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 $\infty$.



Useful notion of unramified Galois representation
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.


