\(L\)-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)

Bases: sage.structure.sage_object.SageObject

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

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)

Bases: sage.modular.abvar.lseries.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

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()
Traceback (most recent call last):
...
NotImplementedError

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

Bases: sage.modular.abvar.lseries.Lseries

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

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()
Traceback (most recent call last):
...
NotImplementedError
sage: L.power_series(3,7)
Traceback (most recent call last):
...
NotImplementedError

prime()

Return the prime \(p\) of this \(p\)-adic \(L\)-series.

EXAMPLES:

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