Arithmetic invariant theory and 2-descent for plane quartic curves. With an appendix by Tasho Kaletha.(English)Zbl 1416.11044

Summary: Given a smooth plane quartic curve $$C$$ over a field $$k$$ of characteristic 0, with Jacobian variety $$J$$, and a marked rational point $$P\in C(k)$$, we construct a reductive group $$G$$ and a $$G$$-variety $$X$$, together with an injection $$J(k)/2J(k)\hookrightarrow G(k)\backslash X(k)$$. We do this using the Mumford theta group of the divisor $$2\Theta$$ of $$J$$, and a construction of Lurie which passes from Heisenberg groups to Lie algebras.
In the appendix a converse to Lurie’s functorial construction of simply laced Lie algebras is given by Tasho Kaletha.

MSC:

 11D25 Cubic and quartic Diophantine equations 11E72 Galois cohomology of linear algebraic groups 14H40 Jacobians, Prym varieties
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