Answer Happy

Let F, be a finite field of characteristic p. Let F € F,[X1,..., Xn] be a homogeneous polynomial of degree d and let V := {a = (a1,...,n) EF": F(a)=0}. For a € F, define G(a) := F(a)-! Show that: (1) #(F\V) = Daer, G(a) (mod p).

(2) For a € Zxo, one has Σα" = 0 (mod p) aer, unless a is a nonzero multiple of q-1. (3) For a = (01,...,On) € (Zo)", write aº = a... One has aº =0 (mod p). aer unless a 1 + ... + ann(-1). (4) If n>d, then G(a) = 0 (mod p), #F}\)=0 (mod p)#V=0 (mod p). aer (5) #C(F,) =q +1 for every conic C defined over Fg. Hint: For (2), express the sum in terms of a generator of F. For (3), observe that I--(£«") (8) - aer EF EF, and apply (2). For (5), use (4) to deduce that C(F) + 0 and apply the bijection between C(F,) and P!(F,) that we have seen in this case in the lectures.