1 results
Perfect points of abelian varieties
- Part of
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- Journal:
- Compositio Mathematica / Volume 159 / Issue 11 / November 2023
- Published online by Cambridge University Press:
- 08 September 2023, pp. 2261-2278
- Print publication:
- November 2023
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- Article
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Let
$p$ be a prime number, 
$k$ a finite field of characteristic 
$p>0$ and 
$K/k$ a finitely generated extension of fields. Let 
$A$ be a 
$K$-abelian variety such that all the isogeny factors are neither isotrivial nor of 
$p$-rank zero. We give a necessary and sufficient condition for the finite generation of 
$A(K^{\mathrm {perf}})$ in terms of the action of 
$\mathrm {End}(A)\otimes \mathbb {Q}_p$ on the 
$p$-divisible group 
$A[p^{\infty }]$ of 
$A$. In particular, we prove that if 
$\mathrm {End}(A)\otimes \mathbb {Q}_p$ is a division algebra, then 
$A(K^{\mathrm {perf}})$ is finitely generated. This implies the ‘full’ Mordell–Lang conjecture for these abelian varieties. In addition, we prove that all the infinitely 
$p$-divisible elements in 
$A(K^{\mathrm {perf}})$ are torsion. These reprove and extend previous results to the non-ordinary case.
