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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.

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