An example of non-cotorsion Selmer group
HTML articles powered by AMS MathViewer
- by King Fai Lai, Ignazio Longhi, Ki-Seng Tan and Fabien Trihan PDF
- Proc. Amer. Math. Soc. 143 (2015), 2355-2364 Request permission
Abstract:
Let $A/K$ be an elliptic curve over a global field of characteristic $p>0$. We provide an example where the Pontrjagin dual of the Selmer group of $A$ over a $\Gamma :=\mathbb {Z}_p$-extension $L/K$ is not a torsion $\mathbb {Z}_p[[\Gamma ]]$-module and show that the Iwasawa Main Conjecture for $A/L$ holds nevertheless.References
- Cristian D. González-Avilés and Ki-Seng Tan, A generalization of the Cassels-Tate dual exact sequence, Math. Res. Lett. 14 (2007), no. 2, 295–302. MR 2318626, DOI 10.4310/MRL.2007.v14.n2.a11
- William E. Lang, Extremal rational elliptic surfaces in characteristic $p$. I. Beauville surfaces, Math. Z. 207 (1991), no. 3, 429–437. MR 1115175, DOI 10.1007/BF02571400
- Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266. MR 444670, DOI 10.1007/BF01389815
- J. S. Milne, On a conjecture of Artin and Tate, Ann. of Math. (2) 102 (1975), no. 3, 517–533. MR 414558, DOI 10.2307/1971042
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804
- Tetsuji Shioda, Some remarks on elliptic curves over function fields, Astérisque 209 (1992), 12, 99–114. Journées Arithmétiques, 1991 (Geneva). MR 1211006
- Christopher Skinner and Eric Urban, The Iwasawa main conjectures for $\rm GL_2$, Invent. Math. 195 (2014), no. 1, 1–277. MR 3148103, DOI 10.1007/s00222-013-0448-1
- Ki-Seng Tan, Modular elements over function fields, J. Number Theory 45 (1993), no. 3, 295–311. MR 1247386, DOI 10.1006/jnth.1993.1079
- Ki-Seng Tan, Selmer groups over $\Bbb {Z}_p^d$-extensions, Math. Ann. 359 (2014), no. 3-4, 1025–1075. MR 3231024, DOI 10.1007/s00208-014-1023-9
- John Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 306, 415–440. MR 1610977
Additional Information
- King Fai Lai
- Affiliation: School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China
- Email: kinglaihonkon@gmail.com
- Ignazio Longhi
- Affiliation: Department of Mathematics, National Taiwan University. Taipei 10764, Taiwan
- Address at time of publication: Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, No. 111 Ren’ai Road, Dushu Lake Higher Education Town, Suzhou Industrial Park, Suzhou 215123 Jiangsu, People’s Republic of China.
- Email: longhi@math.ntu.edu.tw
- Ki-Seng Tan
- Affiliation: Department of Mathematics, National Taiwan University, Taipei 10764, Taiwan
- Email: tan@math.ntu.edu.tw
- Fabien Trihan
- Affiliation: College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter, United Kingdom
- Address at time of publication: Department of Information and Communication Sciences, Faculty of Science and Technology, Sophia University, 4 Yonbancho, Chiyoda-ku, Tokyo 102-0081 Japan
- MR Author ID: 637441
- Email: f-trihan-52m@sophia.ac.jp
- Received by editor(s): August 6, 2013
- Received by editor(s) in revised form: January 21, 2014
- Published electronically: January 21, 2015
- Communicated by: Matthew A. Papanikolas
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 2355-2364
- MSC (2010): Primary 11S40; Secondary 11R23, 11R34, 11R42, 11R58, 11G05, 11G10
- DOI: https://doi.org/10.1090/S0002-9939-2015-12459-8
- MathSciNet review: 3326018