Algebraic curves over fields of prime characteristic

Abstract

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 + x2y2 – t4 = 0 in P2 defined over Fp is ZC(u) = (1 + pu2) (1 + u)2 / (1 – pu) (1 – 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 / p(p – 1) + (p + 2) / p).