  In this paper we obtain explicit formulas for the traces of Hecke operators
on spaces of cusp forms in certain instances related to arithmetic triangle
groups. These expressions are in terms of hypergeometric character sums over
finite fields, a theory developed largely by Greene, Katz,
Beukers-Cohen-Mellit, and Fuselier-Long-Ramakrishna-Swisher-Tu. Our approach,
in contrast to the previous works, is uniform and more geometric, and it works
equally well for forms on elliptic modular curves and Shimura curves. The same
method can be applied to obtain eigenvalues of Hecke operators as well.
