A remark on the arithmetic invariant theory of hyperelliptic curves

Let $C$ be a hyperelliptic curve over a field $k$ of characteristic $0$, and let $P \in C(k)$ be a marked Weierstrass point. As Bhargava and Gross have observed, the 2-descent on the Jacobian of $C$ can be rephrased in terms of the language of arithmetic invariant theory, using the geometry of pencils of quadrics. We give a simple reinterpretation of their construction using instead the geometry of the curve $C$.

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Accepted 24 October 2014

Published 2 April 2015