My past paper question is,
Let X be a smooth projective curve of genus g over the complex numbers C. For p ∈ X let G(p) = {n ∈ N | there is no f ∈ k(X) with $v_p(f) = n$, and $v_q(f)\neq0$ for all $q\neq p$}.
(i) Define ℓ(D), for a divisor D.
(ii) Show that for all p ∈ X,
$ℓ(np) = ℓ((n − 1)p)$ if n ∈ G(p) and $l(np)= ℓ((n − 1)p) + 1$ otherwise.
(iii) Show that G(p) has exactly g elements. [Hint: What happens for large n?]
Clearly (i) is very easy and (ii) follows by considering $ev_p:L(np)\to k$ mapping $f\mapsto (\pi_p^nf)(p)$ where $\pi_p$ is some local parameter.
For (iii) I can show that n is not in G(p) for n>2g-1 by Riemann-Roch. Not sure how to get exactly g points though.
Also noticed that if n is not in G(p) then no multiple can be either but not sure if this is particularly useful.
Any help would be appreciated ! Also the course I'm taking is very much elementary so no scheme theory please !
