From modular curves to Shimura curves: a p-adic approach

Shimura curves arise as a natural generalisation of elliptic curves. As modular curves, they are constructed as Riemann surfaces, and they turn out to have structure of algebraic curve, i.e. they can be described by some algebraic equations with coefficients in some finite extension of Q. Number theorists are interested in the reductions modulo p of these equations. The problem is that these equations are very difficult to compute. I will describe a method to find these reductions without actually knowing the equation.