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Quaternionic Artin representations of ℚ

Published online by Cambridge University Press:  09 September 2016

DAVID E. ROHRLICH
DAVID E. ROHRLICH*
Affiliation:
Department of Mathematics and Statistics, Math and Computer Science Building, 111 Cummington Mall Boston University, Boston, MA 02215, U.S.A. e-mail: rohrlich@math.bu.edu
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Abstract

Isomorphism classes of dihedral Artin representations of ℚ can be counted asymptotically using Siegel's asymptotic averages of class numbers of binary quadratic forms. Here we consider the analogous problem for quaternionic representations. While an asymptotic formula is out of our reach in this case, we show that the asymptotic behaviour in the two cases is quite different.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 1 , July 2017 , pp. 95 - 114
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Alford, W. R., Granville, A. and Pomerance, C. There are infinitely many Carmichael numbers. Ann. Math. 140 (1994), 703–722.Google Scholar
[2] Ambrose, C. On Artin's primitive root conjecture and a problem of Rohrlich. Math. Proc. Camb. Phil. Soc. 157 (2014), 79–99.Google Scholar
[3] Baker, R. C. and Harman, G. Shifted primes without large prime factors. Acta Arith. 83 (1998), 331–361.Google Scholar
[4] Balog, A. p + a without large prime factors. Sém. Théorie des Nombres Bordeaux (1983–84), éxposé 31.Google Scholar
[5] Bateman, P. T. and Diamond, H. G. Analytic Number Theory: an Introductory Course (World Scientific, 2004).Google Scholar
[6] Ellison, W. J. (en collaboration avec M. Mendès France). Les Nombers Premiers (Hermann, 1975).Google Scholar
[7] Erdős, P. On the normal number of prime factors of p - 1 and some related problems concerning Euler's φ-function. Quarterly J. Math. 6 (1935), 205–213.Google Scholar
[8] Friedlander, J. Shifted primes without large prime factors. In: Number Theory and Applications (Kluwer, 1990), 393–401.Google Scholar
[9] Goldfeld, D. and Hoffstein, J. Eisenstein series of $\frac 12$ -integral weight and the mean value of real Dirichlet L-series. Invent. math. 80 (1985), 185–208.Google Scholar
[10] Golomb, S. W. Powerful numbers. American Math. Monthly. 77 (1970), 848–852.Google Scholar
[11] Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. (Oxford University Press, 1979).Google Scholar
[12] Klüners, J. Über die Asymptotik von Zahlkörpern mit vorgegebener Galoisgruppe. (Habilitationsschrift, Universität Kassel, 2005).Google Scholar
[13] Lang, S. Algebraic Number Theory, 2nd ed. (Springer GTM 110, 1994).CrossRefGoogle Scholar
[14] Luca, F. and Sankaranarayanan, A. On the moments of the Carmichael λ function. Acta Arith. 123 (2006) 389–398.Google Scholar
[15] Malle, G. On the distribution of Galois groups, II. Experimental Math. 13 (2004), 129–135.CrossRefGoogle Scholar
[16] Martinet, J. Character theory and Artin L-functions. In: Algebraic Number Fields, Proceedings of the Durham Symposium, Fröhlich, A. ed., (Academic Press, 1977), 1–87.Google Scholar
[17] Pomerance, C. Popular values of Euler's function. Mathematika' 27 (1980), 84–89.Google Scholar
[18] Rohrlich, D. E. Nonvanishing of L-functions for GL(2). Invent. Math. 97 (1989), 381–403.CrossRefGoogle Scholar
[19] Rohrlich, D. E. Self-dual Artin representation. In: Automorphic Represesentations and L-Functions edited by Prasad, D., Rajan, C. S., Sankaranarayanan, A., J., Tata Institute of Fundamental Research Studies in Math. Vol. 22 (2013), 455–499.Google Scholar
[20] Rohrlich, D. E. Artin representations of ℚ of dihedral type. Math. Res. Lett. 22 (2015), 1767–1789.Google Scholar
[21] Siegel, C. L. The average measure of quadratic forms with given determinant and signature. Ann. of Math. 45 (1944), 667–685.CrossRefGoogle Scholar