On canonical subgroups of Hilbert–Blumenthal abelian varieties

Hattori, Shin

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 355-391.

Let $p$ be a rational prime. Let $F$ be a totally real number field which is unramified over $p$. In this paper, we develop a theory of canonical subgroups for Hilbert–Blumenthal abelian varieties with ${𝒪}_F$-actions, in which they are related with Hodge–Tate maps if the $\beta$-Hodge height is less than $(p-1)/p^n$ for every embedding $\beta :F\to {\overline{\mathbb{Q}}}_p$.

Soit $p$ un nombre premier. Soit $F$ un corps totalement réel non ramifié en $p$. Dans cet article, nous développons une théorie de sous-groupes canoniques pour les variétés abéliennes de Hilbert–Blumenthal avec ${𝒪}_F$-actions, dans laquelle ceux-ci sont liés à des applications de Hodge–Tate si la $\beta$-hauteur de Hodge est plus petite que $(p-1)/p^n$ pour tout plongement $\beta :F\to {\overline{\mathbb{Q}}}_p$.

Classification: 11G10, 14L15
Mots-clés : Hilbert–Blumenthal abelian variety, canonical subgroup

[1] Andreatta, Fabrizio; Iovita, Adrian; Pilloni, Vincent $p$-adic families of Siegel modular cuspforms, Ann. Math., Volume 181 (2015) no. 2, pp. 623-697 | DOI | MR | Zbl

[2] Andreatta, Fabrizio; Iovita, Adrian; Pilloni, Vincent On overconvergent Hilbert modular cusp forms, p-adic arithmetic of Hilbert modular forms (Astérisque), Volume 382, Société Mathématique de France, 2016, pp. 163-193 | Zbl

[3] Breuil, Christophe Integral $p$-adic Hodge theory, Algebraic geometry 2000 (Advanced Studies in Pure Mathematics), Volume 36, Mathematical Society of Japan, 2002, pp. 51-80 | DOI | MR | Zbl

[4] Buzzard, Kevin; Calegari, Frank The $2$-adic eigencurve is proper, Doc. Math., Volume extra vol. (2006), pp. 211-232 | MR | Zbl

[5] Buzzard, Kevin; Taylor, Richard Companion forms and weight one forms, Ann. Math., Volume 149 (1999) no. 3, pp. 905-919 | DOI | MR | Zbl

[6] Conrad, Brian Higher-level canonical subgroups in abelian varieties (available at http://math.stanford.edu/ conrad/)

[7] Deligne, Pierre; Pappas, Georgios Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compos. Math., Volume 90 (1994) no. 1, pp. 59-79 | Numdam | Zbl

[8] Fargues, Laurent La filtration canonique des points de torsion des groupes $p$-divisibles, Ann. Sci. Éc. Norm. Supér., Volume 44 (2011) no. 6, pp. 905-961 (avec la collaboration de Yichao Tian) | DOI | Numdam | MR | Zbl

[9] Goren, Eyal Z.; Kassaei, Payman L. Canonical subgroups over Hilbert modular varieties, J. Reine Angew. Math., Volume 670 (2012), pp. 1-63 | DOI | MR | Zbl

[10] Grothendieck, Alexander Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas I, Publ. Math., Inst. Hautes Étud. Sci., Volume 20 (1964), pp. 101-355 | Zbl

[11] Hattori, Shin On a properness of the Hilbert eigenvariety at integral weights: the case of quadratic residue fields (available at http://www.comm.tcu.ac.jp/shinh/)

[12] Hattori, Shin Ramification correspondence of finite flat group schemes over equal and mixed characteristic local fields, J. Number Theory, Volume 132 (2012) no. 10, pp. 2084-2102 | DOI | MR | Zbl

[13] Hattori, Shin Canonical subgroups via Breuil-Kisin modules, Math. Z., Volume 274 (2013) no. 3-4, pp. 933-953 | DOI | MR | Zbl

[14] Hattori, Shin On lower ramification subgroups and canonical subgroups, Algebra Number Theory, Volume 8 (2014) no. 2, pp. 303-330 | DOI | MR | Zbl

[15] Illusie, Luc Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck), Seminar on arithmetic bundles: the Mordell conjecture (Astérisque), Volume 127, Société Mathématique de France, 1985, pp. 151-198 | MR | Zbl

[16] Kassaei, Payman L. A gluing lemma and overconvergent modular forms, Duke Math. J., Volume 132 (2006) no. 3, pp. 509-529 | MR | Zbl

[17] Kassaei, Payman L. Modularity lifting in parallel weight one, J. Am. Math. Soc., Volume 26 (2013) no. 1, pp. 199-225 | DOI | MR | Zbl

[18] Kassaei, Payman L.; Sasaki, Shu; Tian, Yichao Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case, Forum Math. Sigma, Volume 2 (2014), e18, 58 pages (Article ID e18, 58 p.) | MR | Zbl

[19] Katz, Nicholas M. $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Antwerp, 1972) (Lecture Notes in Mathematics), Volume 350, Springer, 1973, pp. 69-190 | DOI | MR | Zbl

[20] Kim, Wansu The classification of $p$-divisible groups over $2$-adic discrete valuation rings, Math. Res. Lett., Volume 19 (2012) no. 1, pp. 121-141 | MR | Zbl

[21] Kisin, Mark Crystalline representations and $F$-crystals, Algebraic geometry and number theory (Progress in Mathematics), Volume 253, Birkhäuser, 2006, pp. 459-496 | DOI | MR | Zbl

[22] Kisin, Mark; Lai, King Fai Overconvergent Hilbert modular forms, Am. J. Math., Volume 127 (2005) no. 4, pp. 735-783 | DOI | MR | Zbl

[23] Lau, Eike Relations between Dieudonné displays and crystalline Dieudonné theory, Algebra Number Theory, Volume 8 (2014) no. 9, pp. 2201-2262 | MR | Zbl

[24] Liu, Tong Torsion $p$-adic Galois representations and a conjecture of Fontaine, Ann. Sci. Éc. Norm. Supér., Volume 40 (2007) no. 4, pp. 633-674 | Numdam | MR | Zbl

[25] Liu, Tong The correspondence between Barsotti-Tate groups and Kisin modules when $p=2$, J. Théor. Nombres Bordx., Volume 25 (2013) no. 3, pp. 661-676 | Numdam | MR | Zbl

[26] Pilloni, Vincent Prolongement analytique sur les variétés de Siegel, Duke Math. J., Volume 157 (2011) no. 1, pp. 167-222 | MR | Zbl

[27] Sasaki, Shu Analytic continuation of overconvergent Hilbert eigenforms in the totally split case, Compos. Math., Volume 146 (2010) no. 3, pp. 541-560 | DOI | MR | Zbl

[28] Tian, Yichao Classicality of overconvergent Hilbert eigenforms: case of quadratic residue degrees, Rend. Semin. Mat. Univ. Padova, Volume 132 (2014), pp. 133-229 | DOI | Numdam | MR | Zbl