How can we construct abelian Galois extensions of basic number fields?

Mazur, Barry

doi:10.1090/S0273-0979-2011-01326-X

How can we construct abelian Galois extensions of basic number fields?
HTML articles powered by AMS MathViewer

by Barry Mazur PDF

Bull. Amer. Math. Soc. 48 (2011), 155-209 Request permission

Abstract:

Irregular primes—37 being the first such prime—have played a great role in number theory. This article discusses Ken Ribet’s construction—for all irregular primes $p$—of specific abelian, unramified, degree $p$ extensions of the number fields $\mathbf {Q}(e^{2\pi i/p})$. These extensions with explicit information about their Galois groups (they are Galois over $\mathbf {Q}$) were predicted to exist ever since the work of Herbrand in the 1930s. Ribet’s method involves a tour through the theory of modular forms; it demonstrates the usefulness of congruences between cuspforms and Eisenstein series, a fact that has inspired, and continues to inspire, much work in number theory.

References

Emil Artin and John Tate, Class field theory, 2nd ed., Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990. MR 1043169
Joël Bellaïche, Relèvement des formes modulaires de Picard, J. London Math. Soc. (2) 74 (2006), no. 1, 13–25 (French, with English summary). MR 2254549, DOI 10.1112/S0024610706022824
J. Bellaïche, Congruences endoscopiques et représentations galoisiennes, Thesis, Université Paris-Sud, 2002.
Joël Bellaïche, À propos d’un lemme de Ribet, Rend. Sem. Mat. Univ. Padova 109 (2003), 45–62 (French, with French and Italian summaries). MR 1997986
Joël Bellaïche and Gaëtan Chenevier, Formes non tempérées pour $\rm U(3)$ et conjectures de Bloch-Kato, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 4, 611–662 (French, with English and French summaries). MR 2097894, DOI 10.1016/j.ansens.2004.05.001
Joël Bellaïche and Gaëtan Chenevier, Families of Galois representations and Selmer groups, Astérisque 324 (2009), xii+314 (English, with English and French summaries). MR 2656025
Jacob Bernoulli, The art of conjecturing, Johns Hopkins University Press, Baltimore, MD, 2006. Together with “Letter to a friend on sets in court tennis”; Translated from the Latin and with an introduction and notes by Edith Dudley Sylla. MR 2195221
R. Brauer and C. Nesbitt, On the modular characters of groups, Ann. of Math. (2) 42 (1941), 556–590. MR 4042, DOI 10.2307/1968918
Joe Buhler, Richard Crandall, Reijo Ernvall, Tauno Metsänkylä, and M. Amin Shokrollahi, Irregular primes and cyclotomic invariants to 12 million, J. Symbolic Comput. 31 (2001), no. 1-2, 89–96. Computational algebra and number theory (Milwaukee, WI, 1996). MR 1806208, DOI 10.1006/jsco.1999.1011
J. Buhler, D. Harvey, Irregular primes to $163$ million, arXiv:0912.2121v2 [math.NT] (2009).
Frank Calegari and Barry Mazur, Nearly ordinary Galois deformations over arbitrary number fields, J. Inst. Math. Jussieu 8 (2009), no. 1, 99–177. MR 2461903, DOI 10.1017/S1474748008000327
J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. Reprint of the 1967 original. MR 911121
N. Tschebotareff, Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören, Math. Ann. 95 (1926), no. 1, 191–228 (German). MR 1512273, DOI 10.1007/BF01206606
Gaëtan Chenevier, Familles $p$-adiques de formes automorphes pour $\textrm {GL}_n$, J. Reine Angew. Math. 570 (2004), 143–217 (French, with English summary). MR 2075765, DOI 10.1515/crll.2004.031
Gaëtan Chenevier, Une correspondance de Jacquet-Langlands $p$-adique, Duke Math. J. 126 (2005), no. 1, 161–194 (French, with English and French summaries). MR 2111512, DOI 10.1215/S0012-7094-04-12615-6
G. Chenevier, The $p$-adic analytic space of pseudo-characters of a profinite group and pseudo-representations over arbitrary rings, arXiv:0809.0415v1 [math.NT] Sept. 2 (2008).
L. Clozel, On Ribet’s level-raising theorem for $\rm U(3)$, Amer. J. Math. 122 (2000), no. 6, 1265–1287. MR 1797662
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013113
P. Deligne, Formes modulaires et représentations ${\ell }$-adiques, Séminaire Bourbaki, Lect. Notes in Math. 1799 Springer, 139-172 (1971).
P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143–316 (French). MR 0337993
Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 379379
Matthew Emerton, The Eisenstein ideal in Hida’s ordinary Hecke algebra, Internat. Math. Res. Notices 15 (1999), 793–802. MR 1710074, DOI 10.1155/S1073792899000409
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
Wee Teck Gan and Nadya Gurevich, CAP representations of $G_2$ and the spin $L$-function of $\textrm {PGSp}_6$, Israel J. Math. 170 (2009), 1–52. MR 2506316, DOI 10.1007/s11856-009-0018-9
Loren Graham and Jean-Michel Kantor, Naming infinity, The Belknap Press of Harvard University Press, Cambridge, MA, 2009. A true story of religious mysticism and mathematical creativity. MR 2526973
J. Herbrand, Sur les classes des corps circulaires, J. Math. Pure Appl. (9) II 417-441 (1932).
Haruzo Hida, On $p$-adic Hecke algebras for $\textrm {GL}_2$ over totally real fields, Ann. of Math. (2) 128 (1988), no. 2, 295–384. MR 960949, DOI 10.2307/1971444
Haruzo Hida, On nearly ordinary Hecke algebras for $\textrm {GL}(2)$ over totally real fields, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 139–169. MR 1097614, DOI 10.2969/aspm/01710139
Haruzo Hida, $p$-adic ordinary Hecke algebras for $\textrm {GL}(2)$, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 5, 1289–1322 (English, with English and French summaries). MR 1313784
H. Hida, Control Theorems and Applications, Lectures at Tata Institute of Fundamental Research (Version of 2/15/00) [See http://www.math.ucla.edu/$\sim$hida/]
Haruzo Hida, Hilbert modular forms and Iwasawa theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2243770, DOI 10.1093/acprof:oso/9780198571025.001.0001
Haruzo Hida, Jacques Tilouine, and Eric Urban, Adjoint modular Galois representations and their Selmer groups, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 21, 11121–11124. Elliptic curves and modular forms (Washington, DC, 1996). MR 1491970, DOI 10.1073/pnas.94.21.11121
Chandrashekhar Khare, Serre’s modularity conjecture: a survey of the level one case, $L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 270–299. MR 2392357, DOI 10.1017/CBO9780511721267.008
Chandrashekhar Khare, Notes on Ribet’s converse to Herbrand, Cyclotomic fields and related topics (Pune, 1999) Bhaskaracharya Pratishthana, Pune, 2000, pp. 273–284. MR 1802388
Chandrashekhar Khare and Jean-Pierre Wintenberger, On Serre’s conjecture for 2-dimensional mod $p$ representations of $\textrm {Gal}(\overline {\Bbb Q}/\Bbb Q)$, Ann. of Math. (2) 169 (2009), no. 1, 229–253. MR 2480604, DOI 10.4007/annals.2009.169.229
Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504. MR 2551763, DOI 10.1007/s00222-009-0205-7
Mark Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math. 153 (2003), no. 2, 373–454. MR 1992017, DOI 10.1007/s00222-003-0293-8
M. Koike, Congruences between cuspforms of weight one and of weight two and a remark on a theorem of Deligne and Serre (Int. Symposium on Algebraic Number Theory, Kyoto, March 1976).
Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723, DOI 10.1007/978-1-4612-0853-2
B. Mazur, Deforming Galois representations, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172, DOI 10.1007/978-1-4613-9649-9_{7}
B. Mazur and A. Wiles, Analogies between function fields and number fields, Amer. J. Math. 105 (1983), no. 2, 507–521. MR 701567, DOI 10.2307/2374266
B. Mazur and A. Wiles, Class fields of abelian extensions of $\textbf {Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI 10.1007/BF01388599
B. Mazur and A. Wiles, On $p$-adic analytic families of Galois representations, Compositio Math. 59 (1986), no. 2, 231–264. MR 860140
I. Piatetski-Shapiro, Two Conjectures on $L$-functions, pp. 519-522 in Wolf Prize in Mathematics Vol 2. (Eds. S.S. Chern and F. Hirzebruch) World Scientific, 2000.
I. I. Piatetski-Shapiro, On the Saito-Kurokawa lifting, Invent. Math. 71 (1983), no. 2, 309–338. MR 689647, DOI 10.1007/BF01389101
F. Pollaczek, Über die irregulären Kreiskörper der $l$-ten und $l^2$-ten Einheitswurzeln, Math. Z. 21 (1924), no. 1, 1–38 (German). MR 1544682, DOI 10.1007/BF01187449
Srinivasa Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (9) 159-184 (1916).
Michel Raynaud, Schémas en groupes de type $(p,\dots , p)$, Bull. Soc. Math. France 102 (1974), 241–280 (French). MR 419467
Kenneth A. Ribet, A modular construction of unramified $p$-extensions of $Q(\mu _{p})$, Invent. Math. 34 (1976), no. 3, 151–162. MR 419403, DOI 10.1007/BF01403065
Kenneth A. Ribet, Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol. 601, Springer, Berlin, 1977, pp. 17–51. MR 0453647
Kenneth A. Ribet and William A. Stein, Lectures on Serre’s conjectures, Arithmetic algebraic geometry (Park City, UT, 1999) IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, 2001, pp. 143–232. MR 1860042, DOI 10.1090/pcms/009/04
David E. Rohrlich, Modular curves, Hecke correspondence, and $L$-functions, Modular forms and Fermat’s last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 41–100. MR 1638476
Jean-Pierre Serre, Zeta and $L$ functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 82–92. MR 0194396
Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
Jean-Pierre Serre, Œuvres. Vol. II, Springer-Verlag, Berlin, 1986 (French). 1960–1971. MR 926690
Jean-Pierre Serre, Œuvres. Vol. III, Springer-Verlag, Berlin, 1986 (French). 1972–1984. MR 926691
Jean-Pierre Serre, Œuvres. Vol. III, Springer-Verlag, Berlin, 1986 (French). 1972–1984. MR 926691
Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI 10.1215/S0012-7094-87-05413-5
Goro Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523–544. MR 318162, DOI 10.2969/jmsj/02530523
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
C. Skinner, Elliptic Curves and Main Conjectures, Kuwait Foundation Lecture 49, May 24, 2005. http://www.dpmms.cam.ac.uk/Seminars/Kuwait/abstracts/L49.pdf
C. M. Skinner and A. J. Wiles, Residually reducible representations and modular forms, Inst. Hautes Études Sci. Publ. Math. 89 (1999), 5–126 (2000). MR 1793414
Christopher Skinner and Eric Urban, Sur les déformations $p$-adiques de certaines représentations automorphes, J. Inst. Math. Jussieu 5 (2006), no. 4, 629–698 (French, with English and French summaries). MR 2261226, DOI 10.1017/S147474800600003X
Christopher Skinner and Eric Urban, Vanishing of $L$-functions and ranks of Selmer groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 473–500. MR 2275606
C. Skinner, E. Urban, The main conjecture for $\mathrm {GL}_2$; See: http://www.math.columbia.edu/$\sim$urban/EURP08.html
William A. Stein, An introduction to computing modular forms using modular symbols, Algorithmic number theory: lattices, number fields, curves and cryptography, Math. Sci. Res. Inst. Publ., vol. 44, Cambridge Univ. Press, Cambridge, 2008, pp. 641–652. MR 2467560, DOI 10.2977/prims/1210167339
H. P. F. Swinnerton-Dyer, On $l$-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 1–55. MR 0406931
H. P. F. Swinnerton-Dyer, On $l$-adic representations and congruences for coefficients of modular forms. II, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol. 601, Springer, Berlin, 1977, pp. 63–90. MR 0498392
E. Urban, On residually reducible representations on local rings, J. Algebra 212 (1999), no. 2, 738–742. MR 1676863, DOI 10.1006/jabr.1998.7635
Eric Urban, Selmer groups and the Eisenstein-Klingen ideal, Duke Math. J. 106 (2001), no. 3, 485–525. MR 1813234, DOI 10.1215/S0012-7094-01-10633-9
E. Urban, Groupes de Selmer et Fonctions $L$ $p$-adiques pour les représentations modulaires adjointes, See: http://www.math.columbia.edu/$\sim$urban/EURP08.html
H. Vandiver, Fermat’s Last Theorem: Its history and the nature of the known results concerning it, Amer. Math. Monthly 53, 555-578 (1946); 60, 164-167 (1953).
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
A. Wiles, On ordinary $\lambda$-adic representations associated to modular forms, Invent. Math. 94 (1988), no. 3, 529–573. MR 969243, DOI 10.1007/BF01394275
A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540. MR 1053488, DOI 10.2307/1971468