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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.

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

1 Answer 1

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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 $l(D)-L(D')\le1$.

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  • $\begingroup$ Thanks! But I can't understand why such that "a" should exist? $\endgroup$
    – Elei
    Jun 29, 2019 at 20:37
  • $\begingroup$ If such that "a" there exist, the proof completes! $\endgroup$
    – Elei
    Jun 29, 2019 at 21:08

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