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Are there any results known about the discriminants of indefinite integral binary quadratic forms admitting automorphisms of order 3 or 6? It seems reasonable to expect that any permissible discriminants ought to be quite small and, if so, one would hope that there might be some classical results on the matter.


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as far as I know the (integral) automorphism group of an indefinite form, discriminant not a square, is infinite cyclic. – Will Jagy Jun 12 '13 at 1:07
Aren't these automorphisms the solutions to the integral Pell equation $u^2+\Delta t^2=1$ where $\Delta$ is the discriminant? So, they are precisely the units in the real quadratic field $\mathbb{Q}(\sqrt{\Delta})$, hence of the form $\mathbb{Z}\times\{\pm 1\}$... – Filippo Alberto Edoardo Jun 12 '13 at 1:27
up vote 2 down vote accepted

The structure of the automorphism group becomes clear when one looks at the Conway topograph of a given form. For an indefinite form not representing 0 the topograph has an infinite periodic river separating the positive and negative values, so there is always an infinite cyclic subgroup of the automorphism group, in addition to the order 2 automorphism $(x,y)\mapsto(-x,-y)$ that acts trivially on the topograph. Some forms have additional order 2 symmetries reflecting across lines perpendicular to the river, giving an infinite dihedral subgroup of the automorphism group. If "anti-automorphisms" that change the sign of the values of a form are allowed as automorphisms, there can also be 180 degree rotational symmetries or glide reflection symmetries. Overall, five of the seven frieze groups are realizable as symmetries of the topograph. The two that aren't realizable involve reflections across the river, which cannot be symmetries of the topograph.

In particular all torsion in the automorphism group is 2-torsion, of order either 2 or 4.

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This is interesting: I wasn't aware of the Conway topograph (I've since hunted down a copy of 'The Sensual Quadratic Form'). Thank you. – MRD1729 Jun 12 '13 at 19:11

no 3 torsion. see Theorem 6.12.4 on pages 132-133, Binary Quadratic Forms by Buchmann and Vollmer. Or pages 31-34 of Binary Quadratic Forms by Buell.

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