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Iwasawa Theory, projective modules, and modular representations
About this Title
Ralph Greenberg, Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 211, Number 992
ISBNs: 978-0-8218-4931-6 (print); 978-1-4704-0609-7 (online)
DOI: https://doi.org/10.1090/S0065-9266-2010-00608-2
Published electronically: September 23, 2010
Keywords: Iwasawa theory for elliptic curves,
Noncommutative Iwasawa theory,
Iwasawa invariants,
Selmer groups,
parity conjecture,
root numbers.
MSC: Primary 11G05, 11R23; Secondary 20C15, 20C20.
Table of Contents
Chapters
- 1. Introduction.
- 2. Projective and quasi-projective modules.
- 3. Projectivity or quasi-projectivity of $X_{E}^{\Sigma _{\mbox {\tiny {0}}}}(K_{\infty })$.
- 4. Selmer atoms.
- 5. The structure of ${{\mathcal H}}_v(K_{\infty }, E)$.
- 6. The case where $\Delta$ is a $p$-group.
- 7. Other specific groups.
- 8. Some arithmetic illustrations.
- 9. Self-dual representations.
- 10. A duality theorem.
- 11. $p$-modular functions.
- 12. Parity.
- 13. More arithmetic illustrations.
Abstract
This paper shows that properties of projective modules over a group ring $\mathbf {Z}_p[\Delta ]$, where $\Delta$ is a finite Galois group, can be used to study the behavior of certain invariants which occur naturally in Iwasawa theory for an elliptic curve $E$. Modular representation theory for the group $\Delta$ plays a crucial role in this study. It is necessary to make a certain assumption about the vanishing of a $\mu$-invariant. We then study $\lambda$-invariants $\lambda _E(\sigma )$, where $\sigma$ varies over the absolutely irreducible representations of $\Delta$. We show that there are non-trivial relationships between these invariants under certain hypotheses.- J. L. Alperin, Local representation theory, Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, Cambridge, 1986. Modular representations as an introduction to the local representation theory of finite groups. MR 860771, DOI 10.1017/CBO9780511623592
- B. J. Birch and N. M. Stephens, The parity of the rank of the Mordell-Weil group, Topology 5 (1966), 295–299. MR 201379, DOI 10.1016/0040-9383(66)90021-8
- John Coates, Takako Fukaya, Kazuya Kato, and Ramdorai Sujatha, Root numbers, Selmer groups, and non-commutative Iwasawa theory, J. Algebraic Geom. 19 (2010), no. 1, 19–97. MR 2551757, DOI 10.1090/S1056-3911-09-00504-9
- J. Coates and R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), no. 1-3, 129–174. MR 1369413, DOI 10.1007/s002220050048
- J. Coates and R. Sujatha, Galois cohomology of elliptic curves, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 88, Published by Narosa Publishing House, New Delhi; for the Tata Institute of Fundamental Research, Mumbai, 2000. MR 1759312
- J. Coates and R. Sujatha, Fine Selmer groups of elliptic curves over $p$-adic Lie extensions, Math. Ann. 331 (2005), no. 4, 809–839. MR 2148798, DOI 10.1007/s00208-004-0609-z
- J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
- P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 501–597 (French). MR 0349635
- Vladimir Dokchitser, Root numbers of non-abelian twists of elliptic curves, Proc. London Math. Soc. (3) 91 (2005), no. 2, 300–324. With an appendix by Tom Fisher. MR 2167089, DOI 10.1112/S0024611505015261
- Tim Dokchitser and Vladimir Dokchitser, Parity of ranks for elliptic curves with a cyclic isogeny, J. Number Theory 128 (2008), no. 3, 662–679. MR 2389862, DOI 10.1016/j.jnt.2007.02.008
- T. Dokchitser, V. Dokchitser, On the Birch-Swinnerton-Dyer quotients modulo squares, Annals of Math. 172 (2010), 567–596.
- Tim Dokchitser and Vladimir Dokchitser, Self-duality of Selmer groups, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 257–267. MR 2475965, DOI 10.1017/S0305004108001989
- Tim Dokchitser and Vladimir Dokchitser, Regulator constants and the parity conjecture, Invent. Math. 178 (2009), no. 1, 23–71. MR 2534092, DOI 10.1007/s00222-009-0193-7
- T. Dokchitser and V. Dokchitser, Computations in non-commutative Iwasawa theory, Proc. Lond. Math. Soc. (3) 94 (2007), no. 1, 211–272. With an appendix by J. Coates and R. Sujatha. MR 2294995, DOI 10.1112/plms/pdl014
- Michael J. Drinen, Iwasawa $\mu$-invariants of elliptic curves and their symmetric powers, J. Number Theory 102 (2003), no. 2, 191–213. MR 1997788, DOI 10.1016/S0022-314X(03)00105-7
- Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636
- Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395. MR 528968, DOI 10.2307/1971116
- Tom Fisher, Descent calculations for the elliptic curves of conductor 11, Proc. London Math. Soc. (3) 86 (2003), no. 3, 583–606. MR 1974391, DOI 10.1112/S0024611502013977
- Cornelius Greither, Some cases of Brumer’s conjecture for abelian CM extensions of totally real fields, Math. Z. 233 (2000), no. 3, 515–534. MR 1750935, DOI 10.1007/s002090050485
- Ralph Greenberg, Iwasawa theory for $p$-adic representations, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97–137. MR 1097613, DOI 10.2969/aspm/01710097
- Ralph Greenberg, Iwasawa theory for motives, $L$-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 211–233. MR 1110394, DOI 10.1017/CBO9780511526053.008
- Ralph Greenberg, Trivial zeros of $p$-adic $L$-functions, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 149–174. MR 1279608, DOI 10.1090/conm/165/01606
- Ralph Greenberg, Iwasawa theory and $p$-adic deformations of motives, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 193–223. MR 1265554, DOI 10.1090/pspum/055.2/1265554
- Ralph Greenberg, Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, 1997) Lecture Notes in Math., vol. 1716, Springer, Berlin, 1999, pp. 51–144. MR 1754686, DOI 10.1007/BFb0093453
- Ralph Greenberg, Introduction to Iwasawa theory for elliptic curves, Arithmetic algebraic geometry (Park City, UT, 1999) IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, 2001, pp. 407–464. MR 1860044, DOI 10.1090/pcms/009/06
- R. Greenberg, Galois representations with open image, preprint.
- R. Greenberg, The image of Galois representations attached to elliptic curves with an isogeny, to appear in Amer. Jour. of Math.
- R. Greenberg, Selmer groups for elliptic curves over $p$-adic Lie extensions, in preparation.
- Ralph Greenberg and Vinayak Vatsal, On the Iwasawa invariants of elliptic curves, Invent. Math. 142 (2000), no. 1, 17–63. MR 1784796, DOI 10.1007/s002220000080
- Li Guo, General Selmer groups and critical values of Hecke $L$-functions, Math. Ann. 297 (1993), no. 2, 221–233. MR 1241803, DOI 10.1007/BF01459498
- Yoshitaka Hachimori and Kazuo Matsuno, An analogue of Kida’s formula for the Selmer groups of elliptic curves, J. Algebraic Geom. 8 (1999), no. 3, 581–601. MR 1689359
- Yoshitaka Hachimori and Romyar T. Sharifi, On the failure of pseudo-nullity of Iwasawa modules, J. Algebraic Geom. 14 (2005), no. 3, 567–591. MR 2129011, DOI 10.1090/S1056-3911-05-00396-6
- Yoshitaka Hachimori and Otmar Venjakob, Completely faithful Selmer groups over Kummer extensions, Doc. Math. Extra Vol. (2003), 443–478. Kazuya Kato’s fiftieth birthday. MR 2046605
- Michael Harris, Systematic growth of Mordell-Weil groups of abelian varieties in towers of number fields, Invent. Math. 51 (1979), no. 2, 123–141. MR 528019, DOI 10.1007/BF01390224
- Lawrence Howe, Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups, Canad. J. Math. 49 (1997), no. 4, 749–771. MR 1471055, DOI 10.4153/CJM-1997-037-7
- Kenkichi Iwasawa, Riemann-Hurwitz formula and $p$-adic Galois representations for number fields, Tohoku Math. J. (2) 33 (1981), no. 2, 263–288. MR 624610, DOI 10.2748/tmj/1178229453
- Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
- Kazuya Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), ix, 117–290 (English, with English and French summaries). Cohomologies $p$-adiques et applications arithmétiques. III. MR 2104361
- Yûji Kida, $l$-extensions of CM-fields and cyclotomic invariants, J. Number Theory 12 (1980), no. 4, 519–528. MR 599821, DOI 10.1016/0022-314X(80)90042-6
- Byoung Du Kim, The parity conjecture for elliptic curves at supersingular reduction primes, Compos. Math. 143 (2007), no. 1, 47–72. MR 2295194, DOI 10.1112/S0010437X06002569
- B. D. Kim, The parity conjecture over totally real fields for elliptic curves at supersingular reduction primes, J. Number Theory 129 (2009), 1149–1160.
- K. Kramer and J. Tunnell, Elliptic curves and local $\varepsilon$-factors, Compositio Math. 46 (1982), no. 3, 307–352. MR 664648
- 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
- B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, DOI 10.1007/BF01390348
- Barry Mazur and Karl Rubin, Finding large Selmer rank via an arithmetic theory of local constants, Ann. of Math. (2) 166 (2007), no. 2, 579–612. MR 2373150, DOI 10.4007/annals.2007.166.579
- Barry Mazur and Karl Rubin, Growth of Selmer rank in nonabelian extensions of number fields, Duke Math. J. 143 (2008), no. 3, 437–461. MR 2423759, DOI 10.1215/00127094-2008-025
- B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61. MR 354674, DOI 10.1007/BF01389997
- A. Movahhedi and T. Nguyễn-Quang-Đỗ, Sur l’arithmétique des corps de nombres $p$-rationnels, Séminaire de Théorie des Nombres, Paris 1987–88, Progr. Math., vol. 81, Birkhäuser Boston, Boston, MA, 1990, pp. 155–200 (French). MR 1042770
- P. Monsky, Generalizing the Birch-Stephens theorem. I. Modular curves, Math. Z. 221 (1996), no. 3, 415–420. MR 1381589, DOI 10.1007/PL00004518
- Jan Nekovář, On the parity of ranks of Selmer groups. II, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 2, 99–104 (English, with English and French summaries). MR 1813764, DOI 10.1016/S0764-4442(00)01808-5
- Jan Nekovář, Selmer complexes, Astérisque 310 (2006), viii+559 (English, with English and French summaries). MR 2333680
- Jan Nekovář, On the parity of ranks of Selmer groups. III, Doc. Math. 12 (2007), 243–274. MR 2350290
- Jan Nekovář, On the parity of ranks of Selmer groups. IV, Compos. Math. 145 (2009), no. 6, 1351–1359. With an appendix by Jean-Pierre Wintenberger. MR 2575086, DOI 10.1112/S0010437X09003959
- Jan Nekovář, Growth of Selmer groups of Hilbert modular forms over ring class fields, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 6, 1003–1022 (English, with English and French summaries). MR 2504111, DOI 10.24033/asens.2087
- A. Nichifor, Iwasawa theory for elliptic curves with cyclic isogenies, University of Washington, Ph.D. thesis, 2004.
- Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR 1737196
- Irving Reiner, Integral representations of cyclic groups of prime order, Proc. Amer. Math. Soc. 8 (1957), 142–146. MR 83493, DOI 10.1090/S0002-9939-1957-0083493-6
- David E. Rohrlich, Elliptic curves and the Weil-Deligne group, Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 125–157. MR 1260960, DOI 10.1090/crmp/004/10
- David E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), no. 3, 311–349. MR 1387669
- David E. Rohrlich, Root numbers of semistable elliptic curves in division towers, Math. Res. Lett. 13 (2006), no. 2-3, 359–376. MR 2231124, DOI 10.4310/MRL.2006.v13.n3.a3
- David E. Rohrlich, Scarcity and abundance of trivial zeros in division towers, J. Algebraic Geom. 17 (2008), no. 4, 643–675. MR 2424923, DOI 10.1090/S1056-3911-08-00462-1
- D. Rohrlich, Galois invariance of local root numbers, to appear in Math. Annalen.
- D. Rohrlich, Inductivity of the global root number, in preparation.
- K. Rubin and A. Silverberg, Families of elliptic curves with constant mod $p$ representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 148–161. MR 1363500
- Peter Schneider, $p$-adic height pairings. II, Invent. Math. 79 (1985), no. 2, 329–374. MR 778132, DOI 10.1007/BF01388978
- Jean-Pierre Serre, Corps locaux, Publications de l’Université de Nancago, No. VIII, Hermann, Paris, 1968 (French). Deuxième édition. MR 0354618
- Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
- Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
- Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
- M. Shuter, Descent on division fields of elliptic curves, University of Cambridge Ph.D. thesis (2006).
- Allan J. Silberger, $\textrm {PGL}_{2}$ over the $p$-adics: its representations, spherical functions, and Fourier analysis, Lecture Notes in Mathematics, Vol. 166, Springer-Verlag, Berlin-New York, 1970. MR 0285673
- Mak Trifković, On the vanishing of $\mu$-invariants of elliptic curves over $\Bbb Q$, Canad. J. Math. 57 (2005), no. 4, 812–843. MR 2152940, DOI 10.4153/CJM-2005-032-9
- Gergely Zábrádi, Characteristic elements, pairings and functional equations over the false Tate curve extension, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 3, 535–574. MR 2418704, DOI 10.1017/S0305004108001114
- G. Zábrádi, Pairings and functional equations over the $GL_2$-extension, to appear in Proc. of the London Math. Soc.