A proof of a conjecture by Lötter on the roots of a supersingular polynomial and its application

In this paper, we prove a conjecture by E. C. Lötter on the roots of a supersingular polynomial. As its application, we present two optimal towers over finite fields corresponding to the sequences of elliptic modular curves $X_0(3\cdot2^{n+2})$ and $X_0(2\cdot3^{n+2})$.