Jtdylktuy

This problem is taken from the exercises in "Number Theory in Function Fields" by M. Rosen, chapter 6, page 76:

Suppose that $omegain Omega_K (0)$ and has a zero $P$ of degree 1,
and that $ord_p (omega)geq g$. Show that $P$ is a weierstrass point.

Here $K$ is some function field, over a ground field $F$. $Omega_K(0)$ is the set of functionals on the adele ring of $K$ which vanishes on $K$ and on adeles $xi=(x_P)$ such that $v_P(x_P)geq 0$ for all prime $P$. $ord_p (omega)$ refers to the order at $P$ of the divisor associated to $omega$: we look at sub-vector spaces of the adeles, of the form $A_K (D)={xi=(x_P):x_Pgeq -ord_P(D),forall p}$. The divisor $(omega)$ is the greatest divisor $D$ such that $omega$ vanishes on $A_K (D)$. In general the book denotes $Omega_K(D)$ the space of functionals vanishing on $K$ and on adeles $xiin A_K(D)$.

I tried to do this: A point is a non-Weierestrass point if all its gaps are consecutive at the beginning. Because $P$ is of degree 1, this happens iff $l(gP)=1$. We know that the Riemman-Roch space $L(gP)congOmega((omega)-gP)$. So, if we could show that this space of differentials has dimension$>1$ we finish. We knowh that $(omega)>(omega)-gP$, hence, $A_K(omega-gP)subset A_K(omega)$ and so $Omega_K(omega)subset Omega_K(omega-gP)$. If this inclusion was not proper, this is it- as $dim_F(Omega_K(omega))=1$ (This follows from Riemman-Roch).

I guess I have 3 questions:

1. Is this the right approach? Any help in proving this result would be great.




2. In general, how can you construct by hand a differential? I thought maybe we could do this using the local components decomposition for a differential, but how do you even find the primes of a function field? EDIT: actually I now understand that because the differentials are a 1 dimensional space over $K$, we only have to find 1 non zero differential and that will give us all the others. But how can we find such a differential, even in a particular cases (say the function field obtained by taking $y^2=x$)?




3. Searching around the site for similar question, and even because of the name "differential", I feel that there's some geometrical intuition to whole of this which I am missing. Is it true? If so, I'll be very glad for any help in seeing this geometrical meaning.