# What is the idea of a monodromy?

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).

• 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.
– Matt
Commented May 22, 2012 at 16:23
• @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, Commented Aug 7, 2012 at 2:50

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.

• Sorry for bothering, but could you provide a reference for the action of topological fundamental group on the singular cohomology of fibers? Commented Mar 19, 2023 at 3:09
• Can you elaborate a bit/give a reference on the "Grothendieck was able to deduce the monodromy theorem in its original geometric context from this [ℓ-adic representation of p-adic Galois group] rather easy"? Thanks a lot! Oh I actually have a vague idea to compare $$1\to I\to G_k\to \widehat{\mathbb{Z}}\to1$$ with $$1\to\pi_1(X_{k^\mathrm{sep}})\to\pi_1(X)\to\widehat{\mathbb{Z}}\to1$$ I guess... But what role does tameness play here? Commented Mar 7 at 22:26