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Isogeny orbits in a family of abelian varieties. (English) Zbl 1382.11045
Summary: We prove that if a curve of a nonisotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety, then it is either torsion or contained in a fiber. This result fits into the context of the Zilber-Pink conjecture. Moreover, by using the polyhedral reduction theory we give a new proof of a result of D. Bertrand [Duke Math. J. 80, No. 1, 223–250 (1995; Zbl 0847.11036)].

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
11G50 Heights
14K10 Algebraic moduli of abelian varieties, classification
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[1] [A]Y. Andr\'{}e, Shimura varieties, subvarieties, and CM points, lectures at the Univ. of Hsinchu, 2001 (with an appendix by C. L. Chai); http://www.math.umd.edu/ \tilde{}yu/notes.shtml.
[2] [AMRY]A. Ash, D. Mumford, M. Rapoport and Y. Tai, Smooth Compactification of Locally Symmetric Varieties, 2nd ed., Cambridge Univ. Press, 2010. · Zbl 0334.14007
[3] [B]D. Bertrand, Minimal heights and polarizations on group varieties, Duke Math. J. 80 (1995), 223–250. · Zbl 0847.11036
[4] [F]G. Faltings, Endlichkeitss\"{}atze f\"{}ur abelsche Variet\"{}aten \"{}uber Zahlk\"{}orpern, Invent. Math. 73 (1983), 349–366. · Zbl 0588.14026
[5] [G]Z. Gao, A special point problem of Andr\'{}e–Pink–Zannier in the universal family of abelian varieties, arXiv:1407.5578v1 (2014). · Zbl 1410.11067
[6] [H]P. Habegger, Special points on fibered powers of elliptic surfaces, J. Reine Angew. Math. 685 (2013), 143–179. · Zbl 1318.14023
[7] [I]J. Igusa, On the graded ring of theta-constants, I, II, Amer. J. Math. 86 (1964), 219–246; 88 (1966), 221–236. · Zbl 0146.31703
[8] [KS]S. Kawaguchi and J. H. Silverman, Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights of abelian varieties, arXiv:1301.4964 (2013). [Mi]J. S. Milne, Abelian varieties, in: Arithmetic Geometry, Springer, 1986, 103– 150. [Mu]D. Mumford, On the equations defining abelian varieties, I–III, Invent. Math. 1 (1966), 287–354; 3 (1967), 75–135; 3 (1967), 215–244.
[9] [O]M. Orr, Families of abelian varieties with many isogenous fibres, arXiv: 1209.3653v3 (2013). [Pa]F. Pazuki, Theta height and Faltings height, Bull. Soc. Math. France 140 (2012), 19–49. [Pi]J. Pila, Rational points of definable sets and results of Andr\'{}e–Oort–Manin– Mumford type, Int. Math. Res. Notices 2009, no. 13, 2476–2507. [Po]B. Poonen, Spans of Hecke points on modular curves, Math. Res. Lett. 8 (2001), 767–770. [Pr]A. Prendergast-Smith, The cone conjecture for abelian varieties, J. Math. Sci. Univ. Tokyo 19 (2012), 243–261. Isogeny orbits in a family of abelian varieties173
[10] [R1]M. Raynaud, Faisceaux amples sur les sch\'{}emas en groupes et les espaces homog‘enes, Lecture Notes in Math. 119, Springer, Berlin, 1970.
[11] [R2]M. Raynaud, Hauteurs et isog\'{}enies, Ast\'{}erisque 127 (1985), 199–234. · Zbl 1182.14049
[12] [S1]J. H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. · Zbl 0505.14035
[13] [S2]J. H. Silverman, Heights and elliptic curves, in: Arithmetic Geometry, Springer, 1986, 253–265.
[14] [Z]Yu. G. Zarhin, A finiteness theorem for unpolarized abelian varieties over number fields with prescribed places of bad reduction, Invent. Math. 79 (1985), 309– 321. · Zbl 0557.14024
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