Geometric Intuition for Big Monodromy - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:22:51Z http://mathoverflow.net/feeds/question/14565 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14565/geometric-intuition-for-big-monodromy Geometric Intuition for Big Monodromy Brandon Levin 2010-02-07T22:35:07Z 2010-02-12T13:01:11Z <p>In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A general setup for such a result in algebraic geometry is:</p> <p>Given a proper, generically smooth map $\pi:X \rightarrow S$ of relative dimension d, say S is connected. This gives rise to an $l$-adic representations of the etale fundamental group $\pi_1(U)$ where $U$ is smooth locus of $\pi$ corresponding to higher pushforward $R^d \pi_* Q_l$. One might say it has "big monodromy" if the Zariski closure of the image is as big as it can be given that it has to respect cup-product, etc.</p> <p>My specific question is what are the geometric consequences of big monodromy? If we know such a result for $\pi$, what does that say about the geometry of the fibration or at the very least is there geometric intuition for what it should mean?</p> <p>I welcome intuition from number theory, algebraic geometry, or complex geometry.</p> <p>I have also heard that "one should expect big monodromy unless there is a reason not to" (for example, complex multiplication). What are other examples of things which inhibit big monodromy? </p> http://mathoverflow.net/questions/14565/geometric-intuition-for-big-monodromy/15102#15102 Answer by JSE for Geometric Intuition for Big Monodromy JSE 2010-02-12T13:01:11Z 2010-02-12T13:01:11Z <p>I found this question a little vague, but let me at least remark on "other examples of things which inhibit big monodromy." Mumford gives an example in section 4 of </p> <p>D. Mumford, “A note of Shimura’s paper “Discontinuous groups and abelian varieties”,” Math. Ann. 181 (1969), 345–351.</p> <p>of an abelian variety A whose Galois representation has image strictly smaller than Sp_{2g}(Z_p), despite the fact that End(A) = Z. The keyword to look up is "Mumford-Tate group", which is in some sense the answer to the question </p> <p>How big COULD the Galois representation on an abelian variety be, subject to all geometric 'things which inhibit big monodromy'?</p> <p>Reference comes from <a href="http://w3.uwyo.edu/~chall14/papers/open%5Fimage.pdf" rel="nofollow">a paper of Chris Hall</a> which shows how to prove big monodromy results in many cases.</p>