Recent Developments in Function Field Arithmetic

Z = {integers} Q = {rational numbers} R = {real numbers} C = {complex numbers} Z+ = { positive integers} q = a power of a prime p Fq = A Finite field with q elements A = Fq[t] A+ = {monics in A} K = Fq(t) K∞ = Fq((1/t)) = completion of K at ∞ C∞ = completion of algebraic closure of K∞ [n] = tq n − t dn = ∏n−1 i=0 (t qn − tqi) `n = ∏n i=1(t− tq i ) deg = function assigning to a ∈ A its degree in t The purpose of this expository talk is to describe some fascinating recent developments in the Function Field Arithmetic and hopefully get some bright students working in this rapidly developing subject area. Though this name ‘Function Field Arithmetic’ of the subject