45 years of Deligne’s Fourier transform

E. Kowalski's blog 2021-11-30

It is exactly 45 years ago today that Deligne, in a letter to David Kazhdan (available here) introduced the $\ell$ -adic version of the classical Fourier transform…

This construction, which operates on sheaves (or better complexes, or objects of the derived category, or…) on the additive group (or more generally a commutative unipotent algebraic group) has a definition that may look utterly bewildering at first for an analyst or an arithmetician, something like

$FT(M)=R\pi_{1,!}(p_2^*M\otimes \mathcal{L}_{\psi(xy)}).$

However, it has the key property that if $t_M$ is the trace function of the input object $M$ , then the trace function of Deligne’s Fourier transform object $FT(M)$ is the discrete Fourier transform of $t_M$ (after fixing a suitable additive character). And this opens the door to the study of these discrete Fourier transforms using all the tools of algebraic geometry, and especially the Riemann Hypothesis of Deligne himself…