It is an open question whether the rank of a curve X/Q (i.e. the Mordell--Weil rank of the group of rational points of the Jacobian J/Q) is bounded in terms of the genus g of X. Shioda extended a construction of Mestre to produce infinite families of g>1 curves over Q with rank at least 4g+7. 

In this talk, I will present a refinement of the Mestre--Shioda construction which leads to some interesting families of curves over Q (and over cyclotomic fields) with rank larger than 4g+7. These families are parametrized by certain highly symmetric rational varieties associated to the Prouhet--Tarry--Escott (PTE) problem, a classical problem in number theory.