The p-parity conjecture for elliptic curves with a p-isogeny
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Kęstutis Česnavičius
Abstract
For an elliptic curve E over a number field K, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell–Weil rank. Assuming finiteness of
Acknowledgements
I thank my advisor Bjorn Poonen for his support and many helpful conversations and suggestions, as well as for reading the manuscript very carefully. I thank Tim Dokchitser for his lectures at the Postech Winter School 2012 in Pohang, South Korea which got me interested in the question answered by this paper, and for pointing out to me Theorem 4.6. Thanks are also due to POSTECH and the organizers of the winter school for an inspiring and hospitable atmosphere. I thank Karl Rubin for telling me that Theorem 1.6 follows from Theorem 1.4. I thank Douglas Ulmer for a very helpful conversation about the technique of twisting. I thank Tim Dokchitser, Jessica Fintzen, Jan Nekovář, and Bjorn Poonen for comments. I thank the anonymous referee for suggestions and a careful reading of the manuscript.
References
[1] Borel A. and Serre J.-P., Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164. 10.1007/BF02566948Search in Google Scholar
[2] Bosch S., Lütkebohmert W. and Raynaud M., Néron models, Ergeb. Math. Grenzgeb. (3) 21, Springer-Verlag, Berlin 1990. 10.1007/978-3-642-51438-8Search in Google Scholar
[3] Breuil C., Groupes p-divisibles, groupes finis et modules filtrés, Ann. of Math. (2) 152 (2000), no. 2, 489–549. 10.2307/2661391Search in Google Scholar
[4] Cassels J. W. S., Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung, J. reine angew. Math. 211 (1962), 95–112. 10.1515/crll.1962.211.95Search in Google Scholar
[5] Coates J., Elliptic curves with complex multiplication and Iwasawa theory, Bull. Lond. Math. Soc. 23 (1991), no. 4, 321–350. 10.1112/blms/23.4.321Search in Google Scholar
[6] Coates J., Fukaya T., Kato K. and Sujatha R., Root numbers, Selmer groups, and non-commutative Iwasawa theory, J. Algebraic Geom. 19 (2010), no. 1, 19–97. 10.1090/S1056-3911-09-00504-9Search in Google Scholar
[7] Deligne P., Courbes elliptiques: formulaire d’après J. Tate, Modular functions of one variable. IV (Antwerp 1972), Lecture Notes in Math. 476, Springer-Verlag, Berlin (1975), 53–73. 10.1007/BFb0097583Search in Google Scholar
[8] Dokchitser T. and Dokchitser V., Parity of ranks for elliptic curves with a cyclic isogeny, J. Number Theory 128 (2008), no. 3, 662–679. 10.1016/j.jnt.2007.02.008Search in Google Scholar
[9] Dokchitser T. and Dokchitser V., Regulator constants and the parity conjecture, Invent. Math. 178 (2009), no. 1, 23–71. 10.1007/s00222-009-0193-7Search in Google Scholar
[10] Dokchitser T. and Dokchitser V., On the Birch–Swinnerton-Dyer quotients modulo squares, Ann. of Math. (2) 172 (2010), no. 1, 567–596. 10.4007/annals.2010.172.567Search in Google Scholar
[11] Dokchitser T. and Dokchitser V., Root numbers and parity of ranks of elliptic curves, J. reine angew. Math. 658 (2011), 39–64. 10.1515/crelle.2011.060Search in Google Scholar
[12] Dokchitser T. and Dokchitser V., Local invariants of isogenous elliptic curves, Trans. Amer. Math. Soc., to appear. 10.1090/S0002-9947-2014-06271-5Search in Google Scholar
[13] Illusie L., Grothendieck’s existence theorem in formal geometry, Fundamental algebraic geometry, Math. Surveys Monogr. 123, American Mathematical Society, Providence (2005), 179–233. Search in Google Scholar
[14] Kim B. D., The parity conjecture for elliptic curves at supersingular reduction primes, Compos. Math. 143 (2007), no. 1, 47–72. 10.1112/S0010437X06002569Search in Google Scholar
[15] Kobayashi S., The local root number of elliptic curves with wild ramification, Math. Ann. 323 (2002), no. 3, 609–623. 10.1007/s002080200318Search in Google Scholar
[16] Liedtke C. and Schröer S., The Néron model over the Igusa curves, J. Number Theory 130 (2010), no. 10, 2157–2197. 10.1016/j.jnt.2010.03.016Search in Google Scholar
[17] Mazur B. and Rubin K., Ranks of twists of elliptic curves and Hilbert’s tenth problem, Invent. Math. 181 (2010), no. 3, 541–575. 10.1007/s00222-010-0252-0Search in Google Scholar
[18] Milne J. S., On the arithmetic of abelian varieties, Invent. Math. 17 (1972), 177–190. 10.1007/BF01425446Search in Google Scholar
[19] Nekovář J., Selmer complexes, Astérisque 310, Société Mathématique de France, Paris 2007. Search in Google Scholar
[20] Nekovář J., On the parity of ranks of Selmer groups. IV, Compos. Math. 145 (2009), no. 6, 1351–1359. 10.1112/S0010437X09003959Search in Google Scholar
[21] Nekovář J., Some consequences of a formula of Mazur and Rubin for arithmetic local constants, Algebra Number Theory 7 (2013), no. 5, 1101–1120. 10.2140/ant.2013.7.1101Search in Google Scholar
[22]
Nekovář J.,
Compatibility of arithmetic and algebraic local constants (the case
[23] Rohrlich D. E., Elliptic curves and the Weil–Deligne group, Elliptic curves and related topics, CRM Proc. Lecture Notes 4, American Mathematical Society, Providence (1994), 125–157. 10.1090/crmp/004/10Search in Google Scholar
[24] Rohrlich D. E., Galois theory, elliptic curves, and root numbers, Compos. Math. 100 (1996), no. 3, 311–349. Search in Google Scholar
[25] Rubin K., Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Arithmetic theory of elliptic curves (Cetraro 1997), Lecture Notes in Math. 1716, Springer-Verlag, Berlin (1999), 167–234. 10.1007/BFb0093455Search in Google Scholar
[26] Schaefer E. F., Class groups and Selmer groups, J. Number Theory 56 (1996), no. 1, 79–114. 10.1006/jnth.1996.0006Search in Google Scholar
[27] Serre J.-P., Local class field theory, Algebraic number theory (Brighton 1965), Thompson, Washington (1967), 128–161. Search in Google Scholar
[28] Serre J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331. 10.1007/978-3-642-39816-2_94Search in Google Scholar
[29] Serre J.-P., Linear representations of finite groups, Grad. Texts in Math. 42, Springer-Verlag, New York 1977. 10.1007/978-1-4684-9458-7Search in Google Scholar
[30] Serre J.-P., Local fields, Grad. Texts in Math. 67, Springer-Verlag, New York 1979. 10.1007/978-1-4757-5673-9Search in Google Scholar
[31] Serre J.-P., Galois cohomology, Springer Monogr. Math., Springer-Verlag, Berlin 2002. Search in Google Scholar
[32] Tate J. T., The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206. 10.1007/BF01389745Search in Google Scholar
[33] Tate J., Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable. IV (Antwerp 1972), Lecture Notes in Math. 476, Springer-Verlag, Berlin (1975), 33–52. 10.1007/BFb0097582Search in Google Scholar
[34] Tate J. and Oort F., Group schemes of prime order, Ann. Sci. Éc. Norm. Supér. (4) 3 (1970), 1–21. 10.24033/asens.1186Search in Google Scholar
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- The algebra and model theory of tame valued fields
- The p-parity conjecture for elliptic curves with a p-isogeny
- A complete solution of P. Samuel’s problem
- Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture
- K-theory of minuscule varieties
- On the normal sheaf of determinantal varieties
- Quantum binary polyhedral groups and their actions on quantum planes
Articles in the same Issue
- Frontmatter
- The algebra and model theory of tame valued fields
- The p-parity conjecture for elliptic curves with a p-isogeny
- A complete solution of P. Samuel’s problem
- Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture
- K-theory of minuscule varieties
- On the normal sheaf of determinantal varieties
- Quantum binary polyhedral groups and their actions on quantum planes
