November  2014, 8(4): 511-519. doi: 10.3934/amc.2014.8.511

Splitting of abelian varieties

1. 

Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Canada M5S 2E4, Canada, Canada

Received  February 2014 Revised  September 2014 Published  November 2014

It is possible that a simple (or absolutely simple) Abelian variety defined over a number field splits modulo every prime of good reduction. This is a new problem that arises in designing crypto systems using Abelian varieties of dimension larger than $1$. We discuss what is behind this phenomenon. In particular, we discuss the question of given an absolutely simple abelian variety over a number field, whether it has simple specializations at a set of places of positive Dirichlet density? A conjectural answer to this question was given by Murty and Patankar, and we explain some recent progress towards proving the conjecture. Our result ([14], Theorem 1.1) is based on the classification of pairs $(G,V)$ consisting of a semi-simple algebraic group $G$ over a non-archimedean local field and an absolutely irreducible representation $V$ of $G$ such that $G$ admits a maximal torus acting irreducibly on $V$.
Citation: V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511
References:
[1]

F. Bogomolov, Sur l'algébricité des représentations $l$-adiques, Comptes Rendus Acad. Sci. Paris, 290 (1980), 701-703.

[2]

N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris, 1975.

[3]

C.-L. Chai and F. Oort, A note on the existence of absolutely simple Jacobians, J. Pure Appl. Algebra, 155 (2001), 115-120. doi: 10.1016/S0022-4049(99)00096-1.

[4]

A. Cojocaru, Reductions of an elliptic curve with almost prime orders, Acta Arith., 119 (2005), 265-289. doi: 10.4064/aa119-3-3.

[5]

C. David and J. Wu, Almost prime values of the order of elliptic curves over finite fields, Forum Math., 24 (2012), 99-119. doi: 10.1515/form.2011.051.

[6]

P. Deligne, Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques, in Proc. Symp. Pure Math. (eds. A. Borel and W. Casselman), AMS, Providence, 1979, 247-289.

[7]

G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73(1983), 349-366. doi: 10.1007/BF01388432.

[8]

A. Grothendieck, Revêtements étales et Groupe Fondamental, Springer, Berlin, 1971.

[9]

N. Katz, Galois properties of torsion points of abelian varieties, Invent. Math., 62 (1981), 481-502. doi: 10.1007/BF01394256.

[10]

N. Koblitz, Primality of the number of points on an elliptic curve over a finite field, Pacific J. Math., 131 (1988),157-165. doi: 10.2140/pjm.1988.131.157.

[11]

S. A. Miri and V. Kumar Murty, An application of sieve methods to elliptic curves, in INDOCRYPT 2001, Springer, Berlin, 2001, 91-98. doi: 10.1007/3-540-45311-3_9.

[12]

V. Kumar Murty, The least prime which does not split completely, Forum Math., 6 (1994), 555-565. doi: 10.1515/form.1994.6.555.

[13]

V. Kumar Murty and V. Patankar, Splitting of abelian varieties, Intl. Math. Res. Notices, 12 (2008), Article ID rnn 033, 27 pp. doi: 10.1093/imrn/rnn033.

[14]

V. Kumar Murty and Y. Zong, Splitting of abelian varieties, elliptic minuscule pairs,, preprint, (). 

[15]

M. Ram Murty, On Artin's conjecture, J. Number Theory, 16 (1983), 147-168. doi: 10.1016/0022-314X(83)90039-2.

[16]

M. Ram Murty, Artin's conjecture and elliptic analogues, in Sieve methods, exponential sums, and their applications in number theory (eds. G.R.H. Greaves, G. Harman and M.N. Huxley), Cambridge Univ. Press, Cambridge, 1996, 325-344. doi: 10.1017/CBO9780511526091.022.

[17]

J. Tate, Classes d'isogénie de variétés abéliennes sur un corps fini (d'après T.Honda), in Séminaire Bourbaki, Springer, Heidelberg, 1971, 95-110.

[18]

J. Tits, Classification of algebraic semisimple groups, in Proc. Sympos. Pure Math. (eds. A. Borel and G. Mostow), AMS, Providence, 1966, 33-62.

[19]

J.-J. Urroz, Almost prime order of CM elliptic cures modulo $p$, in Algorithmic Number Theory, Springer, Berlin, 2008, 74-87. doi: 10.1007/978-3-540-79456-1_4.

show all references

References:
[1]

F. Bogomolov, Sur l'algébricité des représentations $l$-adiques, Comptes Rendus Acad. Sci. Paris, 290 (1980), 701-703.

[2]

N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris, 1975.

[3]

C.-L. Chai and F. Oort, A note on the existence of absolutely simple Jacobians, J. Pure Appl. Algebra, 155 (2001), 115-120. doi: 10.1016/S0022-4049(99)00096-1.

[4]

A. Cojocaru, Reductions of an elliptic curve with almost prime orders, Acta Arith., 119 (2005), 265-289. doi: 10.4064/aa119-3-3.

[5]

C. David and J. Wu, Almost prime values of the order of elliptic curves over finite fields, Forum Math., 24 (2012), 99-119. doi: 10.1515/form.2011.051.

[6]

P. Deligne, Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques, in Proc. Symp. Pure Math. (eds. A. Borel and W. Casselman), AMS, Providence, 1979, 247-289.

[7]

G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73(1983), 349-366. doi: 10.1007/BF01388432.

[8]

A. Grothendieck, Revêtements étales et Groupe Fondamental, Springer, Berlin, 1971.

[9]

N. Katz, Galois properties of torsion points of abelian varieties, Invent. Math., 62 (1981), 481-502. doi: 10.1007/BF01394256.

[10]

N. Koblitz, Primality of the number of points on an elliptic curve over a finite field, Pacific J. Math., 131 (1988),157-165. doi: 10.2140/pjm.1988.131.157.

[11]

S. A. Miri and V. Kumar Murty, An application of sieve methods to elliptic curves, in INDOCRYPT 2001, Springer, Berlin, 2001, 91-98. doi: 10.1007/3-540-45311-3_9.

[12]

V. Kumar Murty, The least prime which does not split completely, Forum Math., 6 (1994), 555-565. doi: 10.1515/form.1994.6.555.

[13]

V. Kumar Murty and V. Patankar, Splitting of abelian varieties, Intl. Math. Res. Notices, 12 (2008), Article ID rnn 033, 27 pp. doi: 10.1093/imrn/rnn033.

[14]

V. Kumar Murty and Y. Zong, Splitting of abelian varieties, elliptic minuscule pairs,, preprint, (). 

[15]

M. Ram Murty, On Artin's conjecture, J. Number Theory, 16 (1983), 147-168. doi: 10.1016/0022-314X(83)90039-2.

[16]

M. Ram Murty, Artin's conjecture and elliptic analogues, in Sieve methods, exponential sums, and their applications in number theory (eds. G.R.H. Greaves, G. Harman and M.N. Huxley), Cambridge Univ. Press, Cambridge, 1996, 325-344. doi: 10.1017/CBO9780511526091.022.

[17]

J. Tate, Classes d'isogénie de variétés abéliennes sur un corps fini (d'après T.Honda), in Séminaire Bourbaki, Springer, Heidelberg, 1971, 95-110.

[18]

J. Tits, Classification of algebraic semisimple groups, in Proc. Sympos. Pure Math. (eds. A. Borel and G. Mostow), AMS, Providence, 1966, 33-62.

[19]

J.-J. Urroz, Almost prime order of CM elliptic cures modulo $p$, in Algorithmic Number Theory, Springer, Berlin, 2008, 74-87. doi: 10.1007/978-3-540-79456-1_4.

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