Fields of moduli and definition of hyperelliptic covers

摘要

In this paper, for all genera g>1, g ≡ 1 mod 4, we construct an explicit hyperelliptic curve whose field of moduli is $ mathbb{Q} $ and such that the minimum subfield of $ mathbb{R} $ over which it can be hyperelliptically defined is a degree three extension of $ mathbb{Q} $ . These examples are related to previous work by Earle, Shimura, and Mestre and to a recent conjecture by Shaska.