I want to understand the existence of the Drinfeld shtuka function but unfortunately I know very little in algebraic geometry.
I am reading Shtukas and Jacobi sums from D. Thakur and I am stucked at the following paragraph:
Fix a transcendental point $\xi \in X(M)$. Then evaluation at $\xi$ induces an embedding of $A$ into $M$. By solving the corresponding equation on the Jacobian of $X$, we see that for some divisor $V$, $V^{(1)}-V+(\xi)-(\bar{\infty})$ is principal. A Drinfeld divisor $V$ relative to $\xi$ is defined to be an effective divisor of degree $g$ such that $V^{(1)}-V+(\xi)-(\bar{\infty})$ is principal. From $0.3.1$ (Drinfeld's vanishing lemma) and Riemann-Roch, it follows that Drinfeld divisor is the unique effective divisor in its divisor class. (In particular, there are $h$ such divisors). Hence there exists a unique function $f=f(V)$ with $\text{sgn}(f) = 1$ and such that $(f)=V^{(1)}-V+(\xi)-(\bar{\infty})$. By abuse of terminology, we call $f$ shtuka.
I get that there exists some $V$ for which the divisor is principal. This comes from the isomorphism between $\text{Pic}$ and $\text{Jac}$ and the surjectivity of $I_d-\text{Frob}$ I guess. But I do not know how to combine Riemann-Roch and Drinfeld's vanishing lemma to obtain that $V$ is effective and further of degree $g$. I am super interested in a "simple" argument for that.
Many thanks!
