Seminar in Algebraic Geometry and Number Theory

Abstract: In this talk I will give an introduction to the Ramanujan conjecture, from its historical setting in terms of the growth rate of Fourier coefficients of modular forms, to its more modern and generalized reformulation in terms of automorphic representations of reductive groups. Since the full Ramanujan conjecture remains open in general, some approximations are often considered in practice such as the Sarnak–Xue density hypothesis which I will also introduce. I will then discuss recent joint work with Shai Evra and Mathilde Gerbelli-Gauthier where we have proved the cohomological Sarnak–Xue density hypothesis for SO_5 and its Gross inner forms.