Dihedral Artin representations and CM fields
HTML articles powered by AMS MathViewer
- by David E. Rohrlich;
- Proc. Amer. Math. Soc. 152 (2024), 3183-3196
- DOI: https://doi.org/10.1090/proc/16793
- Published electronically: June 5, 2024
- HTML | PDF | Request permission
Abstract:
For a fixed CM field $K$ with maximal totally real subfield $F$, we consider isomorphism classes of dihedral Artin representations of $F$ which are induced from $K$, distinguishing between those which are “canonically” induced from $K$ and those which are “noncanonically” induced from $K$. The latter can arise only for Artin representations with image isomorphic to the dihedral group of order 8. We show that asymptotically, the number of noncanonically induced isomorphism classes is always comparable to and in some cases exceeds the number of canonically induced ones.References
- S. Ali Altuğ, Arul Shankar, Ila Varma, and Kevin H. Wilson, The number of $D_4$-fields ordered by conductor, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 8, 2733–2785. MR 4269426, DOI 10.4171/jems/1070
- Paul T. Bateman and Harold G. Diamond, Analytic number theory, Monographs in Number Theory, vol. 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004. An introductory course. MR 2111739, DOI 10.1142/5605
- Manjul Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), no. 2, 1031–1063. MR 2183288, DOI 10.4007/annals.2005.162.1031
- M. Bhargava, A. Shankar, and X. Wang, Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces, To appear.
- Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier, Enumerating quartic dihedral extensions of $\Bbb Q$, Compositio Math. 133 (2002), no. 1, 65–93. MR 1918290, DOI 10.1023/A:1016310902973
- Boris Datskovsky and David J. Wright, Density of discriminants of cubic extensions, J. Reine Angew. Math. 386 (1988), 116–138. MR 936994, DOI 10.1515/crll.1988.386.116
- Matthew Friedrichsen and Daniel Keliher, Comparing the density of $D_4$ and $S_4$ quartic extensions of number fields, Proc. Amer. Math. Soc. 149 (2021), no. 6, 2357–2369. MR 4246788, DOI 10.1090/proc/15358
- 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
- A. M. Odlyzko, Lower bounds for discriminants of number fields, Acta Arith. 29 (1976), no. 3, 275–297. MR 401704, DOI 10.4064/aa-29-3-275-297
- David E. Rohrlich, Self-dual Artin representations, Automorphic representations and $L$-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22, Tata Inst. Fund. Res., Mumbai, 2013, pp. 455–499. MR 3156861
- David E. Rohrlich, Artin representations of $\Bbb Q$ of dihedral type, Math. Res. Lett. 22 (2015), no. 6, 1767–1789. MR 3507261, DOI 10.4310/MRL.2015.v22.n6.a12
- David E. Rohrlich, Quaternionic Artin representations of $\Bbb {Q}$, Math. Proc. Cambridge Philos. Soc. 163 (2017), no. 1, 95–114. MR 3656350, DOI 10.1017/S0305004116000736
- Jean-Pierre Serre, Conducteurs d’Artin des caractères réels, Invent. Math. 14 (1971), 173–183 (French). MR 321908, DOI 10.1007/BF01418887
- Carl Ludwig Siegel, The average measure of quadratic forms with given determinant and signature, Ann. of Math. (2) 45 (1944), 667–685. MR 12642, DOI 10.2307/1969296
Bibliographic Information
- David E. Rohrlich
- Affiliation: Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215
- MR Author ID: 149885
- ORCID: 0009-0006-2763-4267
- Email: rohrlich@math.bu.edu
- Received by editor(s): December 30, 2021
- Received by editor(s) in revised form: February 17, 2023, and January 7, 2024
- Published electronically: June 5, 2024
- Communicated by: Amanda Folsom
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3183-3196
- MSC (2020): Primary 11R32; Secondary 11R20
- DOI: https://doi.org/10.1090/proc/16793
- MathSciNet review: 4767254