Non-spherical Poincaré series, cusp forms and L-functions for GL(3)

Non-spherical Poincaré series, cusp forms and L-functions for GL(3) - Jack Buttcane

Jack Buttcane
University of Buffalo
April 10, 2018

The analytic theory of Poincaré series and Maass cusp forms and their L-functions for $SL(3,Z)$ has, so far, been limited to the spherical Maass forms, i.e. elements of a spectral basis for $L^2(SL(3,Z)\PSL(3,R)/SO(3,R))$. I will describe the Maass cusp forms of $L^2(SL(3,Z)\PSL(3,R))$ which are minimal with respect to the action of the Lie algebra and give a (relatively) simple method for constructing Kuznetsov-type trace formulas by considering Fourier coefficients of certain Poincaré series. In recent work with Valentin Blomer, we have extended our proof of spectral-aspect subconvexity for L-functions of $SL(3,Z)$ Maass forms to the non-spherical case, and I will discuss the structure of that proof, as well.