Heilbronn characters

Foote, Richard; Ginsberg, Hy; Murty, V.

doi:10.1090/bull/1492

Heilbronn characters

Authors: Richard Foote, Hy Ginsberg and V. Kumar Murty
Journal: Bull. Amer. Math. Soc. 52 (2015), 465-496
MSC (2010): Primary 20C15, 11R42
DOI: https://doi.org/10.1090/bull/1492
Published electronically: April 6, 2015
MathSciNet review: 3348444
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Abstract: In a seminal paper in 1972 Hans Heilbronn introduced a virtual character associated to representations of Galois extensions of number fields and Artin’s Conjecture on the holomorphy of $L$-series. His construction has evolved in both application and scope, and may now be applied to produce what are called Heilbronn characters of arbitrary finite groups. This article surveys the inception and development of this concept, weaving together its number-theoretic and group-theoretic dimensions, and culminates in a description of the recent classification of unfaithful minimal Heilbronn characters. Connections with other areas of mathematics, variations on these themes, and possible future directions are also explored.

References

Emil Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. Math. Sem. Univ. Hamburg 5 (1927), no. 1, 353–363 (German). MR 3069486, DOI https://doi.org/10.1007/BF02952531
Michael Aschbacher, Radha Kessar, and Bob Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series, vol. 391, Cambridge University Press, Cambridge, 2011. MR 2848834
A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika 15 (1968), 204–216. MR 258756, DOI https://doi.org/10.1112/S0025579300002588
A. Baker, Imaginary quadratic fields with class number $2$, Ann. of Math. (2) 94 (1971), 139–152. MR 299583, DOI https://doi.org/10.2307/1970739

A. S. Bang, Taltheoretiske Undersølgelser, Tidskrift f. Math. 5 (1886), 70–80 and 130–137.

C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Extensions of $p$-local finite groups, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3791–3858. MR 2302515, DOI https://doi.org/10.1090/S0002-9947-07-04225-0

Carles Broto, Ran Levi, and Bob Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779–856. MR 1992826, DOI https://doi.org/10.1090/S0894-0347-03-00434-X

Carles Broto, Ran Levi, and Bob Oliver, Discrete models for the $p$-local homotopy theory of compact Lie groups and $p$-compact groups, Geom. Topol. 11 (2007), 315–427. MR 2302494, DOI https://doi.org/10.2140/gt.2007.11.315

L. Clozel, Base change for ${\rm GL}(n)$, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 791–797. MR 934282, DOI https://doi.org/10.1215/s0012-7094-87-05525-6

J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339. MR 554399, DOI https://doi.org/10.1112/blms/11.3.308

Everett C. Dade, Accessible characters are monomial, J. Algebra 117 (1988), no. 1, 256–266. MR 955603, DOI https://doi.org/10.1016/0021-8693%2888%2990253-0

P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI https://doi.org/10.2307/1971021

M. Deuring, Die zetafunktion einer algebraischen Kurve vom Geschlechte Eins. I, II, IIII, IV. Nachr. Akad. Wiss., Göttingen, (1953), 85–94; (1955), 13–42; (1956), 37–76; (1957), 55–80.

Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636

Pamela Ferguson and I. M. Isaacs, Induced characters which are multiples of irreducibles, J. Algebra 124 (1989), no. 1, 149–157. MR 1005700, DOI https://doi.org/10.1016/0021-8693%2889%2990156-7

Ramón J. Flores and Jérôme Scherer, Cellularization of classifying spaces and fusion properties of finite groups, J. Lond. Math. Soc. (2) 76 (2007), no. 1, 41–56. MR 2351607, DOI https://doi.org/10.1112/jlms/jdm031

Ramón J. Flores and Richard M. Foote, Strongly closed subgroups of finite groups, Adv. Math. 222 (2009), no. 2, 453–484. MR 2538017, DOI https://doi.org/10.1016/j.aim.2009.05.005

Ramón J. Flores and Richard M. Foote, The cellular structure of the classifying spaces of finite groups, Israel J. Math. 184 (2011), 129–156. MR 2823972, DOI https://doi.org/10.1007/s11856-011-0062-0

R. Foote, Non-monomial characters and Artin’s Conjecture, Trans. Amer. Math. Soc., 321 (1990), 261–272.

Richard Foote, Sylow $2$-subgroups of Galois groups arising as minimal counterexamples to Artin’s conjecture, Comm. Algebra 25 (1997), no. 2, 607–616. MR 1428801, DOI https://doi.org/10.1080/00927879708825877

Richard Foote, A characterization of finite groups containing a strongly closed $2$-subgroup, Comm. Algebra 25 (1997), no. 2, 593–606. MR 1428800, DOI https://doi.org/10.1080/00927879708825876

Richard Foote and V. Kumar Murty, Zeros and poles of Artin $L$-series, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 1, 5–11. MR 966135, DOI https://doi.org/10.1017/S0305004100001316

Richard Foote and David Wales, Zeros of order $2$ of Dedekind zeta functions and Artin’s conjecture, J. Algebra 131 (1990), no. 1, 226–257. MR 1055006, DOI https://doi.org/10.1016/0021-8693%2890%2990173-L

H. Ginsberg, Minimal Heilbronn Characters of Finite Groups, Doctoral Dissertation, University of Vermont, 2010.

Hy Ginsberg, Unfaithful minimal Heilbronn characters of finite groups, J. Algebra 331 (2011), 466–481. MR 2774670, DOI https://doi.org/10.1016/j.jalgebra.2010.09.043

Hy Ginsberg, Unfaithful minimal Heilbronn characters of $L_2(q)$, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 1, 57–69. MR 3021405, DOI https://doi.org/10.1017/S0013091512000168

Hy Ginsberg, Groups with strongly $p$-embedded subgroups and cyclic Sylow $p$-subgroups, J. Group Theory 17 (2014), no. 1, 175–190. MR 3176657, DOI https://doi.org/10.1515/jgt-2013-0039

George Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403–420. MR 202822, DOI https://doi.org/10.1016/0021-8693%2866%2990030-5

Dorian M. Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4, 624–663. MR 450233

David M. Goldschmidt, $2$-fusion in finite groups, Ann. of Math. (2) 99 (1974), 70–117. MR 335627, DOI https://doi.org/10.2307/1971014

David M. Goldschmidt, Strongly closed $2$-subgroups of finite groups, Ann. of Math. (2) 102 (1975), no. 3, 475–489. MR 393223, DOI https://doi.org/10.2307/1971040

Larry Joel Goldstein, Relative imaginary quadratic fields of class number $1$ or $2$, Trans. Amer. Math. Soc. 165 (1972), 353–364. MR 291124, DOI https://doi.org/10.1090/S0002-9947-1972-0291124-8

Daniel Gorenstein and Richard Lyons, The local structure of finite groups of characteristic $2$ type, Mem. Amer. Math. Soc. 42 (1983), no. 276, vii+731. MR 690900, DOI https://doi.org/10.1090/memo/0276

Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592

Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI https://doi.org/10.1007/BF01388809

E. Hecke, Über die $L$-funktionen und den Dirichletschen Primzahlsatz für einen beliebigen Zahlkörper. Gesells. der Wissens. zu Göttingen, Nachrichten (1917), 299–318.

H. Heilbronn, On the class number in imaginary quadratic fields, Quart. J. Math. 5 (1934), 150-160.

H. Heilbronn, Zeta-functions and $L$-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 204–230. MR 0218327

H. Heilbronn, On real simple zeros of Dedekind $\zeta $-functions, Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 108–110. MR 0389785

H. Heilbronn, On real zeros of Dedekind $\zeta $-functions, Canadian J. Math. 25 (1973), 870–873. MR 327719, DOI https://doi.org/10.4153/CJM-1973-090-3

I. Martin Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR 1280461

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

Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341

Joachim König, Solvability of generalized monomial groups, J. Group Theory 13 (2010), no. 2, 207–220. MR 2607576, DOI https://doi.org/10.1515/JGT.2009.047

Joshua M. Lansky and Kevin M. Wilson, A variation on the solvable case of the Dedekind conjecture, J. Ramanujan Math. Soc. 20 (2005), no. 2, 81–90. MR 2169089

Markus Linckelmann, Simple fusion systems and the Solomon 2-local groups, J. Algebra 296 (2006), no. 2, 385–401. MR 2201048, DOI https://doi.org/10.1016/j.jalgebra.2005.09.024

Maruti Ram Murty, On Artin $L$-functions, Class field theory—its centenary and prospect (Tokyo, 1998) Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001, pp. 13–29. MR 1846449, DOI https://doi.org/10.2969/aspm/03010013

M. Ram Murty and A. Raghuram, Some variations on the Dedekind conjecture, J. Ramanujan Math. Soc. 15 (2000), no. 4, 225–245. MR 1801220

M. Ram Murty and V. Kumar Murty, Base change and the Birch-Swinnerton-Dyer conjecture, A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., vol. 143, Amer. Math. Soc., Providence, RI, 1993, pp. 481–494. MR 1210535, DOI https://doi.org/10.1090/conm/143/01014

V. Kumar Murty, Stark zeros in certain towers of fields, Math. Res. Lett. 6 (1999), no. 5-6, 511–519. MR 1739210, DOI https://doi.org/10.4310/MRL.1999.v6.n5.a4

V. Kumar Murty, Class numbers of CM-fields with solvable normal closure, Compositio Math. 127 (2001), no. 3, 273–287. MR 1845038, DOI https://doi.org/10.1023/A%3A1017589432526

Sandra L. Rhoades, A generalization of the Aramata-Brauer theorem, Proc. Amer. Math. Soc. 119 (1993), no. 2, 357–364. MR 1166360, DOI https://doi.org/10.1090/S0002-9939-1993-1166360-8

S. Rhoades, A Character-Theoretic Approach to Artin’s Conjecture, Doctoral Dissertation, University of Vermont, 1993.

Moshe Roitman, On Zsigmondy primes, Proc. Amer. Math. Soc. 125 (1997), no. 7, 1913–1919. MR 1402885, DOI https://doi.org/10.1090/S0002-9939-97-03981-6

H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27. MR 222050

H. M. Stark, A transcendence theorem for class-number problems. II, Ann. of Math. (2) 96 (1972), 174–209. MR 309878, DOI https://doi.org/10.2307/1970897

H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 342472, DOI https://doi.org/10.1007/BF01405166

Josef Stoer and Christoph Witzgall, Convexity and optimization in finite dimensions. I, Die Grundlehren der mathematischen Wissenschaften, Band 163, Springer-Verlag, New York-Berlin, 1970. MR 0286498

Judith Evelyn Steere Sunley, ON THE CLASS NUMBERS OF TOTALLY IMAGINARY QUADRATIC EXTENSIONS OF TOTALLYREAL FIELDS, ProQuest LLC, Ann Arbor, MI, 1971. Thesis (Ph.D.)–University of Maryland, College Park. MR 2621002

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

Robert W. van der Waall, On the structure of the minimal non-$M$-groups, Nederl. Akad. Wetensch. Indag. Math. 40 (1978), no. 3, 398–405. MR 507832