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The local–global principle for symmetric determinantal representations of smooth plane curves

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A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local–global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi–Brauer varieties and mod 2 Galois representations, and prove that the local–global principle holds for conics and cubics. We also construct counterexamples to the local–global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.

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Acknowledgments

Some calculations done by GAP (version 4.7.5) on subgroups of the finite symplectic group \(\mathop {\mathrm {Sp}}\nolimits _{2m}({\mathbb F}_2)\) were very helpful when we studied the action of Galois groups on theta characteristics. The authors would like to thank an anonymous referee for their useful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan

    Yasuhiro Ishitsuka & Tetsushi Ito

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  1. Yasuhiro Ishitsuka
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  2. Tetsushi Ito
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Correspondence to Tetsushi Ito.

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The work of the first author was supported by JSPS KAKENHI Grant Number 13J01450. The work of the second author was supported by JSPS KAKENHI Grant Numbers 20674001 and 26800013.

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Ishitsuka, Y., Ito, T. The local–global principle for symmetric determinantal representations of smooth plane curves. Ramanujan J 43, 141–162 (2017). https://doi.org/10.1007/s11139-016-9775-3

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