Totally real Thue inequalities over imaginary quadratic fields

Let $F(x,y)$ be an irreducible binary form of degree $\geq 3$ with integer coefficients and with real roots. Let $M$ be an imaginary quadratic field, with ring of integers $Z_M$. Let $K>0$. We describe an efficient method how to reduce the resolution of the relative Thue inequalities $$|F(x,y)|\leq K \;\; (x,y\in Z_M)$$ to the resolution of absolute Thue inequalities of type $$|F(x,y)|\leq k \;\; (x,y\in Z).$$ We illustrate our method with an explicit example

On computing Belyi maps

We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French abstract; revised according to referee's suggestion