Georgia Institute of Technology College of Sciences

Given a modular form $f$, one can construct a measure $\mu_f$ on the modular surface $SL(2,\mathbb{Z})\backslash\mathbb{H}$. The celebrated mass equidistribution theorem of Holowinsky and Soundararajan states that as $k\rightarrow\infty$, the measure $\mu_f$ approaches the uniform measure on the surface. Given a maximal order in a quaternion algebra which is non-split over $\mathbb{Q}$, a maximal order leads to a cocompact subgroup of $R^1\subseteq SL(2,\mathbb{Z})$ where the quotient $R^1\backslash\mathbb{H}$ is a Shimura curve. Given a Hecke form $f$ on this Shimura curve, one can construct the analogous measure $\mu_f$, and ask about the limit as $k\rightarrow\infty$. Recent work of Nelson relates this equidistribution problem for the cocompact case to obtaining bounds on sums of Hecke eigenvalues summed over quadratic progressions. In this talk, I will describe this problem in both the cocompact and non-cocompact case while highlighting how differences in algebras lead to differences in geometry. I will then state progress that I have made on bounds that correspond to square root cancellation on average for sums of Hecke eigenvalues summed over quadratic progressions when averaged over a basis of Hecke forms.