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using pari for elliptic and Eisenstein L-series #40465

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a6fe61d
using pari for elliptic and Eisenstein L-series
fchapoton Jul 21, 2025
20529bd
Merge branch 'develop' into Eisenstein_L
fchapoton Aug 26, 2025
af8a5e5
add tolerance for one doctest
fchapoton Aug 26, 2025
abe56d2
Merge branch 'develop' into Eisenstein_L
fchapoton Aug 27, 2025
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57 changes: 8 additions & 49 deletions src/sage/categories/number_fields.py
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Original file line number Diff line number Diff line change
Expand Up @@ -125,7 +125,7 @@ def _call_(self, x):
class ParentMethods:
def zeta_function(self, prec=53,
max_imaginary_part=0,
max_asymp_coeffs=40, algorithm='pari'):
algorithm='pari'):
r"""
Return the Dedekind zeta function of this number field.

Expand All @@ -138,17 +138,11 @@ def zeta_function(self, prec=53,

- ``max_imaginary_part`` -- real (default: 0)

- ``max_asymp_coeffs`` -- integer (default: 40)

- ``algorithm`` -- (default: ``'pari'``) either ``'gp'`` or
``'pari'``
- ``algorithm`` -- ignored

OUTPUT: the zeta function of this number field

If algorithm is ``'gp'``, this returns an interface to Tim
Dokchitser's gp script for computing with `L`-functions.

If algorithm is ``'pari'``, this returns instead an interface to Pari's
This returns an interface to Pari's
own general implementation of `L`-functions.

EXAMPLES::
Expand All @@ -166,51 +160,16 @@ def zeta_function(self, prec=53,
sage: Z(5) # needs sage.rings.number_field sage.symbolic
1.00199015670185

Using the algorithm "pari"::

sage: K.<a> = NumberField(ZZ['x'].0^2 + ZZ['x'].0 - 1) # needs sage.rings.number_field
sage: Z = K.zeta_function(algorithm='pari') # needs sage.rings.number_field sage.symbolic
sage: Z(-1) # needs sage.rings.number_field sage.symbolic
0.0333333333333333

sage: x = polygen(QQ, 'x')
sage: L.<a, b, c> = NumberField([x^2 - 5, x^2 + 3, x^2 + 1]) # needs sage.rings.number_field
sage: Z = L.zeta_function(algorithm='pari') # needs sage.rings.number_field sage.symbolic
sage: Z(5) # needs sage.rings.number_field sage.symbolic
1.00199015670185

TESTS::

sage: QQ.zeta_function() # needs sage.symbolic
PARI zeta function associated to Rational Field
"""
if algorithm == 'gp':
from sage.lfunctions.dokchitser import Dokchitser
r1, r2 = self.signature()
zero = [0]
one = [1]
Z = Dokchitser(conductor=abs(self.absolute_discriminant()),
gammaV=(r1 + r2) * zero + r2 * one,
weight=1,
eps=1,
poles=[1],
prec=prec)
s = 'nf = nfinit(%s);' % self.absolute_polynomial()
s += 'dzk = dirzetak(nf,cflength());'
Z.init_coeffs('dzk[k]', pari_precode=s,
max_imaginary_part=max_imaginary_part,
max_asymp_coeffs=max_asymp_coeffs)
Z.check_functional_equation()
Z.rename('Dokchitser Zeta function associated to %s' % self)
return Z

if algorithm == 'pari':
from sage.lfunctions.pari import lfun_number_field, LFunction
Z = LFunction(lfun_number_field(self), prec=prec)
Z.rename('PARI zeta function associated to %s' % self)
return Z

raise ValueError('algorithm must be "gp" or "pari"')
from sage.lfunctions.pari import lfun_number_field, LFunction
Z = LFunction(lfun_number_field(self), prec=prec,
max_im=max_imaginary_part)
Z.rename(f'PARI zeta function associated to {self}')
return Z

def _test_absolute_disc(self, **options):
r"""
Expand Down
53 changes: 17 additions & 36 deletions src/sage/lfunctions/dokchitser.py
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -100,19 +100,19 @@ class Dokchitser(SageObject):
We compute with the `L`-series of a rank `1` curve. ::

sage: E = EllipticCurve('37a')
sage: L = E.lseries().dokchitser(algorithm='gp'); L
Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: L = E.lseries().dokchitser(algorithm='pari'); L
PARI 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
50
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() # abs tol 1e-19
sage: L.check_functional_equation() # abs tol 1e-17
6.04442711160669e-18

RANK 2 ELLIPTIC CURVE:
Expand All @@ -121,14 +121,13 @@ class Dokchitser(SageObject):
`L`-series of a rank `2` elliptic curve. ::

sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L = E.lseries().dokchitser(algorithm='pari')
sage: L.num_coeffs()
156
163
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
...e-19 + (...e-19)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4)

NUMBER FIELD:

Expand All @@ -140,7 +139,7 @@ class Dokchitser(SageObject):
sage: L.conductor
400
sage: L.num_coeffs()
264
313
sage: L(2)
1.10398438736918
sage: L.taylor_series(2,3)
Expand Down Expand Up @@ -235,15 +234,6 @@ def gp(self):
"""
Return the gp interpreter that is used to implement this Dokchitser
`L`-function.

EXAMPLES::

sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L(2)
0.546048036215014
sage: L.gp()
PARI/GP interpreter
"""
if self.__gp is None:
self._instantiate_gp()
Expand Down Expand Up @@ -354,13 +344,13 @@ def num_coeffs(self, T=1):
EXAMPLES::

sage: E = EllipticCurve('11a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L = E.lseries().dokchitser(algorithm='pari')
sage: L.num_coeffs()
26
27
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L = E.lseries().dokchitser(algorithm='pari')
sage: L.num_coeffs()
568
591
sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1')
sage: L.num_coeffs()
4
Expand Down Expand Up @@ -501,20 +491,11 @@ def __call__(self, s, c=None):
EXAMPLES::

sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser(100, algorithm='gp')
sage: L = E.lseries().dokchitser(100, algorithm='pari')
sage: L(1)
0.00000000000000000000000000000
sage: L(1+I)
-1.3085436607849493358323930438 + 0.81298000036784359634835412129*I
sage: L(1+I, 1.2)
-1.3085436607849493358323930438 + 0.81298000036784359634835412129*I

TESTS::

sage: L(1+I, 0)
Traceback (most recent call last):
...
RuntimeError
"""
self.__check_init()
s = self.__CC(s)
Expand Down Expand Up @@ -558,7 +539,7 @@ def derivative(self, s, k=1):
EXAMPLES::

sage: E = EllipticCurve('389a')
sage: L = E.lseries().dokchitser(algorithm='gp')
sage: L = E.lseries().dokchitser(algorithm='pari')
sage: L.derivative(1,E.rank())
1.51863300057685
"""
Expand Down Expand Up @@ -599,17 +580,17 @@ def taylor_series(self, a=0, k=6, var='z'):
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(algorithm='gp')
sage: L = E.lseries().dokchitser(algorithm='pari')
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, algorithm='gp')
sage: L = E.lseries().dokchitser(200, algorithm='pari')
sage: L.taylor_series(1,3)
...e-82 + (...e-82)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3)
...e-63 + (...e-63)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3)

Check that :issue:`25402` is fixed::

Expand Down
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