I try to solve the following questions:

Give an elementary proof of Riemann-Roch theorem for Riemann sphere $X=\hat{\mathbb{C}}$.


Let $X$ be a torus, and $p\in X$ a point. Show that $$\dim O(np)=\left\{\begin{array}{ll}0& n<0\\1&n=0\\n&n\geq 1\end{array}\right.$$ without using Riemann-Roch theorem.

I combined the two questions because they look similar. If necessary, I will split them to separate questions.

My attempt: The version of Riemann-Roch theorem I learnt is $$\dim O(D)-\dim O(K-D)=\deg (D)+1-g$$ where $g$ is the genus of Riemann surface, $D$ is a divisor and $O(D)$ is given by the subspace of meromorphic functions on the riemann surface $$O(D)=\{f~|~(f)+D\geq 0\}.$$

In the first exercise, $g=0$. Then it is enough to show that $$\dim O(D)-\dim O(K-D)=\deg (D)+1$$ And pick a canonical divisor: on the chart containing $0$, the meromorphic form is $$z~\mathrm{d}z,$$ on the chart containing $\infty$, the form is $$-\frac{\mathrm{d}~w}{w^3}$$ Then we obtain a canonical divisor $$1\cdot 0-3\cdot\infty$$ Also, I know the number of zeros of a meromorphic function on $\hat{\mathbb{C}}$ is equal to the number of its poles. Then I have no idea how to continue.

For the second one, find the canonical divisor on a torus is much more difficult for me.

For both of problems, to avoid the Riemann-Roch theorem, I try to construct the basis of the above vector spaces to verify the equation of dimension. But I don't know how to do it.

Thanks for any hints and answers!

  • $\begingroup$ The second one is easy. $\endgroup$ – Aolong Li Apr 21 '18 at 7:19

For the first part, note that a divisor on a riemann sphere is uniquely determined by its degree up to linear equivalence.

If $d=deg(D) \geq 0$. Without loss of generality, say $D = d\cdot 0$, and $K = -2 \cdot 0$. Then elements $f$ in $O(D)$ are exactly those ones in the following form $$f = z^{-k} h$$ where $0\leq k \leq d$ and $h$ is a polyonomial of degree no more than $k$. Thus, $z^{-k}$ form a basis. So we conclude that $$\ell (D) = d +1$$Similarly, $$\ell (K-D) = 0$$ Since there cannot be a meromorphic function which has zero with multiplicity greater than $d + 2$ at $0$, and be analytic at $\infty$ at the same time.

I'm sure you can work out the rest case when $d < 0$.

The second question needs more work. I believe it can be solved using the theory of elliptic functions.


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