Question

If a plane curve of degree n intersects an elliptic curve in 3n points, then do those points always sum to zero when added using the group law on the points of an elliptic curve ?

Answer 1

I assume that elliptic curve means a cubic curve $E$ with some base point $O$.

Then the answer to your question is yes if and only if the base point O for the group law is a flex point of $E$. (In many texts it is assumed that the base point of $E$ is a flex line, but to define the group law this is completely unnecessary, see Fulton's book on plane curves for the geometric definition of the group law in this case.) Fulton's geometric definition of the group law on a cubic is equivalent with
$P_1,..P_{3n}$ add up to zero if and only if $P_1+..P_{3n}-3n O$ is zero in $Pic^0(E)$, i.e., is a divisor of a function on $E$.

Assume first that $O$ is a flex point. Let g be polynomial defining the flex line, let $f$ be a polynomial defining $C$ and let $P_1,...P_{3n}$ be the $3n$ intersection points of $C$ and $E$. As David commented above we have $div(f/g^n)=P_1+..P_{3n}-3nO$, hence the points add up to zero.

Suppose now that O is not a flex. Let $R$ be the third intersection point of $E$ with the tangent line of $E$ at $O$. Since $O$ is not a flex we have that $R\neq O$. To check whether these points add up to zero we have to calculate $O-O+O-O+R-O$ in $Pic^0(C)$. It is easy to see that this divisor is nonzero in $Pic^0(C)$.

Answer 2

Yes.

Choose a flex as zero point. The sum of three points on a line is zero essentially by construction of the group law. Now the divisor cut on the elliptic curve by a curve of degree $n$ is linearly equivalent to the sum of $n$ hyperplane divisors, so the claim follows.