Divisibility of reduction in groups of rational numbers

Pappalardi, Francesco

doi:10.1090/S0025-5718-2014-02872-X

Divisibility of reduction in groups of rational numbers
HTML articles powered by AMS MathViewer

by Francesco Pappalardi PDF

Math. Comp. 84 (2015), 385-407 Request permission

Abstract:

Given a multiplicative group of nonzero rational numbers and a positive integer $m$, we consider the problem of determining the density of the set of primes $p$ for which the order of the reduction modulo $p$ of the group is divisible by $m$. In the case when the group is finitely generated the density is explicitly computed. Some examples of groups with infinite rank are considered.

References

Leonardo Cangelmi and Francesco Pappalardi, On the $r$-rank Artin conjecture. II, J. Number Theory 75 (1999), no. 1, 120–132. MR 1677559, DOI 10.1006/jnth.1998.2319
Koji Chinen and Leo Murata, On a distribution property of the residual order of $a\pmod p$. IV, Number theory, Dev. Math., vol. 15, Springer, New York, 2006, pp. 11–22. MR 2213825, DOI 10.1007/0-387-30829-6_{2}
Rajiv Gupta and M. Ram Murty, A remark on Artin’s conjecture, Invent. Math. 78 (1984), no. 1, 127–130. MR 762358, DOI 10.1007/BF01388719
Rajiv Gupta and M. Ram Murty, Primitive points on elliptic curves, Compositio Math. 58 (1986), no. 1, 13–44. MR 834046
Helmut Hasse, Über die Dichte der Primzahlen $p$, für die eine vorgegebene ganzrationale Zahl $a\not =0$ von gerader bzw. ungerader Ordnung $\textrm {mod}.p$ ist, Math. Ann. 166 (1966), 19–23 (German). MR 205975, DOI 10.1007/BF01361432
D. R. Heath-Brown, Artin’s conjecture for primitive roots, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 145, 27–38. MR 830627, DOI 10.1093/qmath/37.1.27
Christopher Hooley, On binary cubic forms, J. Reine Angew. Math. 226 (1967), 30–87. MR 213299, DOI 10.1515/crll.1967.226.30
P. Kurlberg and C. Pomerance, On a problem of Arnold: the average multiplicative order of a given integer, Algebra and Number Theory, 7 (2013), no. 4, 981–999. MR 3095233
J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 409–464. MR 0447191
Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
Pieter Moree, Artin’s primitive root conjecture—a survey, Integers 12 (2012), no. 6, 1305–1416. MR 3011564, DOI 10.1515/integers-2012-0043
Pieter Moree, On primes $p$ for which $d$ divides $\textrm {ord}_p(g)$, Funct. Approx. Comment. Math. 33 (2005), 85–95. MR 2274151, DOI 10.7169/facm/1538186603
Pieter Moree, On the distribution of the order over residue classes, Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 121–128. MR 2263073, DOI 10.1090/S1079-6762-06-00168-5
F. Pappalardi, On Hooley’s theorem with weights, Rend. Sem. Mat. Univ. Politec. Torino 53 (1995), no. 4, 375–388. Number theory, II (Rome, 1995). MR 1452393
Francesco Pappalardi, On the $r$-rank Artin conjecture, Math. Comp. 66 (1997), no. 218, 853–868. MR 1377664, DOI 10.1090/S0025-5718-97-00805-3
Francesco Pappalardi, Square free values of the order function, New York J. Math. 9 (2003), 331–344. MR 2028173
Francesco Pappalardi and Andrea Susa, An analogue of Artin’s conjecture for multiplicative subgroups of the rationals, Arch. Math. (Basel) 101 (2013), no. 4, 319–330. MR 3116653, DOI 10.1007/s00013-013-0563-7
Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
Edwin Weiss, Algebraic number theory, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR 0159805
K. Wiertelak, On the density of some sets of primes. IV, Acta Arith. 43 (1984), no. 2, 177–190. MR 736730, DOI 10.4064/aa-43-2-177-190
PARI/GP, version 2.3.4, http://pari.math.u-bordeaux.fr/, Bordeaux, 2009.