Katharine Woo: Sums of Hecke eigenvalues along polynomials and base change

We study sums of absolute values of Hecke eigenvalues of $\mathrm{GL(2)}$ representations that are tempered at all finite places. We show that these sums exhibit logarithmic savings over the trivial bound if and only if the representation is cuspidal. Then, we connect the problem of studying the sums of Hecke eigenvalues along polynomial values to the base change problem for $\mathrm{GL(2)}$. Finally, we will describe some arithmetic applications of bounds on these sums for countng rational points on del Pezzo surfaces.