Curves which cannot be defined over an extension of degree at most two   over the field of moduli
  It has been conjectured that every algebraic curve may be defined either over its field of moduli or over an extension of degree two of it. In this paper we provide a negative answer to it by giving examples of hyperelliptic curves which cannot be defined over an extension of degree at least two over their fields of moduli. 
Ruben A. Hidalgo and Yolanda Fuertes
math.AG math.CV