I'm trying to understand some argument that is being made regarding homogeneous forms on an elliptic curve. In order to do this, I need to understand how we determine the dimension of the space of homogeneous forms of fixed degree $d$ on the elliptic curve $E/\mathbb{Q}$ of rank $r$. I can't see anything online regarding this, and I'm not sure where to start myself. Any advice on how to grasp a hold of this would be appreciated.
2026-03-27 05:37:54.1774589874
Counting Homogeneous Forms on Elliptic Curves
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I solved this recently - it's relatively trivial actually. It simply corresponds to the fact that on a space of dimension $r$ there are at most $\binom{d+r-1}{r-1}$ homogenous functions of degree $d$. This can be seen by looking at homogenous polynomials of degree $d$ among $r$ variables.