Philippe Michel

Algebraic Twists of automorphic forms

The subconvexity problem for $GL(1)$-twists of a fixed $GL(n)$ cusp form is basically equivalent to establishing that Dirichlet characters $\chi$ of prime modulus $q$ does not correlate with the Fourier/Whittaker coefficients of the given $GL(n)$ form in the convexity range $q^{n/2}$. For $n=2$ this problem was solved by Duke-Friedlander-Iwaniec and for $n=3$ by Munshi. In this talk, we discuss the generalisation of this problem when the character $\chi$ is replaced by the trace function of a general $\ell$-adic sheaf. We will discuss the $GL(3)$ case which build on the recent alternative approach of Munshi’s theorem by Holowinsky-Nelson and the $GL(2)\times GL(3)$ case building on a recent preprint of P. Sharma. This is joint work with E. Kowalski, Y. Lin and W. Sawin.