Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T06:10:18.000Z Has data issue: false hasContentIssue false

Determinants of subquotients of Galois representations associated with abelian varieties

Published online by Cambridge University Press:  18 July 2013

Abstract

Given an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell $, the ${\ell }^{n} $-torsion points of $A$ give rise to a representation ${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$. In particular, we get a mod-$\ell $representation ${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$ and an $\ell $-adic representation ${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$. In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.

Applying our results in dimension $g= 1$, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur’s bound on rational isogenies of prime degree to number fields.

Type
Research Article
Copyright
©Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References

Conrad, B., Lifting global representations with local properties, 2010 preprint available at the webpage http://math.stanford.edu/~conrad/papers/locchar.pdf.Google Scholar
Conrad, B., Semistable reduction for abelian varieties, Seminar Lecture Notes available at the webpage http://math.stanford.edu/~akshay/ntslearn.html.Google Scholar
Fontaine, J.-M., Le corps des périodes p-adiques, Périodes p-adiques, Astérisque 223 (1994), 59111.Google Scholar
Fontaine, J.-M., Représentations p-adiques semi-stables, Périodes p-adiques, Astérisque 223 (1994), 113184.Google Scholar
Serre, J.-P., Abelian ℓ-adic representations and elliptic curves, 2nd ed. (A.K. Peters, 1998).Google Scholar

References

Bach, E. and Sorenson, J., Explicit bounds for primes in residue classes, Math. Comp. 65 (216) (1996), 17171735.Google Scholar
David, A., Caractère d’isogénie et critères d’irréductibilité, Preprint available online at http://arxiv.org/abs/1103.3892.Google Scholar
Deligne, P., Travaux de Shimura, in Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Lecture Notes in Math., Volume 244, pp. 123165 (Springer, Berlin, 1971).CrossRefGoogle Scholar
Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (3) (1983), 349366.CrossRefGoogle Scholar
Friedman, E., Analytic formulas for the regulator of a number field, Invent. Math. 98 (3) (1989), 599622.Google Scholar
Grothendieck, A., Modèles de Néron et monodromie. In Séminaire de Géométrie Algébrique, Volume 7, Exposé 9.Google Scholar
Lagarias, J. C. and Odlyzko, A. M., Effective versions of the Chebotarev density theorem, in Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 409464 (Academic Press, London, 1977).Google Scholar
Lenstra, H. W. Jr., Algorithms in algebraic number theory, Bull. Amer. Math. Soc. (N.S.) 26 (2) (1992), 211244.Google Scholar
Mazur, B., Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (2) (1978), 129162.Google Scholar
Merel, L., Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1–3) (1996), 437449.Google Scholar
Milne, J. S., Abelian varieties defined over their fields of moduli. I, Bull. Lond. Math. Soc. 4 (1972), 370372.Google Scholar
Milne, J. S., Complex multiplication (v0.00), 2006. Available at www.jmilne.org/math/.Google Scholar
Momose, F., Isogenies of prime degree over number fields, Compositio Math. 97 (3) (1995), 329348.Google Scholar
Raynaud, M., Schémas en groupes de type $(p, \ldots , p)$ , Bull. Soc. Math. France 102 (1974), 241280.Google Scholar
Rizov, J., Fields of definition of rational points on varieties. 2005. Available athttp://arxiv.org/abs/math/0505364.Google Scholar
Serre, J.-P., Abelian l-adic representations and elliptic curves, McGill University Lecture Notes Written with the Collaboration of Willem Kuyk and John Labute (W. A. Benjamin, Inc., New York, Amsterdam, 1968).Google Scholar
Serre, J.-P., Propriétés Galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (4) (1972), 259331.Google Scholar
Serre, J.-P., Quelques applications du théorème de densité de Chebotarev, Publ. Math. Inst. Hautes Études Sci. (54) (1981), 323401.Google Scholar
Shimura, G., Algebraic number fields and symplectic discontinuous groups, Ann. of Math. (2) 86 (1967), 503592.Google Scholar
Voutier, P., An effective lower bound for the height of algebraic numbers, Acta Arith. 74 (1) (1996), 8195.Google Scholar