Abstract
We study abelian varieties over finitely generated fields K of characteristic zero, whose \(\ell \)-adic Tate modules are isomorphic as Galois modules for all primes \(\ell \).
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Borevich, Z.I., Faddeev, D.K.: Integral representations of quadratic rings. Vestnik Leningrad. Univ. 15(19), 52–64 (1960) (in Russian). MR0153707 (27 #3668)
Chai, C.-L., Conrad, B., Oort, F.: Complex Multiplication and Lifting Problems. Mathematical Surveys and Monographs, vol. 195. American Mathematical Society, Providence (2014)
Conrad, B.: Gross-Zagier revisited. With an appendix by W.R. Mann. In: Darmon, H., Zhang, S. (eds.) Heegner Points and Rankin L-Series. Mathematical Sciences Research Institute Publications, vol. 49, pp. 67–163. Cambridge University Press, Cambridge (2004)
Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73(3), 349–366 (1983)
Faltings, G.: Complements to Mordell. In: Faltings, G., Wüstholz, G., et al. (eds.) Rational Points. Aspects of Mathematics, vol. E6, pp. 203–227. Vieweg, Braunschweig (1986)
Fröhlich, A.: Locally free modules over arithmetic orders. J. Reine Angew. Math. 274/275, 112–124 (1975)
Lang, S.: Abelian Varieties, 2nd edn. Springer, New York (1983)
Lorenzini, D.: An Invitation to Arithmetic Geometry. Graduate Studies in Mathematics, vol. 9. American Mathematical Society, Providence (1996)
Mumford, D.: Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. 2nd edn. Oxford University Press, London (1974)
Patrikis, S., Voloch, F., Zarhin, Yu.G.: Anabelian geometry and descent obstructions on moduli spaces. Algebra Number Theory 10(6), 1191–1219 (2016)
Serre, J.-P.: Abelian \(\ell \)-Adic Representations and Elliptic Curves. Advanced Book Classics, 2nd edn. Addison-Wesley, Redwood City (1989)
Tate, J.: Endomorphisms of abelian varieties over finite fields. Invent. Math. 2, 134–144 (1966)
Zarhin, Yu.G: Hyperelliptic Jacobians without complex multiplication. Math. Res. Lett. 7(1), 123–132 (2000)
Zarhin, Yu.G.: Homomorphisms of abelian varieties. In: Aubry, Y., Lachaud, G. (eds.) Arithmetic, Geometry and Coding Theory. Séminaires & Congres, vol. 11, pp. 189–215. Société Mathématique de France, Paris (2005)
Zarhin, Yu.G.: Homomorphisms of abelian varieties over finite fields. In: Kaledin, D., Tschinkel, Yu. (eds.) Higher-Dimensional Geometry over Finite Fields. NATO Science for Peace and Security Series D, vol. 16, pp. 315–343. IOS, Amsterdam (2008)
Acknowledgements
Part of this work was done in May–June 2015 when the author was visiting Department of Mathematics of the Weizmann Institute of Science (Rehovot, Israel) and in May–June 2016 when the author was a visitor at the Max-Planck-Institut für Mathematik (Bonn, Germany). The hospitality and support of both Institutes are gratefully acknowledged.
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In memoriam of Bill Waterhouse (1941–2016)
The author is grateful to Jiangwei Xue for useful discussions, to Stefan Patrikis and Felipe Voloch for their interest in this paper and to the Simons Foundation for financial and moral support (via Grant # 246625 to Yuri Zarkhin).
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Zarhin, Y.G. Almost isomorphic abelian varieties. European Journal of Mathematics 3, 22–33 (2017). https://doi.org/10.1007/s40879-016-0122-4
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DOI: https://doi.org/10.1007/s40879-016-0122-4