Component Groups

At the primes $p \mid N$ of bad reduction, we study the component group $\Phi_p$ of the Neron model of $A/\mathbb{Z}_p$ . The orders of the groups finite abelian groups $\Phi_p(\mathbb{F}_p)$ are called the Tamagawa numbers $c_p$ . The component group of a representative abelian variety in a class $[A]$ is an isomorphism invariant, which can likewise vary within the isogeny class.

When $p\mid\mid N$ (exactly divides) there is an algorithm to compute $\char93 \Phi_p(\overline{\mathbb{F}}_p)$ (see [KS00,CS01]; and it is implemented in MAGMA). This algorithm can also be used to compute the Tamagawa number $c_p = \char93 \Phi_p(\mathbb{F}_p)$ up to a power of $2$ .

Problem 8.3.1   Find an algorithm to compute $\char93 \Phi_p(\overline{\mathbb{F}}_p)$ when $p^2\mid N$ .

There are standard bounds due to Oort, Lenstra, Lorenzini, etc., on the component group at primes $p$ for which $p^2\mid N$ . These are implemented in MAGMA.

Problem 8.3.2   Find an algorithm to compute $\char93 \Phi_p(\mathbb{F}_p)$ when $\mid\mid N$ . I.e., remove that we currently only know how to compute $\char93 \Phi_p(\mathbb{F}_p)$ up to a power of $2$ .

Problem 8.3.3   Implement in SAGE the Conrad-Kohel-Stein algorithm to compute $\char93 \Phi_p(\overline{\mathbb{F}}_p)$ .