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).