatic', prec=53, init=None)¶ Bases:
sage.structure.sage_object.SageObject
Dokchitser’s \(L\)-functions Calculator
Create a Dokchitser \(L\)-series with
Dokchitser(conductor, gammaV, weight, eps, poles, residues, init, prec)
where
conductor
- integer, the conductorgammaV
- 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 equationpoles
- (default: []) list of points where \(L^*(s)\) has (simple) poles; only poles with \(Re(s)>weight/2\) should be includedresidues
- 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 \(2\) 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
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)
-
check_functional_equation
(T=1.2)¶ 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
-
derivative
(s, k=1)¶ 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
-
gp
()¶ 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
-
init_coeffs
(v, cutoff=1, 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 initLdatamax_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 ticket #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 ticket #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])
-
num_coeffs
(T=1)¶ 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
-
set_coeff_growth
(coefgrow)¶ 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.
EXAMPLE:
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
-
taylor_series
(a=0, k=6, var='z')¶ 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 viewvar
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 expandk
- 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)
-
sage.lfunctions.dokchitser.
reduce_load_dokchitser
(D)¶