class GrossZagierLseries(SageObject):
def __init__(self, E, A, prec=53):
r"""
Class for the Gross-Zagier L-series.
This is attached to a pair `(E,A)` where `E` is an elliptic curve over
`\QQ` and `A` is an ideal class in an imaginary quadratic number field.
For the exact definition, in the more general setting of modular forms
instead of elliptic curves, see section IV of [GrossZagier]_.
INPUT:
- ``E`` -- an elliptic curve over `\QQ`
- ``A`` -- an ideal class in an imaginary quadratic number field
- ``prec`` -- an integer (default 53) giving the required precision
EXAMPLES::
sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries
sage: G = GrossZagierLseries(e, A)
TESTS::
sage: K.<b> = QuadraticField(131)
sage: A = K.class_group().one()
sage: G = GrossZagierLseries(e, A)
Traceback (most recent call last):
...
ValueError: A is not an ideal class in an imaginary quadratic field
"""
self._E = E
self._N = N = E.conductor()
self._A = A
ideal = A.ideal()
K = A.gens()[0].parent()
D = K.disc()
if not(K.degree() == 2 and D < 0):
raise ValueError("A is not an ideal class in an"
" imaginary quadratic field")
Q = ideal.quadratic_form().reduced_form()
epsilon = - kronecker_character(D)(N)
self._dokchister = Dokchitser(N ** 2 * D ** 2,
[0, 0, 1, 1],
weight=2, eps=epsilon, prec=prec)
self._nterms = nterms = Integer(self._dokchister.num_coeffs())
if nterms > 1e6:
# just takes way to long
raise ValueError("Too many terms: {}".format(nterms))
zeta_ord = ideal.number_field().zeta_order()
an_list = gross_zagier_L_series(E.anlist(nterms + 1), Q, N, zeta_ord)
self._dokchister.gp().set('a', an_list[1:])
self._dokchister.init_coeffs('a[k]', 1)
def __call__(self, s, der=0):
r"""
Return the value at `s`.
INPUT:
- `s` -- complex number
- ``der`` -- ? (default 0)
EXAMPLES::
sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries
sage: G = GrossZagierLseries(e, A)
sage: G(3)
-0.272946890617590
"""
return self._dokchister(s, der)
def taylor_series(self, s=1, series_prec=6, var='z'):
r"""
Return the Taylor series at `s`.
INPUT:
- `s` -- complex number (default 1)
- ``series_prec`` -- number of terms (default 6) in the Taylor series
- ``var`` -- variable (default 'z')
EXAMPLES::
sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries
sage: G = GrossZagierLseries(e, A)
sage: G.taylor_series(2,3)
-0.613002046122894 + 0.490374999263514*z - 0.122903033710382*z^2 + O(z^3)
"""
return self._dokchister.taylor_series(s, series_prec, var)
def _repr_(self):
"""
Return the string representation.
EXAMPLES::
sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries
sage: GrossZagierLseries(e, A)
Gross Zagier L-series attached to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field with ideal class Fractional ideal class (2, 1/2*a)
"""
msg = "Gross Zagier L-series attached to {} with ideal class {}"
return msg.format(self._E, self._A)
