This talk will discuss two families, the Legendre family of elliptic curves and the Kloosterman family of exponential sums. While the two families are quite different, they share many of the same arithmetic

properties. For instance, the L-function of symmetric powers of both families are polynomials of known degree and satisfy functional equations once trivial factors are accounted for. This suggests that results on one family may shed light on analogous results for the other.

The aim of this talk is to explore this line of reasoning, with an emphasis on symmetric powers.