Rigid Uniformization vs Grothendieck's Local Monodromy Theory

2012-11-13T02:42:42Z

I've noticed that some interesting results about abelian varieties can each be proven using one of two ways: the theory of rigid uniformization of abelian varieties or Grothendieck's local monodromy theory of SGA 7 1. I'm therefore wondering about possible interconnections between the two.

Detailed Motivation:

The first example of a result referred to above is the fact that CM abelian varieties have (potentially) good reduction everywhere. The proof using rigid uniformization is discussed in Silverman, Advanced Topics in the Arithmetic of Elliptic Curves (ATAoEC) in Chapter V. ATAoEC also gives a proof in Chapter II Section 6 using local class field theory, Neron-Ogg-Shafarevich, and the fact that a pro-$p$ group can only map trivially into a pro-$\ell$ group. I consider this latter proof to be part of Grothendieck's local monodromy theory, as one uses a similar method to prove the local monodromy theorem (at least, as demonstrated to me in Nicholas Katz's course at Princeton this fall; the original should be in the elusive SGA 7 1 Exposé III).

The next example is the following. SGA 7 1 Exposé IX proves that if $A/K$ has semistable reduction over a local field $K$ with inertia group $I$ with dimension $g$ and toric dimension $\mu$, then $T_\ell(A)^f := T_\ell(A)^I$ has rank $2g-\mu$, and $I$ acts trivially on the quotient as well. Furthermore, it has a complement under the Weil pairing (for a fixed polarisation), denoted $T_\ell(A)^t$, of rank $\mu$. See 2.2.5, 2.4, 2.5.4, and 3.5 of the Exposé notes.

Of course, one can prove the same result using rigid uniformization, where $T_\ell(A)^t$ corresponds to the $\ell^n$th roots of unity in $\bar{K}^*$. See Ribet, Galois Action on Division Points of Abelian Varieties with Real Multiplications, Section III, or these notes by Mihran Papikian.

Specific Question: Why do these two theories seem to prove the same results?

This would make sense if I saw similar arguments being used to develop both theories. But I don't see how analyzing the inertia using local class field theory and then looking at profinite groups is the same as writing down $p$-adic power series. While they both have a $p$-adic and $\ell$-adic "flavor," they seem to be very different proofs.

However, please tell me if I'm wrong - could it be that one can trace the arguments developing each theory to find a common thread?

More specifically, can one prove in general that if one can prove a result with one theory, then one can do it with the other? Is one theory strictly stronger than the other? Is there a common generalization?

Rigid Uniformization vs Grothendieck's Local Monodromy Theory - MathOverflow

Rigid Uniformization vs Grothendieck's Local Monodromy Theory