Reciprocity laws and Galois representations: recent breakthroughs
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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 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 10.1112/S0024609300007244
- Avner Ash, Galois representations attached to mod $p$ cohomology of $\textrm {GL}(n,\textbf {Z})$, Duke Math. J. 65 (1992), no. 2, 235–255. MR 1150586, DOI 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 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, DOI 10.1007/978-3-642-52229-1
- 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 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 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, DOI 10.1090/surv/067
- 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 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 $\textrm {GL}_{2/{\Bbb Q}}$, Pure Appl. Math. Q. 2 (2006), no. 2, Special Issue: In honor of John H. Coates., 279–393. MR 2251474, DOI 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 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 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, Annals of Mathematics Studies, No. 83, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. 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 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 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 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 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, DOI 10.1007/978-3-663-09991-8
- 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, DOI 10.1007/978-1-4757-2103-4
- Mark Kisin, The Fontaine-Mazur conjecture for $\textrm {GL}_2$, J. Amer. Math. Soc. 22 (2009), no. 3, 641–690. MR 2505297, DOI 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, DOI 10.1515/9781400881710
- Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR 766911, DOI 10.1007/978-1-4684-0255-1
- 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
- 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 10.1007/s002220100174
- Robert P. Langlands, Base change for $\textrm {GL}(2)$, 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, DOI 10.1007/978-1-4612-0853-2
- 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, DOI 10.1007/978-3-642-51445-6
- 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, DOI 10.1007/978-3-662-03983-0
- I. I. Pjateckii-Sapiro, Zeta-functions of modular curves, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 317–360. MR 0337975
- M. S. Raghunathan, The congruence subgroup problem, Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 4, 299–308. MR 2067695, DOI 10.1007/BF02829437
- Ravi Ramakrishna, Infinitely ramified Galois representations, Ann. of Math. (2) 151 (2000), no. 2, 793–815. MR 1765710, DOI 10.2307/121048
- K. A. Ribet, On modular representations of $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$ arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR 1047143, DOI 10.1007/BF01231195
- Peter Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 245–313. MR 3090258, DOI 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 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, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216
- 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, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 (1966), 209–220. MR 188198, DOI 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 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 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 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, DOI 10.1007/978-1-4612-1934-7
- 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 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 10.2307/2317083
Additional Information
- Jared Weinstein
- Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts
- Received by editor(s): May 18, 2015
- Published electronically: August 25, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 53 (2016), 1-39
- MSC (2010): Primary 11R37, 11R39, 11F80
- DOI: https://doi.org/10.1090/bull/1515
- MathSciNet review: 3403079