On elementary equivalence, isomorphism and isogeny
Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 29-58.

Motivé par un travail récent de Florian Pop, nous étudions les liens entre trois notions d’équivalence pour des corps de fonctions : isomorphisme, équivalence élémentaire et la condition que les deux corps puissent se plonger l’un dans l’autre, ce que nous appelons isogénie. Certains de nos résulats sont purement géométriques : nous donnons une classification par isogénie des variétiés de Severi-Brauer et des quadriques. Ces résultats sont utilisés pour obtenir de nouveaux exemples de “équivalence élémentaire entraine isomorphisme” : pour toutes les courbes de genre zéro sur un corps de nombres et pour certaine courbe de genre un sur un corps de nombres, incluant des courbes qui ne sont pas des courbes elliptiques.

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.

@article{JTNB_2006__18_1_29_0,
     author = {Clark, Pete L.},
     title = {On elementary equivalence, isomorphism and isogeny},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {29--58},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {1},
     year = {2006},
     doi = {10.5802/jtnb.532},
     mrnumber = {2245874},
     zbl = {1106.12003},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.532/}
}
Clark, Pete L. On elementary equivalence, isomorphism and isogeny. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 29-58. doi : 10.5802/jtnb.532. http://www.numdam.org/articles/10.5802/jtnb.532/

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