We show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang’s conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many rational points has only finitely many automorphisms. Moreover, we investigate to what extent finiteness of S-integral points on a variety over a number field persists over finitely generated fields. To this end, we introduce the class of mildly bounded varieties and prove a general criterion for proving this persistence.

中文翻译：

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算术双曲：自同构和持久性

我们证明，如果射影变种的自同构群是扭转，那么它是有限的。受郎对双曲变数有理点的猜想的启发，我们用它来证明只有有限多个有理点的射影变种只有有限的同构性。此外，我们研究了在一个数字场上，一个变体上的S-积分点的有限性在有限生成的场上持续到什么程度。为此，我们介绍了轻度有界品种的类别，并证明了证明这种持久性的一般标准。