Applications of Curves over Finite Fields

The area starts with Galois and Gauss Group theory and expo nential sums were the two application areas then That tradition continues Without Chevalley groups over nite elds there would have been no structure theory of nite simple groups Weil s Riemann Hypothesis for curves over nite elds is a singular event Many error estimates in combinatorics derive from it Research in nite elds requires combinatorial understanding of many ex amples This is true in myriad applications coding theory exceptional poly nomials and covers algorithmic applications of elimination of quanti ers diophantine relations between curves over number elds and their reductions modulo p probabilistic algorithms over nite elds Yet there are powerful general abstract tools Consider two premiere mathematical events from the last twenty ve years Deligne s proof of the general Weil conjectures The classi cation of nite simple groups This conference s papers o er examples of applying such tools to practi cal researcher specialties The sections of this preface divide along the basic themes of the conference The preface makes several connections not appearing directly in the papers Some papers in the conference refer directly to Bernie Dwork not only to his papers This preface see x includes comments giving an overview of his work It compliments the article of Katz and Tate Beyond Weil bounds curves with many rational points Let p be a prime The word curve in this preface always means a one dimensional projective nonsingular algebraic variety The phrase variety over a nite eld means the equations have coe cients in a nite eld Fq The usual notation applies q is p with p a prime and t an integer Then Fq is the unique eld up to isomorphism with q elements For any eld K K is its algebraic closure If X is a curve let its genus be g X g The curve P is the projective line If we use a subscript t as in P t this means t you are given an inhomogenous parameter running over the points on P The Weil bound estimates the number of points on curves It o ers however little for q xed if the curves have large genus Applications with g large include explicit constructions of curves arising in coding theory to compute weight enu merators from Frobenius eigenvalues It also includes applications to graph theory and various problems involving the decomposition types of polynomials Persistent preoccupations include exceptional polynomials Davenport and Schur problems estimates for Kloosterman sums Ramanujan graphs These stipulate a su ciently Date July Mathematics Subject Classi cation Primary F G R Secondary B C D E F NSF DMS and the Alexander von Humboldt Foundation contributed support