let $p$ be an odd prime number, and consider the twisted affine fermat curve $$x^p + y^p = \delta$$ with a rational $\delta.$ a well-known theorem of faltings implies that, for $p\ge 5$, the twisted affine fermat curve has finitely many rational points. when $\delta = 1$, it has just two (trivial) rational points, thanks to wiles' proof of fermat's last theorem. in this talk, we will introduce a different idea to study twisted affine fermat curves. it is based on the connection between the central value of the hasse-weil $l$--function associated to the twisted affine fermat curve and the rank of its jacobian, as predicted by the birch and swinnerton-dyer conjecture. we will give a sufficient (effective) condition for the twisted affine fermat curves to have no rational points in terms of the non-vanishing at the central point of certain $l$--functions. then, using analytic methods, we will conclude that our sufficient condition is satisfied infinitely often, for any prime $p$. (this is joint work with y. tian).