Fourier coefficients of half-integral weight modular forms

In this talk, we give an exposition on some recent work, due to various researchers, on the sign of the Fourier coefficients of modular forms when the Fourier coefficients are real. Then we discuss our results for half-integral weight modular forms in two collaboration projects with the team comprising of Y. -J. Jiang, G. -S. Lü, E. Royer and J. Wu and with my colleague B. Kane respectively. One project is about the sign -change of Fourier coefficients over squarefree numbers. The other is concerned with the halting of an algorithm that constructs supersingular elliptic curves when a maximal order of the quaternion algebra is input into the algorithm.