Skip to main content
SpringerLink
Log in

Intermission

  • Chapter
  • First Online:
The Riemann Hypothesis in Characteristic p in Historical Perspective

Part of the book series: Lecture Notes in Mathematics ((HISTORYMS,volume 2222))

  • 1125 Accesses

Abstract

If one would compare our story with a concert, then Artin’s thesis together with F.K. Schmidt’s paper would pass as the first and second part of the Introduction (Chaps. 3 and 4). Hasse’s work on the elliptic case (Chap. 7) would be the first movement allegro assai with the theme set by Davenport (Chap. 6). Deuring’s theory of correspondences (Chap. 9) would pass as the second movement sostenuto, covering the attempt towards higher genus. The insertion of the virtual proof (Chap. 10) may go as scherzo allegretto.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    According to [DS15] it was Lefschetz who had been instrumental in securing the necessary financial means for a stay of Artin in the USA. Recently I have been informed by Karin Reich that, according to Natascha Artin, also Courant had been active in this direction.

  2. 2.

    And not only Hasse.

  3. 3.

    On the other hand, this note reveals that Hasse’s interest had somewhat shifted, or I should better say expanded, to the theory of function fields over base fields which are algebraic number fields. There are also several other sources which indicate this shift of interest. See, e.g., [Has42a] or the last section of [Has42b].

  4. 4.

    See page 134 ff., Sects. 9.4 and 9.5, and page 154.

  5. 5.

    In some letters Weil also touched personal matters. He asked Hasse for recommendation letters when he applied for academic positions in France. However these applications were not successful at that time. He put his frustration into words in a letter to Hasse: “In France, appointments have little to do with scientific achievements …”

  6. 6.

    In the original we read the date “20.I.38” but the content of the letter shows clearly that this was a misprint and the letter was written in 1939.

  7. 7.

    In view of all what is known of stories about Siegel this may well have been the case.

  8. 8.

    I have found these letters in the archive of Springer Verlag Heidelberg.

  9. 9.

    Later however, in a letter of 20 January 1939, Weil writes: “We have had heavy differences with Julia …” (“Mit Julia haben wir uns verkracht… ”).

  10. 10.

    Charles Pisot (1910–1984) had a research grant to study number theory in Göttingen in the summer semester 1939. He was born in Alsace hence he spoke German as well as French. As Weil wrote to Hasse it was Pisot which he had had in mind for translation of Hasse’s book into French. Pisot became a member of Bourbaki for some time.

  11. 11.

    Gaston Julia had been seriously injured during World War I and since then had health problems throughout his life. Nevertheless, and perhaps just because of this experience, he vehemently advocated a close political cooperation between France and Germany, so that there would be no further war between the two countries …

References

  1. E. Artin, G. Whaples, Axiomatic characterization of fields by the product formula for valuations. Bull. Am. Math. Soc. 51, 469–492 (1945)

    Article  MathSciNet  Google Scholar 

  2. C. Chevalley, Généralisation de la théorie du corps de classes pour les extensions infinies. J. Math. pur. Appl. (9) 15, 359–371 (1936)

    Google Scholar 

  3. C. Chevalley, La théorie du corps de classes. Ann. Math. (2) 41, 394–418 (1940)

    Article  MathSciNet  Google Scholar 

  4. M. Deuring, La teoria aritmetica delle funzioni algebriche di una variabile. Rend. Mat. Appl. V. Ser. 2, 361–412 (1941)

    MathSciNet  MATH  Google Scholar 

  5. D. Dumbaugh, J. Schwermer, Emil Artin and Beyond. Class Field Theory and L-Functions. With Contributions by James Cogdell and Robert Langlands (European Mathematical Society (EMS), Zürich, 2015)

    Google Scholar 

  6. G. Frei, F. Lemmermeyer, P. Roquette, (eds.), Emil Artin and Helmut Hasse. Their Correspondence 1923–1958 English version, revised and enlarged. Contributions in Mathematical and Computational Science, vol. 5 (Springer Basel, Cham, 2014) X + 484 pp.

    MATH  Google Scholar 

  7. H. Hasse, Theorie der Differentiale in algebraischen Funktionenkörpern mit vollkommenem Konstantenkörper. J. Reine Angew. Math. 172, 55–64 (1934)

    MATH  Google Scholar 

  8. H. Hasse, Anwendungen der Theorie der algebraischen Funktionen in der Zahlentheorie. Abh. Ges. Wiss. Göttingen. Math.-Phys. Klasse, III. Folge(Heft 18), 51–55 (1937)

    Google Scholar 

  9. H. Hasse, Der n-Teilungskörper eines abstrakten elliptischen Funktionenkörpers als Klassenkörper, nebst Anwendung auf den Mordell-Weilschen Endlichkeitssatz. Math. Z. 48, 48–66 (1942)

    Article  MathSciNet  Google Scholar 

  10. H. Hasse, Zur arithmetischen Theorie der algebraischen Funktionenkörper. Jahresber. Dtsch. Math.-Ver. 52, 1–48 (1942)

    MATH  Google Scholar 

  11. H. Hasse, Internationale Mathematikertagung in Rom im November 1942. Jber. Deutsche Math. Ver. 53, 21–22 (1943). 2.Abteilung

    Google Scholar 

  12. H. Hasse, Punti razionali sopra curve algebriche a congruenze. Reale Accademia d’Italia, Fondazione Alessandro Volta. Atti dei Convegni 9(1939), pp. 85–140 (1943)

    Google Scholar 

  13. H. Hasse, Überblick über die neuere Entwicklung der arithmetischen Theorie der algebraischen Funktionen. Atti Convegno Mat. Roma 1942, 25–33 (1945)

    MATH  Google Scholar 

  14. H. Hasse, Zetafunktionen und L-Funktionen zu einem arithmetischen Funktionenkörper vom Fermatschen Typus. Abh. Deutsch. Akad. wiss. Berlin, Math.-Naturw. Kl. 1954(4), 70 S. (1955)

    Google Scholar 

  15. K. Hensel, Über den Zusammenhang zwischen den Kongruenzgruppen eines algebraischen Körpers für alle Potenzen eines Primteilers als Modul. J. Reine Angew. Math. 177, 82–93 (1937)

    MathSciNet  MATH  Google Scholar 

  16. E. Kähler, Geppert, Harald, in Neue Deutsche Biographie, vol. 6, p. 247 (1964)

    Google Scholar 

  17. H. Rohrbach, Helmut Hasse und das Crellesche journal. J. Reine Angew. Math. 214–215, 443–444 (1964)

    MathSciNet  MATH  Google Scholar 

  18. F.K. Schmidt, Zur arithmetischen Theorie der algebraischen Funktionen. I. Beweis des Riemann-Rochschen Satzes für algebraische Funktionen mit beliebigem Konstantenkörper. Math. Zeitschr. 41, 415–438 (1936)

    MathSciNet  MATH  Google Scholar 

  19. H.L. Schmid, Kongruenzzetafunktionen in zyklischen Körpern. Abh. Preuß. Akad. Wiss., math.-naturw. Kl. 1941(14), 30 S, (1941)

    Google Scholar 

  20. F. Severi, Trattato di geometria algebrica. Vol.I. Parte I. Geometria delle serie lineari. Zanichelli, Bologna, 1926. 358 pp.

    Google Scholar 

  21. C.L. Siegel, Über einige Anwendungen diophantischer Approximationen. Abhandlungen Akad. Berlin 1929(1), 70pp. (1929)

    Google Scholar 

  22. B.L. van der Waerden, Divisorenklassen in algebraischen Funktionenkörpern. Comment. Math. Helv. 20, 68–80 (1947)

    Article  MathSciNet  Google Scholar 

  23. A. Weil, L’arithmétique sur les courbes algébriques. Acta Math. 52, 281–315 (1929)

    Article  MathSciNet  Google Scholar 

  24. J. Weissinger, Theorie der Divisorenkongruenzen. Abh. math. Sem. Univ. Hamburg 12, 115–126 (1937)

    Article  MathSciNet  Google Scholar 

  25. A. Weil, Généralisation des fonctions abéliennes. J. Math. Pures Appl. 9(17), 47–87 (1938)

    MATH  Google Scholar 

  26. A. Weil, Zur algebraischen Theorie der algebraischen Funktionen. J. Reine Angew. Math. 179, 129–133 (1938)

    MathSciNet  MATH  Google Scholar 

  27. A. Weil, Sur les fonctions algébriques à corps de constantes fini. C. R. Acad. Sci. Paris 210, 592–594 (1940)

    MathSciNet  MATH  Google Scholar 

  28. A. Weil, On the Riemann hypothesis in function fields. Proc. Natl. Acad. Sci. USA 27, 345–347 (1941)

    Article  Google Scholar 

  29. A. Weil, Jacobi sums as “Größencharaktere”. Trans. Am. Math. Soc. 73, 487–495 (1952)

    MATH  Google Scholar 

  30. A. Weil, Adèles et group algébriques. In Sémin. Bourbaki 11 (1958/1959), Exp. No.186 (1959). 9pp.

    Google Scholar 

  31. A. Weil, Œuvres scientifiques. Collected papers. Vol. I (1926–1951). (Springer, Berlin, 1979)

    Google Scholar 

  32. H. Wußing, Zur Emigration von Emil Artin. in Mathematics celestial and terrestrial. Festschrift für Menso Folkerts zum 65. Geburtstag, ed. by J.W. Dauben, S. Kirschner, A. Kühne. Acta Historica Leopoldina, vol. 54, pp. 705–716 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Heidelberg University, Heidelberg, Germany

    Peter Roquette

Authors
  1. Peter Roquette
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Roquette, P. (2018). Intermission. In: The Riemann Hypothesis in Characteristic p in Historical Perspective. Lecture Notes in Mathematics(), vol 2222. Springer, Cham. https://doi.org/10.1007/978-3-319-99067-5_11

Download citation

Publish with us

Policies and ethics

Search

Navigation