Divisibility conditions on the order of the reductions of algebraic numbers

Sgobba, Pietro

doi:10.1090/mcom/3848

Divisibility conditions on the order of the reductions of algebraic numbers
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Math. Comp. 92 (2023), 2281-2305 Request permission

Abstract:

Let $K$ be a number field, and let $G$ be a finitely generated subgroup of $K^\times$. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes $\mathfrak p$ of $K$ such that the order of $(G\bmod \mathfrak p)$ is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes $\mathfrak p$ for which the order is $k$-free, and those for which the order has a prescribed $\ell$-adic valuation for finitely many primes $\ell$. An additional condition on the Frobenius conjugacy class of $\mathfrak p$ may be considered. In order to establish these results, we prove an unconditional version of the Chebotarev density theorem for Kummer extensions of number fields.

References

H. O. Abdullah, A. Ali Mustafa, and F. Pappalardi, Divisibility of reduction in groups of rational numbers II, Int. J. Number Theory 19 (2023), no. 2, 247–260. MR 4547450, DOI 10.1142/S1793042123500124
Christophe Debry and Antonella Perucca, Reductions of algebraic integers, J. Number Theory 167 (2016), 259–283. MR 3504046, DOI 10.1016/j.jnt.2016.03.001
O. Järviniemi, A. Perucca, P. Sgobba, Unified treatment of Artin-type problems II, preprint, submitted for publication (2022), arXiv:2211.15614.
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
H. W. Lenstra Jr., P. Stevenhagen, and P. Moree, Character sums for primitive root densities, Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 3, 489–511. MR 3286520, DOI 10.1017/S0305004114000450
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, Artin’s primitive root conjecture—a survey, Integers 12 (2012), no. 6, 1305–1416. MR 3011564, DOI 10.1515/integers-2012-0043
V. Kumar Murty, Modular forms and the Chebotarev density theorem. II, Analytic number theory (Kyoto, 1996) London Math. Soc. Lecture Note Ser., vol. 247, Cambridge Univ. Press, Cambridge, 1997, pp. 287–308. MR 1694997, DOI 10.1017/CBO9780511666179.019
Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
Francesco Pappalardi, Square free values of the order function, New York J. Math. 9 (2003), 331–344. MR 2028173
Francesco Pappalardi, Divisibility of reduction in groups of rational numbers, Math. Comp. 84 (2015), no. 291, 385–407. MR 3266967, DOI 10.1090/S0025-5718-2014-02872-X
Antonella Perucca, The order of the reductions of an algebraic integer, J. Number Theory 148 (2015), 121–136. MR 3283171, DOI 10.1016/j.jnt.2014.09.022
Antonella Perucca, Reductions of algebraic integers II, Women in numbers Europe II, Assoc. Women Math. Ser., vol. 11, Springer, Cham, 2018, pp. 19–33. MR 3882703
Antonella Perucca and Pietro Sgobba, Kummer theory for number fields and the reductions of algebraic numbers, Int. J. Number Theory 15 (2019), no. 8, 1617–1633. MR 3994150, DOI 10.1142/S179304211950091X
Antonella Perucca and Pietro Sgobba, Kummer theory for number fields and the reductions of algebraic numbers II, Unif. Distrib. Theory 15 (2020), no. 1, 75–92. MR 4221191, DOI 10.2478/udt-2020-0004
Antonella Perucca, Pietro Sgobba, and Sebastiano Tronto, The degree of Kummer extensions of number fields, Int. J. Number Theory 17 (2021), no. 5, 1091–1110. MR 4270875, DOI 10.1142/S1793042121500263
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
SageMath, the Sage Mathematics software system (version 9.2), The Sage Developers, 2021, https://www.sagemath.org.
Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 342472, DOI 10.1007/BF01405166
Peter Stevenhagen, The correction factor in Artin’s primitive root conjecture, J. Théor. Nombres Bordeaux 15 (2003), no. 1, 383–391 (English, with English and French summaries). Les XXIIèmes Journées Arithmetiques (Lille, 2001). MR 2019022, DOI 10.5802/jtnb.408
Jesse Thorner and Asif Zaman, A unified and improved Chebotarev density theorem, Algebra Number Theory 13 (2019), no. 5, 1039–1068. MR 3981313, DOI 10.2140/ant.2019.13.1039
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
Volker Ziegler, On the distribution of the order of number field elements modulo prime ideals, Unif. Distrib. Theory 1 (2006), no. 1, 65–85. MR 2314267