"""
`L`-series of modular abelian varieties
At the moment very little functionality is implemented -- this is mostly a
placeholder for future planned work.
AUTHOR:
- William Stein (2007-03)
TESTS::
sage: L = J0(37)[0].padic_lseries(5)
sage: loads(dumps(L)) == L
True
sage: L = J0(37)[0].lseries()
sage: loads(dumps(L)) == L
True
"""
from sage.structure.sage_object import SageObject
from sage.rings.all import Integer
class Lseries(SageObject):
"""
Base class for `L`-series attached to modular abelian varieties.
"""
def __init__(self, abvar):
"""
Called when creating an L-series.
INPUT:
- ``abvar`` -- a modular abelian variety
EXAMPLES::
sage: J0(11).lseries()
Complex L-series attached to Abelian variety J0(11) of dimension 1
sage: J0(11).padic_lseries(7)
7-adic L-series attached to Abelian variety J0(11) of dimension 1
"""
self.__abvar = abvar
def abelian_variety(self):
"""
Return the abelian variety that this `L`-series is attached to.
OUTPUT:
a modular abelian variety
EXAMPLES::
sage: J0(11).padic_lseries(7).abelian_variety()
Abelian variety J0(11) of dimension 1
"""
return self.__abvar
class Lseries_complex(Lseries):
"""
A complex `L`-series attached to a modular abelian variety.
EXAMPLES::
sage: A = J0(37)
sage: A.lseries()
Complex L-series attached to Abelian variety J0(37) of dimension 2
"""
def __call__(self, s):
"""
Evaluate this complex `L`-series at `s`.
INPUT:
- ``s`` -- complex number
OUTPUT:
a complex number L(A, s).
EXAMPLES:
This is not yet implemented::
sage: L = J0(37).lseries()
sage: L(2)
Traceback (most recent call last):
...
NotImplementedError
"""
raise NotImplementedError
def __cmp__(self, other):
"""
Compare this complex `L`-series to another one.
INPUT:
- ``other`` -- object
OUTPUT:
-1, 0, or 1
EXAMPLES::
sage: L = J0(37)[0].lseries(); M = J0(37)[1].lseries()
sage: cmp(L,M)
-1
sage: cmp(L,L)
0
sage: cmp(M,L)
1
"""
if not isinstance(other, Lseries_complex):
return cmp(type(self), type(other))
return cmp(self.abelian_variety(), other.abelian_variety())
def _repr_(self):
"""
String representation of `L`-series.
OUTPUT:
a string
EXAMPLES::
sage: L = J0(37).lseries()
sage: L._repr_()
'Complex L-series attached to Abelian variety J0(37) of dimension 2'
"""
return "Complex L-series attached to %s"%self.abelian_variety()
def rational_part(self):
"""
Return the rational part of this `L`-function at the central critical
value 1.
NOTE: This is not yet implemented.
EXAMPLES::
sage: J0(37).lseries().rational_part()
Traceback (most recent call last):
...
NotImplementedError
"""
raise NotImplementedError
class Lseries_padic(Lseries):
"""
A `p`-adic `L`-series attached to a modular abelian variety.
"""
def __init__(self, abvar, p):
"""
Create a `p`-adic `L`-series.
EXAMPLES::
sage: J0(37)[0].padic_lseries(389)
389-adic L-series attached to Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37)
"""
Lseries.__init__(self, abvar)
p = Integer(p)
if not p.is_prime():
raise ValueError, "p (=%s) must be prime"%p
self.__p = p
def __cmp__(self, other):
"""
Compare this `p`-adic `L`-series to another one.
First the abelian varieties are compared; if they are the same,
then the primes are compared.
INPUT:
other -- object
OUTPUT:
-1, 0, or 1
EXAMPLES::
sage: L = J0(37)[0].padic_lseries(5); M = J0(37)[1].padic_lseries(5)
sage: K = J0(37)[0].padic_lseries(3)
sage: cmp(L,K)
1
sage: cmp(K,L)
-1
sage: K < L
True
sage: cmp(L,M)
-1
sage: cmp(M,L)
1
sage: cmp(L,L)
0
"""
if not isinstance(other, Lseries_padic):
return cmp(type(self), type(other))
return cmp((self.abelian_variety(), self.__p),
(other.abelian_variety(), other.__p))
def prime(self):
"""
Return the prime `p` of this `p`-adic `L`-series.
EXAMPLES::
sage: J0(11).padic_lseries(7).prime()
7
"""
return self.__p
def power_series(self, n=2, prec=5):
"""
Return the `n`-th approximation to this `p`-adic `L`-series as
a power series in `T`. Each coefficient is a `p`-adic number
whose precision is provably correct.
NOTE: This is not yet implemented.
EXAMPLES::
sage: L = J0(37)[0].padic_lseries(5)
sage: L.power_series()
Traceback (most recent call last):
...
NotImplementedError
sage: L.power_series(3,7)
Traceback (most recent call last):
...
NotImplementedError
"""
raise NotImplementedError
def _repr_(self):
"""
String representation of this `p`-adic `L`-series.
EXAMPLES::
sage: L = J0(37)[0].padic_lseries(5)
sage: L._repr_()
'5-adic L-series attached to Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37)'
"""
return "%s-adic L-series attached to %s"%(self.__p, self.abelian_variety())
