The local-global property for bitangents of plane quartics

20 Apr 2020  ·  Ishitsuka Yasuhiro, Ito Tetsushi, Ohshita Tatsuya, Taniguchi Takashi, Uchida Yukihiro ·

We study the arithmetic of bitangents of smooth quartics over global fields. With the aid of computer algebra systems and using Elsenhans--Jahnel's results on the inverse Galois problem for bitangents, we show that, over any global field of characteristic different from $2$, there exist smooth quartics which have bitangents over every local field, but do not have bitangents over the global field. We give an algorithm to find such quartics explicitly, and give an example over $\mathbb{Q}$. We also discuss a similar problem concerning symmetric determinantal representations. This paper is a summary of the first author's talk at the JSIAM JANT workshop on algorithmic number theory in March 2019. Details will appear elsewhere.

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Number Theory Algebraic Geometry