Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 19J / b / ii

For $a\in {\mathbb{F}}_p^{\ast }$ putConjugation givesSuppose the $\chi$-weight spaceis nonzero. If $0\ne z\in V_{\chi }$, then $t(a)z$ lies in the $\chi (a^{-2}\cdot )$-weight space, becauseHence every character indexed by a nonzero square occurs in ${\mathrm{Res}}_U^{B}V$.

These characters are distinct. Indeed, if $\chi (v\cdot )=\chi (v^{\prime }\cdot )$ with $v\ne v^{\prime }$, multiplication by $v-v^{\prime }$ would show that $\chi$ is trivial on all of $U$, contrary to hypothesis. There are $(p-1)/2$ nonzero squares, sofor at least $(p-1)/2$ values of $v\in {\mathbb{F}}_p^{\ast }$. This is the square orbits of additive characters of a finite field argument.