Rigid Uniformization vs Grothendieck's Local Monodromy Theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:53:03Z http://mathoverflow.net/feeds/question/112245 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112245/rigid-uniformization-vs-grothendiecks-local-monodromy-theory Rigid Uniformization vs Grothendieck's Local Monodromy Theory David Corwin 2012-11-13T02:42:42Z 2012-11-29T06:22:00Z <p>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.</p> <p><strong>Detailed Motivation</strong>:</p> <p>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, <em>Advanced Topics in the Arithmetic of Elliptic Curves</em> (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).</p> <p>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.</p> <p>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, <em>Galois Action on Division Points of Abelian Varieties with Real Multiplications</em>, Section III, or <a href="http://math.stanford.edu/~vakil/snowbird/mihranjun21.pdf" rel="nofollow">these notes</a> by Mihran Papikian.</p> <p><strong>Specific Question</strong>: Why do these two theories seem to prove the same results? </p> <p>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.</p> <p>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?</p> <p>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?</p>