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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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