Abelian varieties over cyclotomic fields with good reduction everywhere. (English) Zbl 1058.11038
J.-M. Fontaine [Invent. Math. 81, 515–538 (1985; Zbl 0612.14043)] and V. A. Abrashkin [Math. USSR, Izv. 31, 1–46 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 4, 691–736 (1987; Zbl 0674.14035)] proved Shafarevich conjectures, namely that there does not exist a non-zero abelian variety over \(\mathbb{Q}\) with good reduction modulo every prime number \(p\). In this paper the author determines those cyclotomic fields \(\mathbb{Q}(\zeta_f)\) to which their theorem can be extended. More precisely, for every conductor \(f\notin \{1,2,4,5,7,8,9,11,12,15\}\) there exist non-zero abelian varieties over \(\mathbb{Q}(\zeta_f)\) with good reduction everywhere. Conversely, for every \(f\) in this set there does not exist a non-zero abelian variety with good reduction everywhere, except possibly for \(f=11,15\). Under the assumption that the Generalized Riemann Hypothesis (GRH) holds, then the same is also true for \(f=11,15\).
The first statement follows from a result of Langlands [Proposition 2, p. 263, B. Mazur, Invent. Math. 76, 179–330 (1984; Zbl 0545.12005)] which implies that certain isogeny factors of the Jacobian variety \(J_1(f)\) of the modular curve \(X_1(f)\) have everywhere good reduction over \(\mathbb{Q}(\zeta_f)\). More specifically, \(J_1(f)\) admits \(\text{Gal}(X_1(f)\to X_0(f))\) as an automorphism group which yields the isogeny decomposition \(J_f\sim \prod_{\psi}J_{f,\psi}\), where \(\psi\) runs through the rational even characters of \((\mathbb{Z}/f\mathbb{Z})^*\). The abelian varieties \(J_{f,\psi}\) are defined over \(\mathbb{Q}\). When the conductor of \(\psi\) is equal to \(f\), then \(J_{f,\psi}\) acquires everywhere good reduction over the maximal real subfield of \(\mathbb{Q}(\zeta_f)\).
So the goal of the paper is the two last statements of the theorem. The proof uses the study of finite flat group schemes and their extensions by one another over \(\mathbb{Z}[\zeta_f]\). First he shows that for a suitable small prime number \(p\), all simple \(p\)-group schemes over \(\mathbb{Z}[\zeta_f]\) have order \(p\). The proof uses Fontaine’s and Abrashkin bounds (loc. cit.) on the ramification of the Galois action and Odlyzko’s [J. Martinet, Journées arithmétiques, Exeter 1980, Lond. Math. Soc. Lect. Note Ser. 56, 151–193 (1982; Zbl 0491.12005)] discriminant bounds. Under the hypothesis of the validity of GRH the latter bounds are much stronger. Then he uses the classification theorem of F. Oort and J. Tate [Ann. Sci. Éc. Norm. Supér. (4) 3, 1–21 (1970; Zbl 0195.50801)] and checks that the only group schemes of order \(p\) over \(\mathbb{Z}[\zeta_f]\) are actually isomorphic to the constant group scheme \(\mathbb{Z}/p\mathbb{Z}\) and to its Cartier dual \(\mu_p\). This implies that every \(p\)-group scheme admits a filtration with closed flat subgroup schemes and subquotients isomorphic to \(\mathbb{Z}/p\mathbb{Z}\) or \(\mu_p\).
The first statement follows from a result of Langlands [Proposition 2, p. 263, B. Mazur, Invent. Math. 76, 179–330 (1984; Zbl 0545.12005)] which implies that certain isogeny factors of the Jacobian variety \(J_1(f)\) of the modular curve \(X_1(f)\) have everywhere good reduction over \(\mathbb{Q}(\zeta_f)\). More specifically, \(J_1(f)\) admits \(\text{Gal}(X_1(f)\to X_0(f))\) as an automorphism group which yields the isogeny decomposition \(J_f\sim \prod_{\psi}J_{f,\psi}\), where \(\psi\) runs through the rational even characters of \((\mathbb{Z}/f\mathbb{Z})^*\). The abelian varieties \(J_{f,\psi}\) are defined over \(\mathbb{Q}\). When the conductor of \(\psi\) is equal to \(f\), then \(J_{f,\psi}\) acquires everywhere good reduction over the maximal real subfield of \(\mathbb{Q}(\zeta_f)\).
So the goal of the paper is the two last statements of the theorem. The proof uses the study of finite flat group schemes and their extensions by one another over \(\mathbb{Z}[\zeta_f]\). First he shows that for a suitable small prime number \(p\), all simple \(p\)-group schemes over \(\mathbb{Z}[\zeta_f]\) have order \(p\). The proof uses Fontaine’s and Abrashkin bounds (loc. cit.) on the ramification of the Galois action and Odlyzko’s [J. Martinet, Journées arithmétiques, Exeter 1980, Lond. Math. Soc. Lect. Note Ser. 56, 151–193 (1982; Zbl 0491.12005)] discriminant bounds. Under the hypothesis of the validity of GRH the latter bounds are much stronger. Then he uses the classification theorem of F. Oort and J. Tate [Ann. Sci. Éc. Norm. Supér. (4) 3, 1–21 (1970; Zbl 0195.50801)] and checks that the only group schemes of order \(p\) over \(\mathbb{Z}[\zeta_f]\) are actually isomorphic to the constant group scheme \(\mathbb{Z}/p\mathbb{Z}\) and to its Cartier dual \(\mu_p\). This implies that every \(p\)-group scheme admits a filtration with closed flat subgroup schemes and subquotients isomorphic to \(\mathbb{Z}/p\mathbb{Z}\) or \(\mu_p\).
Reviewer: Amilcar Pacheco (Rio de Janeiro)
