Abstract
We show that a nontrivial abelian variety over a Hilbertian field in which the weak Mordell–Weil theorem holds admits infinitely many torsors with period any given
Acknowledgements
This work had its genesis in a conversation with Shahed Sharif. In particular, the main idea of the proof of Lemma 7.4 is due to him. We thank Cristian D. González-Avilés for technical (and moral) support. We are grateful to Kestutis Česnavičius for help strengthening the statement and clarifying the proof of Theorem 5.2. We thank the referee for a careful reading.
References
[1] M. Artin and H. P. F. Swinnerton-Dyer, The Shafarevich–Tate conjecture for pencils of elliptic curves on K3 surfaces, Invent. Math. 20 (1973), 249–266. 10.1007/BF01394097Search in Google Scholar
[2] P. L. Clark, Period-index problems in WC-groups II: Abelian varieties, preprint (2004), http://arxiv.org/abs/math/0406135. Search in Google Scholar
[3] P. L. Clark, There are genus one curves of every index over every number field, J. reine angew. Math. 594 (2006), 201–206. 10.1515/CRELLE.2006.040Search in Google Scholar
[4] P. L. Clark, On the indices of curves over local fields, Manuscripta Math. 124 (2007), 411–426. 10.1007/s00229-007-0126-ySearch in Google Scholar
[5] P. L. Clark, The period-index problem in WC-groups IV: A local transition theorem, J. Théor. Nombres Bordeaux 22 (2010), 583–606. 10.5802/jtnb.734Search in Google Scholar
[6]
B. Conrad,
Chow’s
[7] M. D. Fried and M. Jarden, Field arithmetic, 3rd ed., Ergeb. Math. Grenzgeb. (3), Springer, Berlin 2008. Search in Google Scholar
[8] C. D. González-Avilés and K.-S. Tan, A generalization of the Cassels–Tate dual exact sequence, Math. Res. Lett. 14 (2007), 295–302. 10.4310/MRL.2007.v14.n2.a11Search in Google Scholar
[9] B. H. Gross, Lectures on the conjecture of Birch and Swinnerton-Dyer, Arithmetic of L-functions (Park City 2009), IAS/Park City Math. Ser. 18, American Mathematical Society, Providence (2011), 169–209. 10.1090/pcms/018/08Search in Google Scholar
[10] A. Grothendieck, Le groupe de Brauer III, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam (1968), 88–188. Search in Google Scholar
[11] M. Ikeda, Zur Existenz eigentlicher galoisscher Körper beim Einbettungsproblem für galoissche Algebren, Abh. Math. Semin. Univ. Hambg. 24 (1960), 126–131. 10.1007/BF02942025Search in Google Scholar
[12] V. Kolyvagin, On the Mordell–Weil and Shafarevich–Tate groups for Weil elliptic curves, Math. USSR Izv. 33 (1989), 473–499. 10.1070/IM1989v033n03ABEH000853Search in Google Scholar
[13] S. Lang, Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563. 10.2307/2372673Search in Google Scholar
[14] S. Lang and A. Néron, Rational points of abelian varieties over function fields, Amer. J. Math. 81 (1959), 95–118. 10.2307/2372851Search in Google Scholar
[15] S. Lang and J. Tate, Principal homogeneous spaces over abelian varieties, Amer. J. Math. 80 (1958), 659–684. 10.2307/2372778Search in Google Scholar
[16] H. Matsumura, Commutative ring theory, 2nd ed., Cambridge Stud. Adv. Math. 8, Cambridge University Press, Cambridge 1989. Search in Google Scholar
[17] J. S. Milne, The Brauer group of a rational surface, Invent. Math. 11 (1970), 304–307. 10.1007/BF01403184Search in Google Scholar
[18] J. S. Milne, Weil–Châtelet groups over local fields, Ann. Sci. Éc. Norm. Supér. (4) 3 (1970), 273–284. 10.24033/asens.1193Search in Google Scholar
[19] J. S. Milne, On a conjecture of Artin and Tate, Ann. of Math. (2) 102 (1975), 517–533. 10.2307/1971042Search in Google Scholar
[20] J. S. Milne, Arithmetic duality theorems, Perspect. in Math. 1, Academic Press, Boston 1986. Search in Google Scholar
[21] W. R. Scott, Group theory, 2nd ed., Dover Publications, New York 1987. Search in Google Scholar
[22] J.-P. Serre, Lie algebras and lie groups, 2nd ed., Lecture Notes in Math. 1500, Springer, Berlin 1992. 10.1007/978-3-540-70634-2Search in Google Scholar
[23] J.-P. Serre, Cohomologie Galoisienne, 5th ed., Lecture Notes in Math. 5, Springer, Berlin 1994. 10.1007/BFb0108758Search in Google Scholar
[24] I. R. Shafarevich, Exponents of elliptic curves, Dokl. Akad. Nauk SSSR (N.S.) 114 (1957), 714–716. Search in Google Scholar
[25] S. Sharif, Period and index of genus one curves over global fields, Math. Ann. 354 (2012), 1029–1047. 10.1007/s00208-011-0745-1Search in Google Scholar
[26] S. S. Shatz, Profinite groups, arithmetic, and geometry, Ann. of Math. Stud. 67, Princeton University Press, Princeton 1972. 10.1515/9781400881857Search in Google Scholar
[27] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. 10.2969/jmsj/02410020Search in Google Scholar
[28] T. Shioda, On the Mordell–Weil lattices, Comment. Math. Univ. St. Pauli 39 (1990), no. 2, 211–240. Search in Google Scholar
[29] J. Silverman, The arithmetic of elliptic curves, Grad. Texts in Math. 106, Springer, New York 1986. 10.1007/978-1-4757-1920-8Search in Google Scholar
[30] J. Silverman, Advanced topics in the arithmetic of elliptic curves, Grad. Texts in Math. 151, Springer, New York 1994. 10.1007/978-1-4612-0851-8Search in Google Scholar
[31]
W. A. Stein,
There are genus one curves over
[32] J. Tate, WC-groups over p-adic fields, Sem. Bourbaki Exp. 4 (1956–1958), 265–277. Search in Google Scholar
[33] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam (1968), 189–214. Search in Google Scholar
[34] W. C. Waterhouse, Introduction to affine group schemes, Grad. Texts in Math. 66, Springer, New York 1979. 10.1007/978-1-4612-6217-6Search in Google Scholar
[35] S. Willard, General topology, Addison-Wesley Publishing, Reading 1970. Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Modulo p parabolic induction of pro-p-Iwahori Hecke algebra
- There are genus one curves of every index over every infinite, finitely generated field
- Kuranishi-type moduli spaces for proper CR-submersions fibering over the circle
- Petersson inner products of weight-one modular forms
- Severi varieties and Brill–Noether theory of curves on abelian surfaces
- Convergent normal form for real hypersurfaces at a generic Levi-degeneracy
- Semi-continuity of stability for sheaves and variation of Gieseker moduli spaces
- Improved bounds in Weaver and Feichtinger conjectures
- On the ramification of étale cohomology groups
Articles in the same Issue
- Frontmatter
- Modulo p parabolic induction of pro-p-Iwahori Hecke algebra
- There are genus one curves of every index over every infinite, finitely generated field
- Kuranishi-type moduli spaces for proper CR-submersions fibering over the circle
- Petersson inner products of weight-one modular forms
- Severi varieties and Brill–Noether theory of curves on abelian surfaces
- Convergent normal form for real hypersurfaces at a generic Levi-degeneracy
- Semi-continuity of stability for sheaves and variation of Gieseker moduli spaces
- Improved bounds in Weaver and Feichtinger conjectures
- On the ramification of étale cohomology groups