Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the Iwasawa theory of CM fields for supersingular primes


Author: Kâzım Büyükboduk
Journal: Trans. Amer. Math. Soc. 370 (2018), 927-966
MSC (2010): Primary 11G05, 11G07, 11G40, 11R23, 14G10
DOI: https://doi.org/10.1090/tran/7029
Published electronically: August 3, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The goal of this article is two-fold: First, to prove a (two-variable) main conjecture for a CM field $ F$ without assuming the $ p$-ordinary hypothesis of Katz, making use of what we call the Rubin-Stark $ \mathcal {L}$-restricted Kolyvagin systems, which is constructed out of the conjectural Rubin-Stark Euler system of rank $ g$. (We are also able to obtain weaker unconditional results in this direction.) The second objective is to prove the Park-Shahabi plus/minus main conjecture for a CM elliptic curve $ E$ defined over a general totally real field again using (a twist of the) Rubin-Stark Kolyvagin system. This latter result has consequences towards the Birch and Swinnerton-Dyer conjecture for $ E$.


References [Enhancements On Off] (What's this?)

  • [BL15] Kâzım Büyükboduk and Antonio Lei, Coleman-adapted Rubin-Stark Kolyvagin systems and supersingular Iwasawa theory of CM abelian varieties, Proc. Lond. Math. Soc. (3) 111 (2015), no. 6, 1338-1378. MR 3447796, https://doi.org/10.1112/plms/pdv054
  • [BL17] Kâzım Büyükboduk and Antonio Lei, Integral Iwasawa theory of galois representations for non-ordinary primes, Math. Z. 286 (2017), no. 1-2, 361-398. MR 3648502, https://doi.org/10.1007/s00209-016-1765-z
  • [Büy09a] Kâzim Büyükboduk, Kolyvagin systems of Stark units, J. Reine Angew. Math. 631 (2009), 85-107. MR 2542218, https://doi.org/10.1515/CRELLE.2009.042
  • [Büy09b] Kâzım Büyükboduk, Stark units and the main conjectures for totally real fields, Compos. Math. 145 (2009), no. 5, 1163-1195. MR 2551993, https://doi.org/10.1112/S0010437X09004163
  • [Büy10] Kâzim Büyükboduk, On Euler systems of rank $ r$ and their Kolyvagin systems, Indiana Univ. Math. J. 59 (2010), no. 4, 1277-1332. MR 2815034, https://doi.org/10.1512/iumj.2010.59.4237
  • [Büy11] Kâzım Büyükboduk, $ \Lambda $-adic Kolyvagin systems, Int. Math. Res. Not. IMRN 14 (2011), 3141-3206. MR 2817676, https://doi.org/10.1093/imrn/rnq186
  • [Büy14] Kâzım Büyükboduk, Main conjectures for CM fields and a Yager-type theorem for Rubin-Stark elements, Int. Math. Res. Not. IMRN 21 (2014), 5832-5873. MR 3273065
  • [Büy16] Barry Mazur and A celebration of the mathematical work of Glenn Stevens, Ann. Math. Qué. 40 (2016), no. 1, 1-16. MR 3512520, https://doi.org/10.1007/s40316-015-0053-3
  • [CW77] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), no. 3, 223-251. MR 0463176, https://doi.org/10.1007/BF01402975
  • [Hsi12] Ming-Lun Hsieh,
    Iwasawa main conjecture for CM fields, preprint, 2012.
  • [HT93] H. Hida and J. Tilouine, Anti-cyclotomic Katz $ p$-adic $ L$-functions and congruence modules, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 2, 189-259. MR 1209708
  • [HT94] H. Hida and J. Tilouine, On the anticyclotomic main conjecture for CM fields, Invent. Math. 117 (1994), no. 1, 89-147. MR 1269427, https://doi.org/10.1007/BF01232236
  • [IP06] Adrian Iovita and Robert Pollack, Iwasawa theory of elliptic curves at supersingular primes over $ \mathbb{Z}_p$-extensions of number fields, J. Reine Angew. Math. 598 (2006), 71-103. MR 2270567, https://doi.org/10.1515/CRELLE.2006.069
  • [Kat78] Nicholas M. Katz, $ p$-adic $ L$-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199-297. MR 513095, https://doi.org/10.1007/BF01390187
  • [Kat04] Kazuya Kato, $ p$-adic Hodge theory and values of zeta functions of modular forms, Cohomologies $ p$-adiques et applications arithmétiques. III. Astérisque 295 (2004), ix, 117-290. MR 2104361
  • [Kob03] Shin-ichi Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), no. 1, 1-36. MR 1965358, https://doi.org/10.1007/s00222-002-0265-4
  • [Mai08] Fabio Mainardi, On the main conjecture for CM fields, Amer. J. Math. 130 (2008), no. 2, 499-538. MR 2405166, https://doi.org/10.1353/ajm.2008.0019
  • [MR04] Barry Mazur and Karl Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (2004), no. 799, viii+96. MR 2031496, https://doi.org/10.1090/memo/0799
  • [Nek06] Jan Nekovář, Selmer complexes, Astérisque 310 (2006), viii+559. MR 2333680
  • [NQ84] Nguyễn-Quang-ỗ, Formations de classes et modules d'Iwasawa, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 167-185. MR 756093, https://doi.org/10.1007/BFb0099451
  • [NSW08] Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026
  • [Och05] Tadashi Ochiai, Euler system for Galois deformations, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 113-146. MR 2141691
  • [Pol03] Robert Pollack, On the $ p$-adic $ L$-function of a modular form at a supersingular prime, Duke Math. J. 118 (2003), no. 3, 523-558. MR 1983040, https://doi.org/10.1215/S0012-7094-03-11835-9
  • [Pop04] Cristian D. Popescu, Rubin's integral refinement of the abelian Stark conjecture, Stark's conjectures: recent work and new directions, Contemp. Math., vol. 358, Amer. Math. Soc., Providence, RI, 2004, pp. 1-35. MR 2088710, https://doi.org/10.1090/conm/358/06534
  • [PR84] Bernadette Perrin-Riou, Arithmétique des courbes elliptiques et théorie d'Iwasawa, Mém. Soc. Math. France (N.S.) 17 (1984), 130. MR 799673
  • [PR93] Bernadette Perrin-Riou, Fonctions $ L$ $ p$-adiques d'une courbe elliptique et points rationnels, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 4, 945-995. MR 1252935
  • [PR98] Bernadette Perrin-Riou, Systèmes d'Euler $ p$-adiques et théorie d'Iwasawa, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 5, 1231-1307. MR 1662231
  • [PR04] Robert Pollack and Karl Rubin, The main conjecture for CM elliptic curves at supersingular primes, Ann. of Math. (2) 159 (2004), no. 1, 447-464. MR 2052361, https://doi.org/10.4007/annals.2004.159.447
  • [PS11] Jeehoon Park and Shahab Shahabi, Plus/minus $ p$-adic $ L$-functions for Hilbert modular forms, J. Algebra 342 (2011), 197-211. MR 2824537, https://doi.org/10.1016/j.jalgebra.2011.04.033
  • [Roh84] David E. Rohrlich, On $ L$-functions of elliptic curves and cyclotomic towers, Invent. Math. 75 (1984), no. 3, 409-423. MR 735333, https://doi.org/10.1007/BF01388636
  • [Roh89] David E. Rohrlich, Nonvanishing of $ L$-functions for $ {\rm GL}(2)$, Invent. Math. 97 (1989), no. 2, 381-403. MR 1001846, https://doi.org/10.1007/BF01389047
  • [Rub85] Karl Rubin, Elliptic curves and $ {\bf Z}_p$-extensions, Compositio Math. 56 (1985), no. 2, 237-250. MR 809869
  • [Rub87] Karl Rubin, Local units, elliptic units, Heegner points and elliptic curves, Invent. Math. 88 (1987), no. 2, 405-422. MR 880958, https://doi.org/10.1007/BF01388915
  • [Rub91] Karl Rubin, The ``main conjectures'' of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25-68. MR 1079839, https://doi.org/10.1007/BF01239508
  • [Rub92] Karl Rubin, Stark units and Kolyvagin's ``Euler systems'', J. Reine Angew. Math. 425 (1992), 141-154. MR 1151317, https://doi.org/10.1515/crll.1992.425.141
  • [Rub96] Karl Rubin, A Stark conjecture ``over $ \mathbf {Z}$'' for abelian $ L$-functions with multiple zeros, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 33-62. MR 1385509
  • [Rub00] Karl Rubin, Euler systems, Annals of Mathematics Studies, vol. 147, Hermann Weyl Lectures, The Institute for Advanced Study. Princeton University Press, Princeton, NJ, 2000. MR 1749177
  • [Wil78] A. Wiles, Higher explicit reciprocity laws, Ann. Math. (2) 107 (1978), no. 2, 235-254. MR 0480442
  • [Wil95] Andrew Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443-551. MR 1333035, https://doi.org/10.2307/2118559

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G05, 11G07, 11G40, 11R23, 14G10

Retrieve articles in all journals with MSC (2010): 11G05, 11G07, 11G40, 11R23, 14G10


Additional Information

Kâzım Büyükboduk
Affiliation: Department of Mathematics, Koç University, 34450 Sariyer, Istanbul, Turkey
Address at time of publication: School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
Email: kbuyukboduk@ku.edu.tr

DOI: https://doi.org/10.1090/tran/7029
Keywords: Iwasawa theory, Iwasawa's main conjectures, Rubin-Stark elements
Received by editor(s): December 27, 2013
Received by editor(s) in revised form: October 30, 2014, and May 12, 2016
Published electronically: August 3, 2017
Additional Notes: This work was partially supported by Marie Curie IRG grant EC-FP7 230668, TÜBİTAK grant 113F059, EU Horizon 2020 MC-GF Grant CriticalGZ/745691, and the Turkish Academy of Sciences.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society