Abstract
As I have told in Sect. 6.2 Hasse originally was interested in the estimate of solutions of diophantine congruences. It was Artin who in November of 1932, when Hasse was visiting Hamburg, told him that the estimating problem was pointing to the RHp. It appears that at that time Hasse did not yet believe in the general validity of the RHp for all function fields with higher genus gā>ā1. (See page 65.)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In the summer term of 1933 Davenport stayed in Gƶttingen with a stipend he had got from his College. Thus he became a witness of the dissolution of the Gƶttingen mathematical scene due to the disastrous policy of the Nazi government. On weekends he often went to Marburg, which is not far from Gƶttingen, in order to meet Hasse. But sometimes there were letters exchanged.
- 2.
It appears that Hasse had those automorphisms in mind which respect the āadditionā of two point sets. He knew already from the elliptic case that there are more field automorphisms, namely the translations. See page 95.
- 3.
President of the University.
- 4.
The official organization of mathematics students which at that time was dominated by Nazi students.
- 5.
An additional reason for removing Tornier from Gƶttingen may have been that he permanently produced financial trouble. He had pawned a large part of his future salary and also had asked for and received high sums from the university as advance payment. It appears that he had fallen back to his addiction to drugs (morphine) which Hasse had mentioned in a letter to Fraenkel of 10 July 1927. (At that time Hasse had believed that Tornier had been able to overcome this.) But at Berlin University Tornier did not stop this behavior and in 1938, when he was caught in financial fraud he had to leave the university and also the Nazi party. More precisely, in order to avoid public scandal he was āadvisedā to leave the party and the university on his own application. He then retreated to a psychiatric clinic in Silesia. These facts are documented in the personal files for Tornier at the archive of Humboldt University in Berlin.
- 6.
That was what apparently many people at those times still expected or at least hoped. And not only at those times.
- 7.
As to the terminology and the description of the situation see page 96 ff.
- 8.
Among them was the work of Weilās student Elizabeth Lutz at Strassbourg on the structure of the group of rational points of an elliptic curve over a \(\mathfrak {p}\)-adic field (\(\mathfrak {p}\)-adic uniformization). Weil had proposed to have Lutzās paper published in Gƶttingen as a āsign of continued cooperationā. Hasse gladly agreed; he accepted Lutzās paper for Crelleās Journal [Lut37].
- 9.
I have found this expression in Schappacherās survey [Sch07].
- 10.
Rohrbach had held a position at Berlin University but had got there into trouble because of political reasons.
- 11.
She too came from Berlin. There she did not like the Nazi dominated atmosphere and therefore she had asked Rohrbach where to go for her Ph.D. Rohrbach suggested to come to Gƶttingen too and study with Hasse. The latter proposed to her to work on the foundation of arithmetics in higher dimensional function fields. But one year later von Caemmerer suddenly left Gƶttingen and went to Britain where she married Bernhard Neumann with whom she had been secretly engaged in Berlin already. He was Jewish and therefore had to leave Germany. Both Hanna and Bernhard grew to become well known group theorists.
References
M. Deuring, Algebren. Erg. d. Math. u. ihrer Grenzgebiete. Julius (Springer, Berlin, 1935), 143 pp.
M. Deuring, Arithmetische Theorie der Korrespondenzen algebraischer Funktionenkƶrper. I. J. Reine Angew. Math. 177, 161ā191 (1937)
M. Deuring, Arithmetische Theorie der Korrespondenzen algebraischer Funktionenkƶrper. II. J. Reine Angew. Math. 183, 25ā36 (1940)
M. Deuring, Reduktion algebraischer Funktionenkƶrper nach Primdivisoren des Konstantenkƶrpers. Math. Z. 47, 643ā654 (1942)
G. Frei, P. Roquette (eds.), Emil Artin and Helmut Hasse. Their correspondence 1923ā1934. With contributions of Franz Lemmermeyer and an introduction in English. (UniversitƤtsāVerlag, Gƶttingen, 2008), 497 pp.
H. Hasse, Bericht Ć¼ber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkƶrper. II: ReziprozitƤtsgesetz. Jahresber. Dtsch. Math.-Ver., 6(ErgƤnzungsband), 1930. IV + 204 pp.
H. Hasse, Theory of cyclic algebras over an algebraic number field. Trans. Am. Math. Soc. 34, 171ā214 (1932)
H. Hasse, Beweis des Analogons der Riemannschen Vermutung fĆ¼r die Artinschen und F. K. Schmidtschen Kongruenzzetafunktionen in gewissen elliptischen FƤllen. VorlƤufige Mitteilung. Nachr. Ges. Wiss. Gƶttingen, Math.āPhys. Kl. I 1933(42), 253ā262 (1933)
H. Hasse, Zur Theorie der abstrakten elliptischen Funktionenkƶrper. Nachr. Ges. Wiss. Gƶttingen I. N.F. 1, 119ā129 (1935)
H. Hasse, Die Gruppe der p n-primƤren Zahlen fĆ¼r einen Primteiler \({\mathfrak {p}}\) von p. J. Reine Angew. Math. 176, 174ā183 (1937)
H. Hasse, Ćber die Riemannsche Vermutung in Funktionenkƶrpern, in C. R. du CongrĆØs Internat. Math. Oslo 1936, vol. 1, pp. 189ā206 (1937)
D. Hilbert, On the theory of the relative abelian number fields. (Ćber die Theorie der relativ-Abelāschen Zahlkƶrper.). Gƶtt. Nachr. 1898, 370ā399 (1898)
A. Hurwitz, Ćber algebraische Korrespondenzen und das verallgemeinerte Korrespondenzprinzip. Berichte v.d. sƤchsischen Gesellsch. d. Wissenschaften, mathematisch-physikalische Klasse. 1886, 10ā38 (1886). Wiederabdruck in Math. Ann. Bd.28 (1887) 561ā585
W. Krull, Idealtheorie. Erg. d. Math. u. ihrer Grenzgebiete. Julius (Springer, Berlin, 1935). V, 172 pp.
S. Lefschetz, Correspondences between algebraic curves, in Selected topics in Algebraic Geometry, V. Snyder et al. Bulletin of the National Research Council, vol. 63 (1928), pp. 381ā448
E. Lutz, Sur lāequation y 2ā=āx 3āāāAxāāāB dans les corps \(\mathfrak {p}\)-adiques. J. Reine Angew. Math. 177, 238ā247 (1937)
P. Roquette, The Brauer-Hasse-Noether Theorem in Historical Perspective.. Schriftenreihe der Heidelberger Akademie der Wissenschaften, vol. 15 (Springer, Berlin, 2005). I, 77 pp.
O.F.G Schilling, Ćber gewisse Beziehungen zwischen der Arithmetik hyperkompler Zahlsysteme und algebraischer Zahlkƶrper. Math. Ann. 111, 372ā398 (1935)
H.L. Schmid, Zyklische algebraische Funktionenkƶrper vom Grade p n Ć¼ber endlichem Konstantenkƶrper der Charakteristik p. J. Reine Angew. Math. 175, 108ā123 (1935)
O.F.G. Schilling, The Theory of Valuations. Mathematical Surveys, vol. IV (American Mathematical Society, Providence, RI, 1950), 253 pp.
N. Schappacher, Das mathematische Institut der UniversitƤt Gƶttingen 1929ā1950, in Die UniversitƤt Gƶttingen unter dem Nationalsozialismus, ed. by H. Becker, andere, pp. 345ā373. K. G. Saur (1987)
N. Schappacher, A historical sketch of B. L. van der Waerdenās work in algebraic geometry: 1926ā1946. In Episodes in the history of modern algebra (1800ā1950). Based on a workshop held at MSRI, Berkeley, CA, USA, April 2003 (American Mathematical Society, Providence, RI, 2007); (London Mathematical Society, London 2007), pp. 245ā283
F. Severi, Trattato di geometria algebrica. Vol.I. Parte I. Geometria delle serie lineari. Zanichelli, Bologna, 1926. 358 pp.
I. Shafarevic, Das allgemeine ReziprozitƤtsgesetz. Mat. Sbornik, n.S. 26(68), 113ā146 (1951). Russisch
N. Schappacher, E. Scholz (eds.), Oswald TeichmĆ¼ller ā Leben und Werk. Jahresber. Dtsch. Math.-Ver. 94, 1ā39 (1992)
B.L. van der Waerden, Zur algebraischen Geometrie. Selected Papers. (Springer, Berlin, 1983), VII + 479 pp.
A. Weil, LāarithmĆ©tique sur les courbes algĆ©briques. Acta Math. 52, 281ā315 (1929)
A. Weil, Åuvres scientifiques. Collected papers. Vol. I (1926ā1951). (Springer, Berlin, 1979)
E. Witt, Zyklische Kƶrper und Algebren der Charakteristik p vom Grad p n. Struktur diskret bewerteter perfekter Kƶrper mit vollkommenem Restklassenkƶrper der Charakteristik p. J. Reine Angew. Math. 176, 126ā140 (1936)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
Ā© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Roquette, P. (2018). Towards Higher Genus. 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_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-99067-5_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-99066-8
Online ISBN: 978-3-319-99067-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)