Take a projective curve of arithmetic genus one over a finite field $k=\mathbb{F}_{p^n}$. The curve is allowed to have nodes but not cusps. How large can the automorphism group of the curve be? For elliptic curves it should be a semidirect product of the group of rational points and the group of automorphism preserving a fixed rational point but I have no mental picture for singular curves.
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$\begingroup$ If you add rational tails you can make the automorphism as large as you want. $\endgroup$– AngeloCommented Aug 9, 2019 at 18:42
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$\begingroup$ @Angelo assume the curve is geometrically irreducible. $\endgroup$– user144105Commented Aug 9, 2019 at 21:20
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1$\begingroup$ If it's geometrically irreducible and has arithmetic genus 1, it is either smooth, or it is a geometrically rational curve with a single node. This last case is not very hard to analyze. $\endgroup$– AngeloCommented Aug 13, 2019 at 10:15
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