= Solution
A <divisor on an algebraic curve> is a finite formal integer sum $D=\sum_pn_pp$. Its degree is $\sum_pn_p$. The <Riemann-Roch space> $L(D)$ consists of zero and the meromorphic functions $h$ with $(h)+D\geq0$, and $\ell(D)=\dim L(D)$. The <Riemann-Roch theorem> states
$$
\ell(D)-\ell(K-D)=\deg D+1-g,
$$
where $K$ is a <canonical divisor>, of degree $2g-2$. In <genus> one, $\deg(K-D)=-n<0$ for $\deg D=n>0$. A negative-degree <divisor on an algebraic curve> has no nonzero section, since a <principal divisor> has degree zero and an <effective divisor> has nonnegative degree. Hence
$$
\boxed{\ell(D)=n\quad(g=1,\ n>0).}
$$
A <nonsingular plane cubic> has <genus> one by the <genus-degree formula>. Therefore $\ell(p+q)=2$, and there is a nonconstant $f\in L(p+q)$. It has at most simple poles at $p,q$ and no other poles. It cannot have no poles, since a <holomorphic function> on a compact <Riemann surface> is constant. It cannot have just one simple pole either, by the allowed fact, or by $\ell(p)=\ell(q)=1$. Thus \b[$f$ has simple poles at both $p$ and $q$, and no others].
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