Algebraic curves over fields of prime characteristic

The theory of Algebraic curves was mostly developed in the 19-th century. Chapter 2 determines the zeta function for a famous curve that was mentioned in the last entry of Gauss's journal. We find that for p ≡ 3 mod 4 the zeta function of the curve C : x2t2 + y2t2 + x 2y2 - t 4 = 0 in P2 defined over Fp is ZCu= 1+pu2 1+u21- pu1-u . Algebraic curves are covers of the projective line. Every curve has a birational invariant associated to it known as the genus. Let X be a smooth projective curve that is an An-Galois covering of the projective line branched only at infinity. Chapter 3 investigates what possibilities there are for the genus of X. For example let d2 = gcd(p - 1, p + 2). There exists a curve X that is an Ap+2-Galois cover of the projective line branched only at infinity with the genus of X being g=1+Ap+2 2-1-d2 pp-1+ p+2p .