(a) State the Riemann-Roch theorem.

(b) Let $E$ be a smooth projective curve of genus 1 over an algebraically closed field $k$, with char $k \neq 2,3$. Show that there exists an isomorphism from $E$ to the plane cubic in $\mathbf{P}^{2}$ defined by the equation

$$y^{2}=\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right)\left(x-\lambda_{3}\right)$$

for some distinct $\lambda_{1}, \lambda_{2}, \lambda_{3} \in k$.

(c) Let $Q$ be the point at infinity on $E$. Show that the map $E \rightarrow C l^{0}(E), P \mapsto[P-Q]$ is an isomorphism.

Describe how this defines a group structure on $E$. Denote addition by $\boxplus$. Determine all the points $P \in E$ with $P \boxplus P=Q$ in terms of the equation of the plane curve in part (b).