I will discuss some unlikely-intersection problems for curves in elliptic surfaces, defined over a number field. We introduce a new tool: an equidistribution theorem for heights associated to R-divisors, extending known results for Q-divisors of Chambert-Loir, Thuillier, and Yuan (2008). As applications, we obtain a new proof of a result of Barroero and Capuano (2016) and prove a case of a conjecture of Zhang (1998).