Let $C$ be a compact Riemann surface of genus $1$, and $p\in C$, and $w$ be a local holomorphic coordinate on $C$ near $p$ with $w=0$ at $p$. As usual, for a divisor $D$ denote by $L(D)$ the vector space of meromorphic functions $f$ (together with $f\equiv 0$) such that $(f) + D$ is an effective divisor; and $l(D) = \dim L(D)$.

One can easily show (by the Riemann-Roch theorem and $\deg D < 0 \Rightarrow l(D)=0$) that $l(mp) = 0$ for $m\leq 0$ and $m$ for $m>0$ (where $m\in \mathbb{Z}$). In particular, we can deduce that $L(p) = \langle 1 \rangle$, $L(2p) = \langle 1,f \rangle$ and $L(3p) = \langle 1,f,g \rangle$ where $f(w) = w^{-2} + O(w^{-1})$ and $g(w) = w^{-3} + O(w^{-2})$ near $p$. Then by considering $L(6p)$, which contains the $\bf{seven}$ maps $g^2, f^3, fg, f^2, g, f, 1$, we deduce that $g^2 = f^3 + Afg + Bf^2 + Cg + Df + E$ for unique scalars $A,\ldots,E$ (the coefficients at $g^2$ and $f^3$ must be equal by comparing $w^{-6}$ terms on both sides of the equation).

And so we remain with the following two:

a) Hence construct a non-trivial holomorphic map of Riemann surfaces $\Phi: C \to D$ where $D$ is the cubic curve $$ D = \{[x,y,z] \in \mathbb{CP^{2}}: y^2z = x^3 + Axyz + Bx^2z + Cyz^2 + Dxz^2 + Ez^3\} $$

b) Now suppose $\sigma: C \to C$ is an isomorphism of Riemann surfaces with $\sigma^k = id_C$ for some $k\geq 2$ and $p$ is an isolated fixed point of $\sigma$, and the coordinate $w$ is chosen so that $\sigma: w \to e^{2\pi i/k}w$ near $p$. Then the pullback $\sigma^{*}$ maps $L(mp) \to L(mp)$ by $\sigma^{*}: h \to h \circ \sigma$ for each $m\in \mathbb{Z}$ and defines a representation of $\mathbb{Z}_k = \langle 1, \sigma, \ldots, \sigma^{k-1} \rangle$ on $L(mp)$. We choose $f,g$ from above to be eigenvectors of $\sigma^{*}$. Show that:

i) If $k=2$ then $A=C=0$

ii) If $k=3$ then $A=B=D=0$

iii) If $k=4$ then $A=B=C=E=0$

iv) If $k=6$ then $A=B=C=D = 0$

v) If $k\neq 2,3,4,6$ then $A=B=C=D=E = 0$

By considering $g/f$, show that case v) cannot happen.

So in a) I guess it should be something like $\Phi(p) = [f,g,1]$ but how to write it down properly and check it is holomorphic?

In b) - absolutely no ideas.

Any help appreciated!

  • $\begingroup$ I will point out that the tag (geometry) is deprecated on MathOverflow and it should not be used on this site, see the tag-info. Perhaps there is some geometry related tag which would be more suitable for the question. (After all, the question is also missing a top-level tag.) $\endgroup$ – Martin Sleziak Jul 12 '19 at 6:55
  • $\begingroup$ Applying Riemann-Roch to the divisor of a meromorphic 1-form gives an holomorphic 1-form $\omega$ and $p \mapsto \int_{p_0}^p \omega$ is a biholomorphic map $C \to \Bbb{C}/L$ where $L$ is integrating $\omega$ on closed-loops. On complex tori your spaces have a concrete definition in term of $\wp_L$. $\endgroup$ – reuns Jul 12 '19 at 15:09
  • $\begingroup$ @reuns Sorry, but I do not understand anything from this :( $\endgroup$ – DesmondMiles13 Jul 12 '19 at 17:40
  • $\begingroup$ Any compact Riemann surface of genus $1$ is biholomorphic to a complex torus $\Bbb{C}/(u \Bbb{Z}+v\Bbb{Z})$ whose holomorphic 1-form is $dz$ $\endgroup$ – reuns Jul 12 '19 at 18:10
  • $\begingroup$ @reuns Isn't there a more natural way to construct $\Phi$? And do your arguments help for b), I at least do not see how? $\endgroup$ – DesmondMiles Jul 13 '19 at 1:55

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