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
class sage.modular.abvar.lseries.Lseries(abvar)

Base class for L-series attached to modular abelian varieties.

__init__(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
__weakref__
list of weak references to the object (if defined)
abelian_variety()

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
class sage.modular.abvar.lseries.Lseries_complex(abvar)

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
__call__(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)
...
NotImplementedError
__cmp__(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
_repr_()

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'
rational_part()

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()
...
NotImplementedError
class sage.modular.abvar.lseries.Lseries_padic(abvar, p)

A p-adic L-series attached to a modular abelian variety.

__cmp__(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
__init__(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)
_repr_()

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)'
power_series(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()
...
NotImplementedError
sage: L.power_series(3,7)
...
NotImplementedError
prime()

Return the prime p of this p-adic L-series.

EXAMPLES:

sage: J0(11).padic_lseries(7).prime()
7

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