Independence of \(\ell\) and local terms

Let \(k\) be an algebraically closed field and let \(c:C\rightarrow X\times X\) be a correspondence. Let \(\ell \) be a prime invertible in \(k\) and let \(K\in D^b_c(X, \overline {\mathbb Q}_\ell )\) be a complex. An action of \(c\) on \(K\) is by definition a map \(u:c_1^*K\rightarrow c_2^!K\). For such an action one can define for each proper component \(Z\) of the fixed point scheme of \(c\) a local term \(\text{lt}_Z(K, u)\in \overline {\mathbb Q}_\ell \). In this talk I will discuss various techniques for studying these local terms and some independence of \(\ell \) results for them. I will also discuss consequences for traces of correspondences acting on cohomology.