Arithmetic exponent pairs

The theory of classical exponent pairs for analytic exponential sums has been extensively developed since 1920’s thanks to van der Corput, Philipps, Bombieri, Iwaniec, Bourgain, et al, motivated by various applications to analytic number theory. Recently, we were able to develop the so-called arithmetic exponent pairs for general Frobenius trace functions composited by $\ell$-adic sheaves over finite fields. These allow us to conclude a series of non-trivial estimates for very short algebraic exponential sums. We explain the underlying ideas in this talk. Some applications will also be mentioned if time permits.