A proof of a conjecture of Y. Morita (Q2918781)

Let \(A\) be an abelian variety over a number field \(E\), and, \(G_A\) its Mumford-Tate group. The author proves that if \(G_A\) has no nontrivial \(\mathbb Q\)-rational unipotent elements, than \(A\) has potentially good reduction at every finite prime of \(E\). This had been conjectured by \textit{Y. Morita} [J. Fac. Sci., Univ. Tokyo, Sect. I A 22, 437--447 (1975; Zbl 0322.14015)] for the special case of abelian varieties of PEL-type. The main ingredients of the proof are a criterion of \textit{F. Paugam} on good reduction of abelian varieties [Math. Ann. 329, No. 1, 119--160 (2004); erratum ibid. 332, No. 4, 937 (2005; Zbl 1078.11042)]; Vasiu's work on Morita's conjecture [\textit{A. Vasiu}, J. Reine Angew. Math. 618, 51--75 (2008; Zbl 1230.11079)]; and, a local-global principle of isotropy for Mumford-Tate groups.