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Units from 3- and 4-torsion on Jacobians of curves of genus 2. (English) Zbl 0828.11033

Let \(C\) be a curve of genus 2 defined over a number field \(K\) and let \(J\) be its Jacobian. For \(n > 0\) denote the \(n\)-torsion on \(J\) by \(J[n]\). Further assume that \(C\) has a Weierstraß point defined over \(K\). Then \(C\) has a model of the form \(y^2 = q(x)\) where \(q \in K [x]\) is of degree 5. Also assume that \(C\) has everywhere potentially good reduction. The author now explicitly gives a function that evaluates to units at all \(u \in J[3] \backslash \{O\}\) (the origin of \(J)\) in the extension \(K(J[3])\). This function also evaluates to units at all \(u \in J[4] \backslash J[2]\) in the extension \(K(J[4])\) except possibly at primes dividing 2.

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H45 Special algebraic curves and curves of low genus
14H40 Jacobians, Prym varieties
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References:

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