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Let $\mathbf C(t)$ be the field of rational functions and let $\overline{\mathbf C(t)}$ be an algebraic closure. Let $G$ be the Galois group of $\overline {\mathbf C(t)}$ over $\mathbf C(t)$.

Let $\rho:G \to GL_d(\mathbf Q_\ell)$ be a Galois representation.

Is there a useful notion of "unramified" at $x$, where $x$ is a point of $\mathbf P^1_{\mathbf C}$?

I'm trying to mimick the arithmetic situation (replacing $\mathbf C(t)$ by $\mathbf{Q}$ and $x$ by a prime number $p$), but this gives me that $\rho$ is unramified everywhere. Does anybody have any suggestions? I'd really be interested in knowing whether there is a useful notion of ramification in this setting.

Here's what I would like such a notion of ramification to satisfy.

Let $X$ be a smooth projective variety (say of general type) over $\mathbf{C}(t)$ which has a smooth projective model over $\mathbf{A}^1$, but not over $\mathbf{P}^1$. Then, I would like the associated Galois representation (via etale cohomology of $X_{\overline{\mathbf C(T)}}$) to be unramified at all $x$ in $\mathbf{A}^1$ and to be ramified at $\infty$.

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3 Answers 3

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Yes. For $t$ a local coordinate at a pont $P$, choose an embedding $\overline{\mathbb C(t)} \subset \overline {\mathbb C((t))}$ that sends $t$ to $t$. This turns every representation of the absolute Galois group of $\mathbb C(t)$ into a representation of the absolute Galois group of $\mathbb C((t))$. Define the Galois representation to be unramified at $P$ if the second representation is trivial.

Here $\mathbb C((t))$ is playing a role analogous to the maximal unramified extension of the $p$-adic numbers. One could equivalently use the field of fractions of the etale local ring at $P$, which is $\mathbb C((t)) \cap \overline{\mathbb C(t)}$.

To prove that this has the property you want, we use the smooth and proper base change, which together imply that your Galois representation is the generic fiber of a sheaf on $\mathbb P^1$ that is lisse on some open set $U$. But a sheaf is lisse on $U$ exactly when it trivializes modulo $l^n$ on some etale cover of $U$ for each $n$. The generic point of that etale cover is a field extension of $\mathbb C(t)$ which is contained in the field of fractions of the etale local ring of each point in $P \in U$, so it's contained in $\mathbb C((t))$. Since the sheaf is trivial on that generic point, the mod $l^n$ Galois representation is trivial on that field, so it's trivial on $\mathbb C((t))$, so the whole Galois representation is trivial on $\mathbb C((t))$.

Note that there are no Galois representations unramified on $\mathbb A^1$ but ramified at the last point, because there are no covers unramified on $\mathbb A^1$ but ramified at the last point. Indeed, you will rarely find a family that is bad at just one point of $\mathbb P^1$. But if you removed two or three points, your criterion would make perfect sense.

Of course, you will never know for certainty it will be ramified at the bad point - e.g., $H^0$ won't be.

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The standard definition of "unramified" applies to the function field case, so any finite branched cover of the projective line yields a Galois representation that is ramified in finitely many places. In particular, a representation of the absolute Galois group of $\mathbb{C}(t)$ that is unramified on $\mathbb{A}^1$ is just a representation of the fundamental group of $\mathbb{A}^1_{\mathbb{C}}$, i.e., the trivial group.

However, there is a way to get a nontrivial object by following a conjectural dictionary between Galois representations on characteristic $p$ varieties and $D$-modules on complex varieties. In this dictionary, wild ramification (which doesn't occur in characteristic zero) corresponds to irregular singularities (which don't appear in the usual Riemann-Hilbert correspondence). In particular, the Artin-Schreier sheaf on a characteristic $p$ line is tied to the $D$-module on the complex line whose global solutions are exponential functions $k e^z$. They are similar in the sense that the Artin-Schreier representation is unramified away from infinity but has wild ramification there, and $e^z$ is entire but has an essential singularity at infinity.

In conclusion, your smooth projective variety over $\mathbb{C}(t)$ may yield something interesting if you do some kind of $D$-module pushforward instead of looking at the Galois representation.

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What does a dictionary mean in this context? –  Will Sawin Dec 20 '12 at 3:59
    
A dictionary is a set of phenomenological analogies with some unifying theme. It can be useful for transferring conjectures to new domains. –  S. Carnahan Dec 20 '12 at 5:44
    
Examples include Weil's zeta function dictionary for curves, Grothendieck's fonctions-faisceaux dictionary, Vojta's dictionary between diophantine approximation and Nevanlinna theory, and lots of dualities in theoretical physics. –  S. Carnahan Dec 20 '12 at 7:45

I think the points of ramification are the points around which there is monodromy. A point $x$ is ramified if the element of the Galois group corresponding to a loop around $x$ has a non-trivial image under $\rho$. In your example, if $X_t$ represents the fiber above a point $t$ in the affine line and you start at $t_0$ and go around a loop (around infinity) paying attention how a basis for the cohomology of $X_{t_0}$ changes as you analytically continue it, and when you get back to $t_0$ the basis changed, then infinity is ramified.

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