I've been reading Szamuely's Galois Groups and Fundamental Groups and my question is based on the material in there. Let $X$ be a compact, connected Riemann surface, so that $\mathcal{M}(X)$ is a field. If $Y$ is another compact, connected Riemann surface and $f\colon Y\rightarrow X$ is a non-constant, holomorphic map, then pullback along $f$ induces a map $\mathcal{M}(X)\rightarrow\mathcal{M}(Y)$ that we regard as a field extension. I am interested in the absolute Galois group $\operatorname{Gal}(\overline{\mathcal{M}(X)}/\mathcal{M}(X))$, which is already partially described in the book.
For any cofinite subset $X^{\prime}\subseteq X$, one can consider the composite $K_{X^{\prime}}\subseteq\overline{\mathcal{M}(X)}$ of all extensions thus induced by branched coverings $f\colon Y\rightarrow X$ that are unbranched over $X^{\prime}$. These are Galois extensions and the inverse limit of the system of the $\operatorname{Gal}(K_{X^{\prime}}/\mathcal{M}(X))$ and restrictions is isomorphic to $\operatorname{Gal}(\overline{\mathcal{M}(X)}/\mathcal{M}(X))$. On the other hand, $\operatorname{Gal}(K_{X^{\prime}}/\mathcal{M}(X))$ is isomorphic to the profinite completion $\widehat{\pi_1(X^{\prime},x_0)}$ for any $x_0\in X^{\prime}$ and. If $x_0\in X^{\prime\prime}\subseteq X^{\prime}$, the restriction $\operatorname{Gal}(K_{X^{\prime\prime}}/\mathcal{M}(X))\rightarrow\operatorname{Gal}(K_{X^{\prime}}/\mathcal{M}(X))$ corresponds to the homomorphism $\widehat{\pi_1(X^{\prime\prime},x_0)}\rightarrow\widehat{\pi_1(X^{\prime},x_0)}$ induced by inclusion.
In particular, these two results together imply that the absolute Galois group $\operatorname{Gal}(\overline{\mathcal{M}(X)}/\mathcal{M}(X))$ is a topological invariant of $X$ (up to isomorphism, of course). On the other hand, the classification of surfaces and the fact that complex manifolds are always orientable readily implies that the genus is a complete topological invariant of Riemann surfaces. So, does $\operatorname{Gal}(\overline{\mathcal{M}(X)}/\mathcal{M}(X))$ remember the genus? If not, what does it remember?