Reciprocity laws and Galois representations: recent breakthroughs

Weinstein, Jared

doi:10.1090/bull/1515

Reciprocity laws and Galois representations: recent breakthroughs

Author: Jared Weinstein
Journal: Bull. Amer. Math. Soc. 53 (2016), 1-39
MSC (2010): Primary 11R37, 11R39, 11F80
DOI: https://doi.org/10.1090/bull/1515
Published electronically: August 25, 2015
MathSciNet review: 3403079
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a polynomial $f(x)$ with integer coefficients, a reciprocity law is a rule which determines, for a prime $p$, whether $f(x)$ modulo $p$ is the product of distinct linear factors. We examine reciprocity laws through the ages, beginning with Fermat, Euler and Gauss, and continuing through the modern theory of modular forms and Galois representations. We conclude with an exposition of Peter Scholze’s astonishing work on torsion classes in the cohomology of arithmetic manifolds.

References

Avner Ash, Darrin Doud, and David Pollack, Galois representations with conjectural connections to arithmetic cohomology, Duke Math. J. 112 (2002), no. 3, 521–579. MR 1896473, DOI https://doi.org/10.1215/S0012-9074-02-11235-6
Avner Ash and Robert Gross, Generalized non-abelian reciprocity laws: a context for Wiles’ proof, Bull. London Math. Soc. 32 (2000), no. 4, 385–397. MR 1760802, DOI https://doi.org/10.1112/S0024609300007244
Avner Ash, Galois representations attached to mod $p$ cohomology of ${\rm GL}(n,{\bf Z})$, Duke Math. J. 65 (1992), no. 2, 235–255. MR 1150586, DOI https://doi.org/10.1215/S0012-7094-92-06510-0
Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI https://doi.org/10.1090/S0894-0347-01-00370-8
S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961
Armand Borel and Lizhen Ji, Compactifications of symmetric and locally symmetric spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. MR 2189882
Spencer Bloch, Book Review: Étale cohomology, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 235–239. MR 1567311, DOI https://doi.org/10.1090/S0273-0979-1981-14894-1
Johan Bosman, Polynomials for projective representations of level one forms, Computational aspects of modular forms and Galois representations, Ann. of Math. Stud., vol. 176, Princeton Univ. Press, Princeton, NJ, 2011, pp. 159–172. MR 2857091
A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. MR 387495, DOI https://doi.org/10.1007/BF02566134
Joe P. Buhler, Icosahedral Galois representations, Lecture Notes in Mathematics, Vol. 654, Springer-Verlag, Berlin-New York, 1978. MR 0506171
A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR 1721403
J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
Frank Calegari and Matthew Emerton, Completed cohomology—a survey, Non-abelian fundamental groups and Iwasawa theory, London Math. Soc. Lecture Note Ser., vol. 393, Cambridge Univ. Press, Cambridge, 2012, pp. 239–257. MR 2905536
Laurent Clozel, Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 77–159 (French). MR 1044819
David A. Cox, Primes of the form $x^2 + ny^2$, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322
Pierre Deligne, Formes modulaires et représentations $l$-adiques, Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363, Lecture Notes in Math., vol. 175, Springer, Berlin, 1971, pp. Exp. No. 355, 139–172 (French). MR 3077124
Pierre Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 247–289 (French). MR 546620
Darrin Doud and Michael W. Moore, Even icosahedral Galois representations of prime conductor, J. Number Theory 118 (2006), no. 1, 62–70. MR 2220262, DOI https://doi.org/10.1016/j.jnt.2005.08.008
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, A local-global compatibility conjecture in the $p$-adic Langlands programme for ${\rm GL}_{2/{\Bbb Q}}$, Pure Appl. Math. Q. 2 (2006), no. 2, Special Issue: In honor of John H. Coates., 279–393. MR 2251474, DOI https://doi.org/10.4310/PAMQ.2006.v2.n2.a1
Gerd Faltings, $p$-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255–299. MR 924705, DOI https://doi.org/10.1090/S0894-0347-1988-0924705-1
Gerd Faltings, Almost étale extensions, Astérisque 279 (2002), 185–270. Cohomologies $p$-adiques et applications arithmétiques, II. MR 1922831
Jean-Marc Fontaine and William Messing, $p$-adic periods and $p$-adic étale cohomology, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 179–207. MR 902593, DOI https://doi.org/10.1090/conm/067/902593
Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41–78. MR 1363495
Jean-Marc Fontaine, Représentations $p$-adiques semi-stables, Astérisque 223 (1994), 113–184 (French). With an appendix by Pierre Colmez; Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293972
Gerhard Frey, Links between stable elliptic curves and certain Diophantine equations, Ann. Univ. Sarav. Ser. Math. 1 (1986), no. 1, iv+40. MR 853387
Stephen S. Gelbart, Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. Annals of Mathematics Studies, No. 83. MR 0379375
Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219. MR 733692, DOI https://doi.org/10.1090/S0273-0979-1984-15237-6
Stephen Gelbart, Three lectures on the modularity of $\overline \rho _{E,3}$ and the Langlands reciprocity conjecture, Modular forms and Fermat’s last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 155–207. MR 1638479
Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. MR 0342495
Mark Goresky, Compactifications and cohomology of modular varieties, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 551–582. MR 2192016
E. Hecke, Zur Theorie der elliptischen Modulfunktionen, Math. Ann. 97 (1927), no. 1, 210–242 (German). MR 1512360, DOI https://doi.org/10.1007/BF01447866
Florian Herzig, The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations, Duke Math. J. 149 (2009), no. 1, 37–116. MR 2541127, DOI https://doi.org/10.1215/00127094-2009-036

M. Harris, K.-W. Lan, R. Taylor, and J. Thorne, On the rigid cohomology of certain Shimura varieties, Preprint.

Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802

R. Huber, Continuous valuations, Math. Z. 212 (1993), no. 3, 455–477. MR 1207303, DOI https://doi.org/10.1007/BF02571668

Roland Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996. MR 1734903

Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716

Mark Kisin, The Fontaine-Mazur conjecture for ${\rm GL}_2$, J. Amer. Math. Soc. 22 (2009), no. 3, 641–690. MR 2505297, DOI https://doi.org/10.1090/S0894-0347-09-00628-6

Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569

Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR 766911

Chandrashekhar Khare and Jean-Pierre Wintenberger, On Serre’s conjecture for 2-dimensional mod $p$ representations of ${\rm Gal}(\overline {\Bbb Q}/\Bbb Q)$, Ann. of Math. (2) 169 (2009), no. 1, 229–253. MR 2480604, DOI https://doi.org/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 https://doi.org/10.1007/s00222-009-0205-7

Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1–241 (French, with English and French summaries). MR 1875184, DOI https://doi.org/10.1007/s002220100174

Robert P. Langlands, Base change for ${\rm GL}(2)$, Annals of Mathematics Studies, No. 96, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 574808

Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723

G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825

James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531

James S. Milne, Lie algebras, algebraic groups, and Lie groups, 2013, Available at www.jmilne.org/math/.

Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859

I. I. Pjateckii-Sapiro, Zeta-functions of modular curves, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 317–360. Lecture Notes in Math., Vol. 349. MR 0337975

M. S. Raghunathan, The congruence subgroup problem, Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 4, 299–308. MR 2067695, DOI https://doi.org/10.1007/BF02829437

Ravi Ramakrishna, Infinitely ramified Galois representations, Ann. of Math. (2) 151 (2000), no. 2, 793–815. MR 1765710, DOI https://doi.org/10.2307/121048

K. A. Ribet, On modular representations of ${\rm Gal}(\overline {\bf Q}/{\bf Q})$ arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR 1047143, DOI https://doi.org/10.1007/BF01231195

Peter Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 245–313. MR 3090258, DOI https://doi.org/10.1007/s10240-012-0042-x

Peter Scholze, $p$-adic Hodge theory for rigid-analytic varieties, Forum Math. Pi 1 (2013), e1, 77. MR 3090230, DOI https://doi.org/10.1017/fmp.2013.1

Peter Scholze, Perfectoid spaces: a survey, Current developments in mathematics 2012, Int. Press, Somerville, MA, 2013, pp. 193–227. MR 3204346

Peter Scholze, Torsion in the cohomology of locally symmetric spaces, Preprint, Bonn, 2013.

M. H. Sengün, Arithmetic aspects of Bianchi groups, Computations with Modular Forms: Proceedings of a summer school and conference, Heidelberg, August/September 2011, Contributions in Mathematical and Compuational Sciences, vol. 6, Springer, 2014, pp. 279–315.

J.-P. Serre, A course in arithmetic, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR 0344216

Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de ${\rm Gal}(\overline {\bf Q}/{\bf Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI https://doi.org/10.1215/S0012-7094-87-05413-5

Goro Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 (1966), 209–220. MR 188198, DOI https://doi.org/10.1515/crll.1966.221.209

R. Sujatha, H. N. Ramaswamy, and C. S. Yogananda (eds.), Math unlimited, Science Publishers, Enfield, NH; distributed by CRC Press, Boca Raton, FL, 2012. Essays in mathematics. MR 2885277

J. T. Tate, $p$-divisible groups, Proc. Conf. Local Fields (Driebergen, 1966) Springer, Berlin, 1967, pp. 158–183. MR 0231827

John Tate, Rigid analytic spaces, Invent. Math. 12 (1971), 257–289. MR 306196, DOI https://doi.org/10.1007/BF01403307

Jerrold Tunnell, Artin’s conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 173–175. MR 621884, DOI https://doi.org/10.1090/S0273-0979-1981-14936-3

Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI https://doi.org/10.2307/2118560

Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575

André Weil, Adeles and algebraic groups, Progress in Mathematics, vol. 23, Birkhäuser, Boston, Mass., 1982. With appendices by M. Demazure and Takashi Ono. MR 670072

Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI https://doi.org/10.2307/2118559

B. F. Wyman, What is a reciprocity law?, Amer. Math. Monthly 79 (1972), 571–586; correction, ibid. 80 (1973), 281. MR 308084, DOI https://doi.org/10.2307/2317083