Abstract
For any \(n\ge 2\) we study the group algebra decomposition of an \(([\frac{n}{2}]+1)\)-dimensional family of principally polarized abelian varieties of dimension \(n\) with an action of the dihedral group of order \(2n\). For any odd prime \(p, n=p\) and \(n=2p\) we compute the induced polarization on the isotypical components of these varieties and some other distinguished subvarieties. In the case of \(n=p\) the family contains a one-dimensional family of Jacobians. We use this to compute a period matrix for Klein’s icosahedral curve of genus 5.
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The second author was supported by Fondecyt Grant 1100767, the third author by Fondecyt Grant 1100113.
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Lange, H., Rodríguez, R.E. & Rojas, A.M. Polarizations on abelian subvarieties of principally polarized abelian varieties with dihedral group actions. Math. Z. 276, 397–420 (2014). https://doi.org/10.1007/s00209-013-1206-1
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DOI: https://doi.org/10.1007/s00209-013-1206-1