# Solving a problem without using the Riemann - Roch theorem

I am studying elliptic curves and I faced with the concept of the divisors and Riemann Roch Theorem. My reference is " The Arithmetic of The Elliptic Curves " of Silverman . I tried to solve one of the exercises that stats:

Let C be smooth curve and let $$D \in Div(C)$$. With out using the Riemann - Roch, Prove the following :

1 - $$L(D)$$ is a $$\bar K -$$vector space.

2- If $$Deg(D) \geq 0,$$ then : $$l(D) \leq deg (D) + 1$$ I could solve the first part, but I can not solve the second. I was wondering if somebody helps me to prove it or give me a hint.

• Do you mean $l(D)\le\deg(D)+1$? Jun 29, 2019 at 14:31
• yes. Sorry, I corrected it.
– Elei
Jun 29, 2019 at 14:33

If $$D$$ has negative degree, then $$l(D)=0$$. So start an induction on the degree starting with $$\deg D=-1$$. In general, write $$D=D'+(P)$$ for some point $$P$$. It suffices to prove $$l(D)\le l(D')+1$$. Otherwise one must have $$L(D)$$ properly containing $$l(D')$$. Let $$f$$, $$g\in L(D)-L(D')$$. Then $$f$$ and $$g$$ have the same order at $$P$$, and so $$f-ag$$ has degree at one higher for a suitable scalar, so $$f-ag\in L(D')$$. This shows $$\dim(L(D)/L(D'))\le1$$, that is $$l(D)-L(D')\le1$$.