On the history of Artin’s L-functions and conductors Seven letters from Artin to Hasse in the year 1930

In the year 1923 Emil Artin introduced his new L-functions belonging to Galois characters. But his theory was still incomplete in two respects. First, the theory depended on the validity of the General Reciprocity Law which Artin was unable at that time to prove in full generality. Secondly, in the explicit definition of L-functions the ramified primes could not be taken into account; hence that definition was of provisional character only whereas the final definition could be given in a rather indirect way only. In later years Artin filled both of these gaps: In 1927 he proved the General Reciprocity Law, and in 1930 he gave a complete definition of his L-functions, including the ramified and the infinite primes; at the same time he introduced his theory of conductors for Galois characters. This development is well documented in the correspondence between Artin and Hasse of those years. In the present paper we discuss seven letters from Artin to Hasse, written in the year 1930, where he expounds his ideas about the final definition of the L-functions and about his conductors. We also discuss some letters from Emmy Noether to Hasse of the same time which are directly inspired by Artin’s. ∗This copy contains some minor corrections of the published version.