At the primes
of bad reduction, we study the component
group
of the Neron model of
. The orders of the
groups finite abelian groups
are called the Tamagawa
numbers
. The component group of a representative abelian
variety in a class
is an isomorphism invariant, which can
likewise vary within the isogeny class.

When
(exactly divides) there is an algorithm to compute
(see [KS00,CS01]; and it is implemented in MAGMA). This
algorithm can also be used to compute the Tamagawa number
up to a power of
.

Problem 8.3.1
Find an algorithm to compute
when
.

There are standard bounds due to Oort, Lenstra, Lorenzini, etc.,
on the component group at primes
for which
. These
are implemented in MAGMA.

Problem 8.3.2
Find an algorithm to compute
when
.
I.e., remove that we currently only know how to compute
up to a power of
.

Problem 8.3.3
Implement in SAGE the Conrad-Kohel-Stein algorithm to compute
.