Stark’s Basic Conjecture

Some Notation. In these notes, k denotes a number field, i.e., a finite extension of the field Q of rationals, and Ok is its ring of integers. If I is an ideal in Ok, we denote its number of residue classes |Ok/I| by NI. This number is also the positive generator of the ideal NK/QI ⊂ Z, and is a multiplicative function of ideals. By a place v of k we mean an equivalence class of non-trivial absolute values on k. These are of three types, finite, real and complex, corresponding, respectively, to a prime ideal P in Ok, to an embedding of k into R, and to a pair of distinct complex conjugate embeddings of k into C. We denote the number of real places by r1 = r1(k) and the number of complex ones by r2 = r2(k). For each finite place v we denote the corresponding prime ideal by Pv and the residue field by Fv. We sometimes denote the cardinality |Fv| of Fv by qv = NPv.