In this paper, we determine the automorphism groups of the hyperelliptic modular curves $X_{1}(N)$, and determine explicit forms for the actions of all automorphisms on certain defining equations of $X_{1}(N)$.

It is known that a positive integer $n$ is the area of a right triangle with rational sides if and only if the elliptic curve $E^{(n)}: y^{2} = x(x^{2}-n^{2})$ has a rational point of order different than 2. A generalization of this result states that a positive integer $n$ is the area of a triangle with rational sides if and only if there is a nonzero rational number $\tau$ such that the elliptic curve $E^{(n)}_{\tau}: y^{2} = x(x-n\tau)(n+n\tau^{-1})$ has a rational point of order different than 2. Such $n$ are called $\tau$-congruent numbers. It is shown that for a given integer $m>1$, each congruence class modulo $m$ contains infinitely many distinct $\tau$-congruent numbers.

We give a simple proof of a theorem of C. Houdayer that an amenable von Neumann subalgebra in a tracial free product von Neumann algebra $M=M_{1}*M_{2}$ is contained in $M_{1}$ whenever it has a diffuse intersection with $M_{1}$.