staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Pete L. Clark
On elementary equivalence, isomorphism and isogeny
Journal de théorie des nombres de Bordeaux, 18 no. 1 (2006), p. 29-58, doi: 10.5802/jtnb.532
Article PDF | Reviews MR 2245874 | Zbl 1106.12003

Résumé - Abstract

Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero curves over a number field, and for certain genus one curves over a number field, including some which are not elliptic curves.

Bibliography

[1] S. A. Amitsur, Generic splitting fields of central simple algebras. Annals of Math. (2) 62 (1955), 8–43.  MR 70624 |  Zbl 0066.28604
[2] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete 21, Springer-Verlag, 1990.  MR 1045822 |  Zbl 0705.14001
[3] E. Bombieri, D. Mumford, Enriques’ classification of surfaces in char. p. III. Invent. Math. 35 (1976), 197–232.  MR 491720 |  Zbl 0336.14010
[4] J. W. S. Cassels, Diophantine equations with special reference to elliptic curves. J. London Math. Soc. 41 (1966), 193–291.  MR 199150 |  Zbl 0138.27002
[5] P.L. Clark, Period-index problems in WC-groups I: elliptic curves. J. Number Theory 114 (2005), 193–208.  MR 2163913 |  Zbl 02207377
[6] J.-L. Duret, Equivalence éleméntaire et isomorphisme des corps de courbe sur un cors algebriquement clos. J. Symbolic Logic 57 (1992), 808–923.  MR 1187449 |  Zbl 0774.12011
[7] R. Hartshorne, Algebraic geometry. Springer GTM 52, 1977.  MR 463157 |  Zbl 0367.14001
[8] D. Hoffmann, Isotropy of 5-dimensional quadratic forms over the function field of a quadric. Proc. Sympos. Pure Math. 58, Part 2, Amer. Math. Soc., Providence, 1995.  MR 1327299 |  Zbl 0824.11023
[9] S. Iitaka, An introduction to birational geometry of algebraic varieties. Springer GTM 76, 1982.  MR 637060 |  Zbl 0491.14006
[10] C. U. Jensen, H. Lenzing, Model-theoretic algebra with particular textitasis on fields, rings and modules. Algebra, Logic and Applications 2, Gordon and Breach Science Publishers, 1989.  MR 1057608 |  Zbl 0728.03026
[11] D. Krashen, Severi-Brauer varieties of semidirect product algebras. Doc. Math. 8 (2003), 527–546.  MR 2029172 |  Zbl 1047.16011
[12] T.-Y. Lam, The algebraic theory of quadratic forms. W. A. Benjamin, 1973.  MR 396410 |  Zbl 0259.10019
[13] Yu. I. Manin, Cubic Forms. Algebra, geometry, arithmetic. North-Holland, 1986.  MR 833513 |  Zbl 0582.14010
[14] H. Nishimura, Some remarks on rational points. Mem. Coll. Sci. Univ. Kyoto, Ser A. Math. 29 (1955), 189–192. Article |  MR 95851 |  Zbl 0068.14802
[15] J. Ohm, The Zariski problem for function fields of quadratic forms. Proc. Amer. Math. Soc. 124 (1996), no. 6., 1649–1685.  MR 1307553 |  Zbl 0859.11027
[16] D. Pierce, Function fields and elementary equivalence. Bull. London Math. Soc. 31 (1999), 431–440.  MR 1687564 |  Zbl 0959.03022
[17] F. Pop, Elementary equivalence versus isomorphism. Invent. Math. 150 (2002), no. 2, 385–408.  MR 1933588 |  Zbl 01965448
[18] W. Scharlau, Quadratic and Hermitian forms. Grundlehren 270, Springer-Verlag, 1985.  MR 770063 |  Zbl 0584.10010
[19] J. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics 106, Springer-Verlag, 1986.  MR 817210 |  Zbl 0585.14026
[20] A. Wadsworth, Similarity of quadratic forms and isomorphism of their function fields. Trans. Amer. Math. Soc. 208 (1975), 352–358.  MR 376527 |  Zbl 0336.15013
[21] E. Witt, Uber ein Gegenspiel zum Normensatz. Math. Z. 39 (1934), 462–467.  MR 1545510 |  Zbl 0010.14901 |  JFM 60.0915.01