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Is there a connexion between :

1) The monodromy group of a topological space.

2) The $\ell$-adic monodromy theorem of Grothendieck.

3) The $p$-adic monodromy conjecture of Fontaine (which is now proved).

I am mainly interested in the link between 2) and 3).

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    $\begingroup$ Yes. Roughly speaking the monodromy of a topological space was the motivation behind $\ell$-adic monodromy. I assume $p$-adic monodromy just means $\ell$-adic when $\ell$ is the characteristic of the field. I wish I had more time right now, because explaining this would be a good exercise for me. $\endgroup$
    – Matt
    May 22, 2012 at 16:23
  • $\begingroup$ @Matt: Dear Matt, The $p$-adic monodromy conjecture of Fontaine is something much more specific: that de Rham representations of Galois groups of $p$-adic fields are necessarily potentially semistable. Regards, $\endgroup$
    – Matt E
    Aug 7, 2012 at 2:50

1 Answer 1

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A topological space does not have a monodromy group (unless someone is abusing terminology). It has a fundamental group (more precisely, once we fix a base-point, it has a fundamental group relative to that base-point).

If $f:X \to S$ is a fibre bundle, then the cohomology spaces of the fibres of $X$ (say with $\mathbb Q$ coefficients, just to fix ideas, although any other coefficients would be okay too; and in some fixed degree $i$) glue together to form a local system over $S$ (i.e., a locally constant sheaf of $\mathbb Q$-vector spaces), which (once we fix a base-point $s$), we can identify with a representation of $\pi_1(S,s)$; indeed, the representation is on the vector space $H^i(X_s, \mathbb Q),$ where $X_s := f^{-1}(s)$ is the fibre over $s$.

Intuitively, if $c$ is a cohomology class on $X_s$, and $\gamma$ is a loop based at $s$, then you can move $c$ through the fibres $X_{s'}$ as $s'$ moves along $\gamma$, until you get back to $X_s$.

To understand this, you will need to think about examples. A good one to start with is the fibre bundle $S^2 \to \mathbb R P^2$, taking $i = 0$, so that $H^0(X_s)$ is just the $\mathbb Q$-vector space of dimension $2$ spanned by the two points of $S^2$ lying over a point $s \in \mathbb R P^2$.

A harder example, but more directly relevant to algebraic geometry, is the Legendre family of elliptic curves $y^2 := x(x-1)(x-\lambda)$ (I mean the projective curves, although following tradition I am just writing down the affine equations) parameterized by $\lambda \in S = \mathbb C P^1 \setminus \{0,1,\infty\}.$

Here the interesting case is $i = 1$, i.e. the family of $H^1$'s of the fibres.


Ehresmann's theorem says that any smooth proper map of varieties $f: X \to S$ over $\mathbb C$ is topologically a fibre bundle, so this gives lots of examples of monodromy arising from algebraic geometry.

If the base $S$ is an algebraic curve, and $D^{\times}$ is any copy of the punctured disk sitting inside $S$ (you should think of $S$ as being a punctured Riemann surface, like the above example of $\mathbb C P^1 \setminus \{0,1,\infty\}$, and $D^{\times}$ as being a neighbourhood of one of the punctures), then you can pull back $X$ to $D^{\times}$, and consider the action of $\pi_1(D^{\times}) \cong \mathbb Z$ on the local system of $H^i$. (This is the local monodromy around the puncture.)

Grothendieck's monodromy theorem says that this local monodromy action is always quasi-unipotent, i.e. some power of the generator of $\pi_1(D^{\times})$ acts unipotently.


There is a variant of all of the above working with $\ell$-adic cohomology in the etale topology rather than usual cohomology in the setting of complex varieties, which makes sense over any ground field.

This leads one to think about $\ell$-adic representations of $p$-adic Galois groups (such as $G_{\mathbb Q_p}$) in geometric terms. In this context, the analogue of Grothendieck's monodromy theorem is that the tame inertia acts quasi-unipotently; this follows from the famous relation $\varphi N = p N \varphi$ (where $\varphi$ is Frobenius and $N$ is the log of a generator of tame inertia). (Note that Grothendieck was able to deduce the monodromy theorem in its original geometric context from this rather easy and general theorem about $\ell$-adic reps. of $p$-adic Galois groups.)


In Fontaine's $p$-adic Hodge theory, the analogue, for a $p$-adic representation of a $p$-adic Galois group, of tame inertia acting quasi-unipotently, is that the $p$-adic representation should be potentially semi-stable. This is not true of all $p$-adic representations, but Fontaine conjectured that it was true for those that are de Rham. This is his monodromy conjecture, now proved by Andre, Kedlaya, Mebkhout, and Berger.

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