Thursday, February 11, 2010

Tamagawa numbers and components groups at additive primes

I had an idea while driving back from snowboarding today that could lead to a nice paper if worked out. The problem is to find an algorithm to compute Tamagawa numbers of optimal newform modular abelian varieties $A$ at primes~$p$ of additive reduction, say quotients of $J_0(N)$ with $p^2\mid N$.

Idea. We have $A^{\vee} \to J_0(N) \to A$ is a morphism with computable degree (basically the modular degree). We can compute the component group of $J_0(N)$ explicitly as a $\T$-module, I think, by work of Raynaud. Using functoriality of the component group, we can thus probably in many cases locate the prime-to-the-modular degree part of the image of the component group of $A$ in $J_0(N)$. Since $p$ is an additive prime, there is a pretty tight bound on the primes that can divide the component group's order, so in some (many?) cases this may already be enough to nail down the component group.

When it isn't we can also look for congruences with lower level, and if there aren't any, then conclude in many cases (when representation is irreducible) by Ribet's theorem that those primes don't divide the component group.

Another thought is that we should be able to use the modular degree and computation of the component group of $J_0(N)$ at $p$ to give a bound on the powers of primes that can divide $c_{A,p}$, which is also new.

I'm sure putting together everything above would (a) yield some very interesting tables, and (b) be easily publishable as a paper.