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Torsion of abelian varieties and Lubin-Tate extensions. (English) Zbl 1468.11136

Summary: We show that, for an abelian variety defined over a \(p\)-adic field \(K\) which has potential good reduction, its torsion subgroup with values in the composite field of \(K\) and a certain Lubin-Tate extension over a \(p\)-adic field is finite.

MSC:

11G10 Abelian varieties of dimension \(> 1\)
11G25 Varieties over finite and local fields
11S31 Class field theory; \(p\)-adic formal groups
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[1] Bourbaki, N., Algèbre. Chapitre 5, Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse, Livre II: Actualités Sci. Ind., vol. 1261 (1958), Hermann: Hermann Paris · Zbl 0102.27203
[2] Chiarellotto, B.; Le Stum, B., Sur la pureté de la cohomologie cristalline, C. R. Acad. Sci. Paris Sér. I Math., 8, 961-963 (1998) · Zbl 0936.14016
[3] Coates, J.; Sujatha, R.; Wintenberger, J.-P., On the Euler-Poincaré characteristics of finite dimensional \(p\)-adic Galois representations, Publ. Math. Inst. Hautes Études Sci., 93, 107-143 (2001) · Zbl 1143.11021
[4] Colmez, P., Espaces de Banach de dimension finie, J. Inst. Math. Jussieu, 1, 331-439 (2002) · Zbl 1044.11102
[5] Conrad, B., Lifting global representations with local properties (2011), preprint, available at
[6] Deligne, P., La conjecture de Weil I, Publ. Math. Inst. Hautes Études Sci., 43, 273-308 (1974) · Zbl 0287.14001
[7] Deligne, P., La conjecture de Weil II, Inst. Hautes Études Sci. Publ. Math., 52, 137-252 (1980) · Zbl 0456.14014
[8] Faltings, G., \(p\)-adic Hodge theory, J. Amer. Math. Soc., 1, 255-299 (1988) · Zbl 0764.14012
[9] Faltings, G., Crystalline cohomology and \(p\)-adic Galois-representations, (Algebraic Analysis, Geometry, and Number Theory. Algebraic Analysis, Geometry, and Number Theory, Baltimore MD (1988)), 25-80 · Zbl 0805.14008
[10] Fontaine, J.-M., Le corps des périodes \(p\)-adiques, Astérisque, 223, 59-111 (1994), with an appendix by Pierre Colmez, Périodes \(p\)-adiques (Bures-sur-Yvette, 1988) · Zbl 0940.14012
[11] Fontaine, J.-M., Représentations \(p\)-adiques semi-stables, Astérisque, 223, 113-184 (1994), with an appendix by Pierre Colmez, Périodes \(p\)-adiques (Bures-sur-Yvette, 1988) · Zbl 0865.14009
[12] Imai, H., A remark on the rational points of abelian varieties with values in cyclotomic \(Z_p\)-extensions, Proc. Japan Acad., 51, 12-16 (1975) · Zbl 0323.14010
[13] Katz, N.; Messing, W., Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math., 23, 73-77 (1974) · Zbl 0275.14011
[14] Kubo, Y.; Taguchi, Y., A generalization of a theorem of Imai and its applications to Iwasawa theory, Math. Z., 275, 1181-1195 (2013) · Zbl 1286.11085
[15] Mattuck, A., Abelian varieties over \(p\)-adic ground fields, Ann. of Math. (2), 62, 92-119 (1955) · Zbl 0066.02802
[16] Nakkajima, Y., \(p\)-adic weight spectral sequences of log varieties, J. Math. Sci. Univ. Tokyo, 12, 513-661 (2005) · Zbl 1108.14015
[17] Niziol, W., Crystalline conjecture via \(K\)-theory, Ann. Sci. Éc. Norm. Supér. (4), 31, 659-681 (1998) · Zbl 0929.14009
[18] Serre, J.-P., Local Fields, Graduate Texts in Mathematics, vol. 67 (1979), Springer-Verlag, translated from the French by Marvin Jay Greenberg · Zbl 0423.12016
[19] Serre, J.-P., Abelian \(l\)-Adic Representations and Elliptic Curves, Advanced Book Classics (1989), Addison-Wesley Publishing Company Advanced Book Program: Addison-Wesley Publishing Company Advanced Book Program Redwood City, CA, with the collaboration of Willem Kuyk and John Labute · Zbl 0709.14002
[20] Serre, J.-P.; Tate, J., Good reduction of abelian varieties, Ann. of Math. (2), 8, 492-517 (1986) · Zbl 0172.46101
[21] Tsuji, T., \(p\)-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math., 137, 233-411 (1999) · Zbl 0945.14008
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