During a course on Riemann surfaces, the professor mentions and outlines a proof that all compact Riemann surfaces can be embedded in a projective space, which uses Poincare series, running as follows:
Only compact Riemann surfaces $X$ of genus 2 are considered, since the cases of genus 0 and 1 have been dealt with before in class. $X$ has universal cover $\Delta$, the unit disc, and deck transformation group $\Gamma$. We are able to construct holomorphic functions $f_0,f_1,...,f_n$ on $\Delta$ that are in the form of Poincare series, which are never simultaneously zero on $X$, and will give a embedding $[f_0:...:f_n]:X \to P^n$.
Since he will prove the result using line bundles and more algebraic tools, he did not elaborate on this more analytic approach further. However I am quite interested in looking into this proof, I would be happy if anyone can supply a reference directing me towards a proof of the stated result using this approach, since I am unable to locate one myself. Thank you in advance!