Sharp endpoint extension inequalities for the moment curve on finite fields

We investigate the sharp endpoint extension inequality for the moment curve in finite fields. We determine the optimal constant and characterize the maximizers in two complementary regimes: (i) low dimensions $d\le 20$; (ii) large field cardinality $q\ge \frac{d(d-1)}{2\log 6}+\frac{(2d-1)}{3}$. Our proof strategy relies on an intriguing interplay between analysis, algebra and combinatorics.