On the Northcott property and local degrees

Checcoli, S.; Fehm, A.

doi:10.1090/proc/15411

On the Northcott property and local degrees
HTML articles powered by AMS MathViewer

by S. Checcoli and A. Fehm

Proc. Amer. Math. Soc. 149 (2021), 2403-2414

DOI: https://doi.org/10.1090/proc/15411

Published electronically: March 26, 2021

PDF | Request permission

Abstract:

We construct infinite Galois extensions $L$ of $\mathbb {Q}$ that satisfy the Northcott property on elements of small height, and where this property can be deduced solely from the local behavior of $L$ at the different prime numbers. We also give examples of Galois extensions of $\mathbb {Q}$ which have finite local degree at all prime numbers and do not satisfy the Northcott property.

References

Francesco Amoroso and Roberto Dvornicich, A lower bound for the height in abelian extensions, J. Number Theory 80 (2000), no. 2, 260–272. MR 1740514, DOI 10.1006/jnth.1999.2451
Francesco Amoroso, Sinnou David, and Umberto Zannier, On fields with Property (B), Proc. Amer. Math. Soc. 142 (2014), no. 6, 1893–1910. MR 3182009, DOI 10.1090/S0002-9939-2014-11925-3
Francesco Amoroso and Umberto Zannier, A relative Dobrowolski lower bound over abelian extensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 3, 711–727. MR 1817715
Francesco Amoroso and Umberto Zannier, A uniform relative Dobrowolski’s lower bound over abelian extensions, Bull. Lond. Math. Soc. 42 (2010), no. 3, 489–498. MR 2651944, DOI 10.1112/blms/bdq008
Enrico Bombieri and Umberto Zannier, A note on heights in certain infinite extensions of $\Bbb Q$, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 12 (2001), 5–14 (2002) (English, with English and Italian summaries). MR 1898444
Sara Checcoli, A note on Galois groups and local degrees, Manuscripta Math. 159 (2019), no. 1-2, 1–12. MR 3936131, DOI 10.1007/s00229-018-1039-7
Sara Checcoli and Martin Widmer, On the Northcott property and other properties related to polynomial mappings, Math. Proc. Cambridge Philos. Soc. 155 (2013), no. 1, 1–12. MR 3065255, DOI 10.1017/S0305004113000042
Pierre Dèbes and Nour Ghazi, Galois covers and the Hilbert-Grunwald property, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 3, 989–1013 (English, with English and French summaries). MR 3013814, DOI 10.5802/aif.2714
V. Dimitrov, A proof of the Schinzel-Zassenhaus conjecture on polynomials, arXiv:1912.12545 [math.NT] (2019).
Roberto Dvornicich and Umberto Zannier, On the properties of Northcott and of Narkiewicz for fields of algebraic numbers. part 1, Funct. Approx. Comment. Math. 39 (2008), no. part 1, 163–173. MR 2490096, DOI 10.7169/facm/1229696562
Arno Fehm, Three counterexamples concerning the Northcott property of fields, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 2, 309–314. MR 3797986, DOI 10.4171/RLM/807
Paul Fili, On the heights of totally $p$-adic numbers, J. Théor. Nombres Bordeaux 26 (2014), no. 1, 103–109 (English, with English and French summaries). MR 3232769
L. Frey, Explicit Small Heights in Infinite Non-Abelian Extensions, arXiv:1712.04214 [math.NT] (2017).
Aurélien Galateau, Small height in fields generated by singular moduli, Proc. Amer. Math. Soc. 144 (2016), no. 7, 2771–2786. MR 3487213, DOI 10.1090/proc/13058
P. Habegger, Small height and infinite nonabelian extensions, Duke Math. J. 162 (2013), no. 11, 2027–2076. MR 3090783, DOI 10.1215/00127094-2331342
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
Jürgen Klüners and Gunter Malle, Counting nilpotent Galois extensions, J. Reine Angew. Math. 572 (2004), 1–26. MR 2076117, DOI 10.1515/crll.2004.050
Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, 3rd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. MR 2078267, DOI 10.1007/978-3-662-07001-7
Jürgen Neukirch, On solvable number fields, Invent. Math. 53 (1979), no. 2, 135–164. MR 560411, DOI 10.1007/BF01390030
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
J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields (Second Edition), Springer (2008).
Arnaud Plessis, Minoration de la hauteur de Weil dans un compositum de corps de rayon, J. Number Theory 205 (2019), 246–276 (French, with English and French summaries). MR 3996352, DOI 10.1016/j.jnt.2019.05.008
Lukas Pottmeyer, Heights and totally $p$-adic numbers, Acta Arith. 171 (2015), no. 3, 277–291. MR 3416948, DOI 10.4064/aa171-3-5
Lukas Pottmeyer, A note on extensions of $\Bbb Q^{tr}$, J. Théor. Nombres Bordeaux 28 (2016), no. 3, 735–742 (English, with English and French summaries). MR 3610695
A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number, Acta Arith. 24 (1973), 385–399. MR 360515, DOI 10.4064/aa-24-4-385-399
Alexandra Shlapentokh, First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers, Israel J. Math. 226 (2018), no. 2, 579–633. MR 3819703, DOI 10.1007/s11856-018-1708-y
Xavier Vidaux and Carlos R. Videla, A note on the Northcott property and undecidability, Bull. Lond. Math. Soc. 48 (2016), no. 1, 58–62. MR 3455748, DOI 10.1112/blms/bdv089
Carlos R. Videla, Definability of the ring of integers in pro-$p$ Galois extensions of number fields, Israel J. Math. 118 (2000), 1–14. MR 1776073, DOI 10.1007/BF02803513
Arnold Walfisz, Zur additiven Zahlentheorie. II, Math. Z. 40 (1936), no. 1, 592–607 (German). MR 1545584, DOI 10.1007/BF01218882
Martin Widmer, On certain infinite extensions of the rationals with Northcott property, Monatsh. Math. 162 (2011), no. 3, 341–353. MR 2775852, DOI 10.1007/s00605-009-0162-7