Abelian varieties without a prescribed Newton Polygon reduction

Xue, Jiangwei; Yu, Chia-Fu

doi:10.1090/S0002-9939-2015-12483-5

Abelian varieties without a prescribed Newton Polygon reduction
HTML articles powered by AMS MathViewer

by Jiangwei Xue and Chia-Fu Yu PDF

Proc. Amer. Math. Soc. 143 (2015), 2339-2345 Request permission

Abstract:

In this article we construct for each integer $g\ge 2$ an abelian variety $A$ of dimension $g$ defined over a number field for which there exists a symmetric integral slope sequence of length $2g$ that does not appear as the slope sequence of $\widetilde {A}$ for any good reduction $\widetilde {A}$ of $A$.

References

Fedor A. Bogomolov and Yuri G. Zarhin, Ordinary reduction of $K3$ surfaces, Cent. Eur. J. Math. 7 (2009), no. 2, 206–213. MR 2506961, DOI 10.2478/s11533-009-0013-8
Noam D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over $\textbf {Q}$, Invent. Math. 89 (1987), no. 3, 561–567. MR 903384, DOI 10.1007/BF01388985
Taira Honda, On the Jacobian variety of the algebraic curve $y^{2}=1-x^{l}$ over a field of characteristic $p>0$, Osaka Math. J. 3 (1966), 189–194. MR 225777
Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723, DOI 10.1007/978-1-4612-0853-2
Ju. I. Manin, Theory of commutative formal groups over fields of finite characteristic, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 3–90 (Russian). MR 0157972
Rutger Noot, Abelian varieties—Galois representation and properties of ordinary reduction, Compositio Math. 97 (1995), no. 1-2, 161–171. Special issue in honour of Frans Oort. MR 1355123
Rutger Noot, Abelian varieties with $l$-adic Galois representation of Mumford’s type, J. Reine Angew. Math. 519 (2000), 155–169. MR 1739726, DOI 10.1515/crll.2000.010
A. Ogus, Hodge cycles and crystalline cohomology, Lecture Notes in Math., vol. 1, Springer-Verlag, 900 (1982), 357–414.
Bjorn Poonen, Drinfeld modules with no supersingular primes, Internat. Math. Res. Notices 3 (1998), 151–159. MR 1606391, DOI 10.1155/S1073792898000130
Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
Goro Shimura and Yutaka Taniyama, Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, Mathematical Society of Japan, Tokyo, 1961. MR 0125113
S. G. Tankeev, On the weights of an $l$-adic representation and the arithmetic of Frobenius eigenvalues, Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 1, 185–224 (Russian, with Russian summary); English transl., Izv. Math. 63 (1999), no. 1, 181–218. MR 1701843, DOI 10.1070/im1999v063n01ABEH000233
John Tate, Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363, Lecture Notes in Math., vol. 175, Springer, Berlin, 1971, pp. Exp. No. 352, 95–110 (French). MR 3077121
Chia-Fu Yu, The isomorphism classes of abelian varieties of CM-type, J. Pure Appl. Algebra 187 (2004), no. 1-3, 305–319. MR 2027907, DOI 10.1016/S0022-4049(03)00144-0
Chia-Fu Yu, Endomorphism algebras of QM abelian surfaces, J. Pure Appl. Algebra 217 (2013), no. 5, 907–914. MR 3003315, DOI 10.1016/j.jpaa.2012.09.022