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The classification (up to isogeny) of simple abelian varieties over finite fields with only one rational point has been given by Madan-Pal [MR:0439848]. It consists of an infinite family of examples over $\F_2$, together with six sporadic examples over $\F_2$ and one each over $\F_3$ and $\F_4$.

The infinite family consists of, for each positive integer $m \neq 1,2,7,30$, an abelian variety of dimension $\varphi(m)$ with Weil polynomial $\prod_{1 \leq k \leq m/2; (k,m) = 1} \left( 1 - \left(\left( 1 - \cos \frac{2 \pi k}{m} \right)\pm \sqrt{ \left(2 + \cos \frac{2\pi k}{m} \right)^2 - 1 }\right) x + 2x^2 \right).$

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• Review status: beta
• Last edited by Kiran S. Kedlaya on 2019-09-01 23:44:57
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