QM abelian varieties, hypergeometric character sums and modular forms

This is a report on recent work, with Wen-Ching Winnie Li and Ling Long. In that work explicit formulas are given, involving hypergeometric character sums, for the traces of Hecke operators $T_p$ acting spaces of cusp forms $S_k(\mathrm{\Gamma })$ of weight $k$ for certain arithmetically defined Fuchsian subgroups $\mathrm{\Gamma }\subset {\mathrm{S}\mathrm{L}}_2(\mathbf{R})$. In particular we consider the groups attached to the quaternion division algebra $B_6$ over $\mathbf{Q}$ of discriminant 6.