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Let $C$ be compact Riemann surface of high genus. Let $p$ be a general point on $C$ and $z$ a local coordinate around $p$.

Given a meromorphic function on $C$, regular outside $p$, we can look at its Laurant expansion in $z$. Let us call $A(C)$ the algebra of these Laurant expansions.

At least if one knows the genus, it is possible to reconstruct $C$ out of $A(C)$. Indeed, take $n$ big enough e.g. bigger than $2g+1$, let $V$ the vector subspace of $A(C)$ of functions with poles of order at most $n$, then $C$ embeds into $\mathbb{P}V$ via the line bundle $\mathcal{O}_C(nP)$, and one can read off the relations among sections inside $A(C)$.

Is it possible to characterize the subalgebras of $\mathbb{C}[[z]][z^{-1}]$ which are of the form $A(C)$ for some $C$?

Is it possible to extract "easily" information about $C$, such as the genus or if $C$ is hyperelliptic, out of $A(C)$?

More generally, do you have any reference, key-word or standard name for the algebra $A(C)$?

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    $\begingroup$ $A(C)$ is isomorphic to the algebra of algebraic functions on the affine algebraic curve $C\smallsetminus p$. In particular $C\smallsetminus p=\operatorname{Spec} A(C) $, and $C$ is its natural compactification. $\endgroup$
    – abx
    Commented Jun 25, 2019 at 11:27
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    $\begingroup$ Sort of related. You might be interested in Weierstrass gap sequences. They have been studied and they tell you about the existence of functions with prescribed order of poles at only p. $\endgroup$
    – meh
    Commented Jun 25, 2019 at 14:48

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