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Nonmonomial characters and Artin's conjecture


Author: Richard Foote
Journal: Trans. Amer. Math. Soc. 321 (1990), 261-272
MSC: Primary 11R42; Secondary 11R32
DOI: https://doi.org/10.1090/S0002-9947-1990-0987161-9
MathSciNet review: 987161
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Abstract: If $ E/F$ is a Galois extension of number fields with solvable Galois group $ G$, the main result of this paper proves that if the Dedekind zeta-function of $ E$ has a zero of order less than $ {\mathcal{M}_G}$ at the complex point $ {s_0} \ne 1$, then all Artin $ L$-series for $ G$ are holomorphic at $ {s_0}$ -- here $ {\mathcal{M}_G}$ is the smallest degree of a nonmonomial character of any subgroup of $ G$. The proof relies only on certain properties of $ L$-functions which are axiomatized to give a purely character-theoretic statement of this result.


References [Enhancements On Off] (What's this?)

  • [1] C. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Wiley, New York, 1966. MR 1013113 (90g:16001)
  • [2] E. Dade, Characters of groups with normal extra special subgroups, Math. Z. 152 (1976), 1-31. MR 0486085 (58:5873)
  • [3] W. Feit, Characters of finite groups, Benjamin, New York, 1967. MR 0219636 (36:2715)
  • [4] R. Foote and V. K. Murty, Zeros and poles of Artin $ L$-series, Math. Proc. Cambridge Philos. Soc. 105 (1989), 5-11. MR 966135 (89k:11109)
  • [5] R. Foote and D. Wales, Zeros of order $ 2$ of Dedekind zeta-functions and Artin's conjecture J. Algebra 125 (1989) (to appear). MR 1055006 (91b:11132)
  • [6] D. Gorenstein, Finite groups, Harper and Row, New York, 1968. MR 0231903 (38:229)
  • [7] H. Heilbronn, Zeta-functions and $ L$-functions, Algebraic Number Theory (J. Cassels and A. Frölich, eds.), Chapter VIII, Academic Press, London, 1967. MR 0218327 (36:1414)
  • [8] -, On real zeros of Dedekind $ \zeta $-functions, The collected papers of Hans Arnold Heilbronn, Wiley, New York, 1988.
  • [9] B. Huppert, Endliche Gruppen. I, Springer-Verlag, 1967. MR 0224703 (37:302)
  • [10] D. T. Price, Character ramification and $ M$-groups, Math. Z. 130 (1973), 325-337. MR 0320129 (47:8670)
  • [11] R. W. van der Waall, Minimal non-$ M$-groups. III, Indag. Math. 86 (1983), 483-492. MR 731831 (85h:20024)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0987161-9
Article copyright: © Copyright 1990 American Mathematical Society

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