Abstract
The Narasimhan–Nori conjecture asks for a closed formula for the number of non-isomorphic principal polarizations of any given abelian variety. In this paper, we introduce a new algorithm that gives a lower bound on the number of non-isomorphic principal polarizations on any given abelian variety. We show, for example, that the Jacobian of the genus four underlying curve of Schoen’s I-WP minimal surface has at least 9 non-isomorphic principal polarizations. We also explore the Jacobians of Klein’s quartic, Fermat’s quartic, Bring’s curve, and more.
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Notes
This is because \(\text {End}({{\,\mathrm{Jac}\,}}(\text {I-WP})) \otimes {\mathbb {Q}}\simeq M_4(K)\), where K is imaginary quadratic. Thus, \({{\,\mathrm{Jac}\,}}(\text {I-WP})\) is isogenous to the product of elliptic curves with complex multiplication by K.
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Acknowledgements
The authors would like to thank Magma and Sage contributors Edgar Costa, Nicolas Mascot, John Voight, and above all Jeroen Sijsling, who generously offered incredibly detailed and consistent help in computing the automorphism groups of Jacobians. Lee would also like to thank Matthias Weber for his guidance at the beginning of this project. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while Lee was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2019 semester. Ray is partially supported the National Science Foundation GRFP under Grant Number DGE 1842165.
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Lee, D., Ray, C. Automorphisms of abelian varieties and principal polarizations. Rend. Circ. Mat. Palermo, II. Ser 71, 483–494 (2022). https://doi.org/10.1007/s12215-020-00590-7
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DOI: https://doi.org/10.1007/s12215-020-00590-7