# Rigid Uniformization vs Grothendieck's Local Monodromy Theory

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?

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The generalization from elliptic curves to abelian varieties involves the dual abelian variety in an essential way that you appear to have completely missed (e.g., your statement of the orthogonality theorem of SGA7 is wrong). This causes you to overlook serious aspects such as that in case of semistable reduction, the toric parts of the reductions have canonically dual geometric character lattices; this fact lies far deeper than anything about the monodromy operator and is made vivid in rigid uniformization. I recommend more experience with abelian varieties before asking such questions – user27056 Nov 13 '12 at 3:54
@Will: What exactly is the meaning of your statement #1? There are statements seen via rigid uniformization for which there isn't any evident means of proof in terms of monodromy formalism; e.g., the canonical duality between character lattices of toric parts in the reductions (as noted in my comment). In particular, the rigid uniformization is so much richer than the monodromy operator (which only records inertial action, and so can be extracted from the rigid uniformization). – user27056 Nov 13 '12 at 4:02
@Davidac897: It is not true that "if one fixes a polarization" then the role of the dual abelian variety can be suppressed. Plenty of abelian varieties do not admit a polarization with degree coprime to $\ell$, and in such cases you're not going to get the orthogonality theorem integrally (i.e., without inverting $\ell$) in the way you wish (with the dual hidden behind the polarization). The dual AV is a key ingredient in the proof of the orthogonality theorem. If you acquire more experience with abelian varieties then you will be in better position to understand the ideas in Expose IX. – user27056 Nov 13 '12 at 4:12
@Davidac897: In situations where both viewpoints prove a common result it is reasonable to ask for a unified perspective, but the final paragraph of your question and the "Specific Question" go beyond this, into a realm that is disproved by the duality aspect with toric parts of the reductions. The rigid uniformization subsumes the monodromy operator. (As an aside, IMHO even for AV's admitting a principal polarization, it is a conceptual error to think about the orthogonality theorem without always keeping in mind the role of the dual, just like for Weil pairings in higher dimensions.) – user27056 Nov 13 '12 at 4:57
@xbnv: $R^1 \pi_* \mathbb Q_l$, etale category. I'm interested in seeing the difference between the monodromy and rigid-analytic picture. The specific claim I made was that any sheaf satisfying some natural set of conditions, come from a rigid-analytic group in this way. Thus reasoning sheaf-theoretically from those conditions and reasoning with rigid analysis would be equivalent, at least for solving problems one can state in terms of that sheaf. – Will Sawin Nov 13 '12 at 6:25