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

Abstract

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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.

How to cite

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Clark, 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

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