Field theory for function fields of plane quartic curves (Q1975907)

Let \(K\) be the rational function field of a smooth plane curve \(C\) of degree \( d\) (\(d\geq 2)\) defined over an algebraic number field \(k\) of characteristic zero. If \(K_m\) denotes a maximal rational subfield of \(K\), the authors of the paper under review answer, in the case where \(C\) is a quartic curve, the following questions. (1) When is the extension \(K/K_m\) Galois? (2) Let \(L\) be the Galois closure of \(K/K_m\). What could we say about \(L\)? (3) What is the Galois group \(\text{Gal}(L/K_m)\)? The characterisation is dependent on the point \(P\), which is the center of the projection of the curve \(C\) to a line \(l\), and of the genus \(g(P)\) of the curve \(\grave{C}\), which corresponds to the field \(L\).