# On elementary equivalence, isomorphism and isogeny

Pete L. Clark^{[1]}

- [1] 1126 Burnside Hall Department of Mathematics and Statistics McGill University 805 Sherbrooke West Montreal, QC, Canada H3A 2K6

Journal de Théorie des Nombres de Bordeaux (2006)

- Volume: 18, Issue: 1, page 29-58
- ISSN: 1246-7405

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topClark, Pete L.. "On elementary equivalence, isomorphism and isogeny." Journal de Théorie des Nombres de Bordeaux 18.1 (2006): 29-58. <http://eudml.org/doc/249643>.

@article{Clark2006,

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.},

affiliation = {1126 Burnside Hall Department of Mathematics and Statistics McGill University 805 Sherbrooke West Montreal, QC, Canada H3A 2K6},

author = {Clark, Pete L.},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {elementary equivalence; isomorphism; isogeny; function field; Severi-Brauer variety; quadric; elliptic curve; Jacobian},

language = {eng},

number = {1},

pages = {29-58},

publisher = {Université Bordeaux 1},

title = {On elementary equivalence, isomorphism and isogeny},

url = {http://eudml.org/doc/249643},

volume = {18},

year = {2006},

}

TY - JOUR

AU - Clark, Pete L.

TI - On elementary equivalence, isomorphism and isogeny

JO - Journal de Théorie des Nombres de Bordeaux

PY - 2006

PB - Université Bordeaux 1

VL - 18

IS - 1

SP - 29

EP - 58

AB - 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.

LA - eng

KW - elementary equivalence; isomorphism; isogeny; function field; Severi-Brauer variety; quadric; elliptic curve; Jacobian

UR - http://eudml.org/doc/249643

ER -

## References

top- S. A. Amitsur, Generic splitting fields of central simple algebras. Annals of Math. (2) 62 (1955), 8–43. Zbl0066.28604MR70624
- S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete 21, Springer-Verlag, 1990. Zbl0705.14001MR1045822
- E. Bombieri, D. Mumford, Enriques’ classification of surfaces in char. p. III. Invent. Math. 35 (1976), 197–232. Zbl0336.14010MR491720
- J. W. S. Cassels, Diophantine equations with special reference to elliptic curves. J. London Math. Soc. 41 (1966), 193–291. Zbl0138.27002MR199150
- P.L. Clark, Period-index problems in WC-groups I: elliptic curves. J. Number Theory 114 (2005), 193–208. Zbl1087.11036MR2163913
- J.-L. Duret, Equivalence éleméntaire et isomorphisme des corps de courbe sur un cors algebriquement clos. J. Symbolic Logic 57 (1992), 808–923. Zbl0774.12011MR1187449
- R. Hartshorne, Algebraic geometry. Springer GTM 52, 1977. Zbl0367.14001MR463157
- 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. Zbl0824.11023MR1327299
- S. Iitaka, An introduction to birational geometry of algebraic varieties. Springer GTM 76, 1982. Zbl0491.14006MR637060
- 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. Zbl0728.03026MR1057608
- D. Krashen, Severi-Brauer varieties of semidirect product algebras. Doc. Math. 8 (2003), 527–546. Zbl1047.16011MR2029172
- T.-Y. Lam, The algebraic theory of quadratic forms. W. A. Benjamin, 1973. Zbl0259.10019MR396410
- Yu. I. Manin, Cubic Forms. Algebra, geometry, arithmetic. North-Holland, 1986. Zbl0582.14010MR833513
- H. Nishimura, Some remarks on rational points. Mem. Coll. Sci. Univ. Kyoto, Ser A. Math. 29 (1955), 189–192. Zbl0068.14802MR95851
- J. Ohm, The Zariski problem for function fields of quadratic forms. Proc. Amer. Math. Soc. 124 (1996), no. 6., 1649–1685. Zbl0859.11027MR1307553
- D. Pierce, Function fields and elementary equivalence. Bull. London Math. Soc. 31 (1999), 431–440. Zbl0959.03022MR1687564
- F. Pop, Elementary equivalence versus isomorphism. Invent. Math. 150 (2002), no. 2, 385–408. Zbl1162.12302MR1933588
- W. Scharlau, Quadratic and Hermitian forms. Grundlehren 270, Springer-Verlag, 1985. Zbl0584.10010MR770063
- J. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics 106, Springer-Verlag, 1986. Zbl0585.14026MR817210
- A. Wadsworth, Similarity of quadratic forms and isomorphism of their function fields. Trans. Amer. Math. Soc. 208 (1975), 352–358. Zbl0336.15013MR376527
- E. Witt, Uber ein Gegenspiel zum Normensatz. Math. Z. 39 (1934), 462–467. Zbl0010.14901MR1545510

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