At the moment very little functionality is implemented – this is mostly a placeholder for future planned work.
AUTHOR:
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
Base class for -series attached to modular abelian varieties.
Called when creating an L-series.
INPUT:
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
Return the abelian variety that this -series is attached to.
EXAMPLES:
sage: J0(11).padic_lseries(7).abelian_variety()
Abelian variety J0(11) of dimension 1
A complex -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
Evaluate this complex -series at
.
INPUT:
EXAMPLES: This is not yet implemented:
sage: L = J0(37).lseries()
sage: L(2)
...
NotImplementedError
Compare this complex -series to another one.
INPUT:
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
String representation of -series.
EXAMPLES:
sage: L = J0(37).lseries()
sage: L._repr_()
'Complex L-series attached to Abelian variety J0(37) of dimension 2'
Return the rational part of this -function at the central critical
value 1.
NOTE: This is not yet implemented.
EXAMPLES:
sage: J0(37).lseries().rational_part()
...
NotImplementedError
A -adic
-series attached to a modular abelian variety.
Compare this -adic
-series to another one.
First the abelian varieties are compared; if they are the same, then the primes are compared.
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
Create a -adic
-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)
String representation of this -adic
-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 the -th approximation to this
-adic
-series as
a power series in
. Each coefficient is a
-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()
...
NotImplementedError
sage: L.power_series(3,7)
...
NotImplementedError
Return the prime of this
-adic
-series.
EXAMPLES:
sage: J0(11).padic_lseries(7).prime()
7