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
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show all references

References:
[1]

Comptes Rendus Acad. Sci. Paris, 290 (1980), 701-703.  Google Scholar

[2]

Hermann, Paris, 1975.  Google Scholar

[3]

J. Pure Appl. Algebra, 155 (2001), 115-120. doi: 10.1016/S0022-4049(99)00096-1.  Google Scholar

[4]

Acta Arith., 119 (2005), 265-289. doi: 10.4064/aa119-3-3.  Google Scholar

[5]

Forum Math., 24 (2012), 99-119. doi: 10.1515/form.2011.051.  Google Scholar

[6]

in Proc. Symp. Pure Math. (eds. A. Borel and W. Casselman), AMS, Providence, 1979, 247-289.  Google Scholar

[7]

Invent. Math., 73(1983), 349-366. doi: 10.1007/BF01388432.  Google Scholar

[8]

Springer, Berlin, 1971.  Google Scholar

[9]

Invent. Math., 62 (1981), 481-502. doi: 10.1007/BF01394256.  Google Scholar

[10]

Pacific J. Math., 131 (1988),157-165. doi: 10.2140/pjm.1988.131.157.  Google Scholar

[11]

in INDOCRYPT 2001, Springer, Berlin, 2001, 91-98. doi: 10.1007/3-540-45311-3_9.  Google Scholar

[12]

Forum Math., 6 (1994), 555-565. doi: 10.1515/form.1994.6.555.  Google Scholar

[13]

Intl. Math. Res. Notices, 12 (2008), Article ID rnn 033, 27 pp. doi: 10.1093/imrn/rnn033.  Google Scholar

[14]

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

[15]

J. Number Theory, 16 (1983), 147-168. doi: 10.1016/0022-314X(83)90039-2.  Google Scholar

[16]

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.  Google Scholar

[17]

in Séminaire Bourbaki, Springer, Heidelberg, 1971, 95-110.  Google Scholar

[18]

in Proc. Sympos. Pure Math. (eds. A. Borel and G. Mostow), AMS, Providence, 1966, 33-62.  Google Scholar

[19]

in Algorithmic Number Theory, Springer, Berlin, 2008, 74-87. doi: 10.1007/978-3-540-79456-1_4.  Google Scholar

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