Weil numbers in finite extensions of ℚ^{𝕒𝕓}: the Loxton-Kedlaya phenomenon

Stan, Florin; Zaharescu, Alexandru

doi:10.1090/S0002-9947-2014-06414-3

Weil numbers in finite extensions of $\mathbb {Q}^{ab}$: the Loxton-Kedlaya phenomenon
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by Florin Stan and Alexandru Zaharescu; with an appendix by Kiran S. Kedlaya PDF

Trans. Amer. Math. Soc. 367 (2015), 4359-4376 Request permission

Abstract:

A finiteness phenomenon described by Loxton and later by Kedlaya states that, for any fixed $m$, there exist (modulo multiplication by roots of unity) only finitely many $m$-Weil numbers in $\mathbb {Q}^{ab}$. In the present paper we show that this phenomenon extends to all finite extensions of $\mathbb {Q}^{ab}$.

References

Julián Aguirre and Juan Carlos Peral, The trace problem for totally positive algebraic integers, Number theory and polynomials, London Math. Soc. Lecture Note Ser., vol. 352, Cambridge Univ. Press, Cambridge, 2008, pp. 1–19. With an appendix by Jean-Pierre Serre. MR 2428512, DOI 10.1017/CBO9780511721274.003
Bruno Anglès and Tatiana Beliaeva, On Weil numbers in cyclotomic fields, Int. J. Number Theory 5 (2009), no. 5, 871–884. MR 2553513, DOI 10.1142/S1793042109002432
J. W. S. Cassels, On a conjecture of R. M. Robinson about sums of roots of unity, J. Reine Angew. Math. 238 (1969), 112–131. MR 246852, DOI 10.1515/crll.1969.238.112
A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520
Christopher Deninger, Exchanging the places $p$ and $\infty$ in the Leopoldt conjecture, J. Number Theory 114 (2005), no. 1, 1–17. MR 2163903, DOI 10.1016/j.jnt.2005.03.006
Taira Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83–95. MR 229642, DOI 10.2969/jmsj/02010083
Jean-François Jaulent, Plongements $l$-adiques et $l$-nombres de Weil, J. Théor. Nombres Bordeaux 20 (2008), no. 2, 335–351 (French, with English and French summaries). MR 2477507
E. Kowalski, Weil numbers generated by other Weil numbers and torsion fields of abelian varieties, J. London Math. Soc. (2) 74 (2006), no. 2, 273–288. MR 2269629, DOI 10.1112/S0024610706023040
L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173 - 175.
Yanhua Liang and Qiang Wu, The trace problem for totally positive algebraic integers, J. Aust. Math. Soc. 90 (2011), no. 3, 341–354. MR 2833305, DOI 10.1017/S1446788711001030
J. H. Loxton, On the maximum modulus of cyclotomic integers, Acta Arith. 22 (1972), 69–85. MR 309896, DOI 10.4064/aa-22-1-69-85
J. H. Loxton, On two problems of R. W. Robinson about sums of roots of unity, Acta Arith. 26 (1974/75), 159–174. MR 371852, DOI 10.4064/aa-26-2-159-174
David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
A. Greaves and R. W. K. Odoni, Weil-numbers and CM-fields. I, J. Reine Angew. Math. 391 (1988), 198–212. MR 961169
James McKee and Chris Smyth, Salem numbers of trace $-2$ and traces of totally positive algebraic integers, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 3076, Springer, Berlin, 2004, pp. 327–337. MR 2137365, DOI 10.1007/978-3-540-24847-7_{2}5
R. W. K. Odoni, Weil numbers and CM fields. II, J. Number Theory 38 (1991), no. 3, 366–377. MR 1114485, DOI 10.1016/0022-314X(91)90025-7
Raphael M. Robinson, Conjugate algebraic integers on a circle, Math. Z. 110 (1969), 41–51. MR 246826, DOI 10.1007/BF01114639
I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 1 (1918), no. 4, 377–402 (German). MR 1544303, DOI 10.1007/BF01465096
Carl Ludwig Siegel, The trace of totally positive and real algebraic integers, Ann. of Math. (2) 46 (1945), 302–312. MR 12092, DOI 10.2307/1969025
C. J. Smyth, The mean values of totally real algebraic integers, Math. Comp. 42 (1984), no. 166, 663–681. MR 736460, DOI 10.1090/S0025-5718-1984-0736460-5
Christopher Smyth, Totally positive algebraic integers of small trace, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 3, 1–28 (English, with French summary). MR 762691
Florin Stan and Alexandru Zaharescu, Siegel’s trace problem and character values of finite groups, J. Reine Angew. Math. 637 (2009), 217–234. MR 2599088, DOI 10.1515/CRELLE.2009.097
Florin Stan and Alexandru Zaharescu, The Siegel norm of algebraic numbers, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 55(103) (2012), no. 1, 69–77. MR 2951973
J. Tate, Classes d’isogénie de variétés abéliennes sur un corps fini (d’après T. Honda), Sém. Bourbaki, 21 , 1968/69, Exp. 352.