Let $X$ be a complex smooth compact curve and $K$ - its field of rational functions. Let $K^{\times}_\mathbb{Q}$ be the multiplicative group of $K$ tensored with $\mathbb{Q}.$ In which ways one can nicely present it via generators and relations?

The question is trivial for $\mathbb{P}^1.$ Let $E$ be a plane elliptic curve. Denote by $l_{x,y}$ a linear function on $\mathbb{P}^2$ passing through points $x$ and $y$ of the curve and by $l_{x}$ - passing throgh $x$ and $-x$. One can show that functions $\dfrac{l_{x,y}}{l_{x+y}}$ generate $K^{\times}_\mathbb{Q}$. They satisfy the following cocycle relation: $\dfrac{l_{x,y}}{l_{x+y}} \dfrac{l_{x+y,z}}{l_{x+y+z}}=\dfrac{l_{y,z}}{l_{y+z}} \dfrac{l_{y+z,x}}{l_{x+y+z}}.$