Emil Artin's unpublished generalization of his dissertation
Emil Artin's dissertation on `Quadratic Function Fields' [Quadratische Körper im Gebiete der höheren Kongruenzen, Math. Z. 19, 153-206, 207-246 (1924; JFM 50.0107.01)] developed the theory of quadratic extensions of \(\mathbb F_p[X]\) in analogy to the theory of quadratic extensions of \(\mathbb Q\), and studied ideals, unique factorization, the reciprocity law, and zeta functions; in particular, Artin came up with the Riemann conjecture for elliptic and certain hyperelliptic curves, and verified them in special cases. The generalization of these results to arbitrary function fields in one variable over finite fields was initiated by Hasse and F. K. Schmidt, and led up to the Weil conjectures and their proof by Deligne.NEWLINENEWLINENEWLINEThis article describes the content of a planned sequel to Artin's dissertation, namely the generalization of its results from \(\mathbb F_p\) to arbitrary finite fields (which, as Artin observed, is straightforward), as well as additional results on the behavior of the zeta functions under extension of the field of constants. Artin also gave a very nice proof of \textit{E. Jacobsthal}'s result [Anwendungen einer Formel aus der Theorie der quadratischen Reste; Diss. Berlin (1906; JFM 37.0226.01)] on the number of \(\mathbb F_p\)-rational points on the elliptic curve \(y^2 = x(x^2-1)\), as well as a related result for \(y^2 = x^3-1\).NEWLINENEWLINENEWLINEThe main part of this article, however, deals with the personal situation of Artin at the time of his dissertation, and tries to explain why Artin never published these generalizations. Apparently, there are two major reasons for this: one is the fact that Hilbert called most of Artin's results `trivialities' when Artin gave a talk in Göttingen on Nov. 22, 1921, although he seemed impressed by Artin's derivation of Jacobsthal's formulas (Artin heard of Jacobsthal's results only the week after his lecture, and they were probably also unknown to Hilbert; Artin also overlooked that \textit{L. v. Schrutka} [J. Reine Angew. Math. 140, 252-265 (1911; JFM 42.0220.02)] had taken care of the corresponding result for \(y^2 = x^3-1\)) and later even offered Artin the publication of his results in the Annalen. The second reason seems to have been Artin's discovery, possibly influenced by lectures of Siegel, that Hecke's articles contained a `colossal amount' of mathematics, which -- in addition to Artin's isolation in Göttingen -- eventually made him move to Hamburg, where Hecke was a full professor. The rest, as they say, is history.
scientific article; zbMATH DE number 1690693