# Function field of a curve as a multiplicative group

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: $\dfrac{l_{x,y}}{l_{x+y}} \dfrac{l_{x+y,z}}{l_{x+y+z}}=\dfrac{l_{y,z}}{l_{y+z}} \dfrac{l_{y+z,x}}{l_{x+y+z}}.$

• What do you need besides the standard exact sequence $0 \to \mathbb{C}^{\times} \to K^{\times} \to Div^0(X) \to Pic^0(X) \to 0$ where $Div^0(X)$ is the subgroup of the free abelian group on the points of $X$ with coefficients adding to zero (and is itself a free abelian group) and $Pic^0(X)$ is defined as the quotient and is an abelian variety and, for an elliptic curve $E$, is $E$ itself? – Felipe Voloch Oct 2 '16 at 18:44