Description
Abstract. The analytic theory of Poincaré series and Maass cusp forms and their $L$-functions for $\mathrm{SL}(3,\mathbb{Z})$ has, so far, been limited to the spherical Maass forms, i.e. elements of a spectral basis for $L^2(\mathrm{PSL}(3,\mathbb{Z})\backslash\mathrm{PSL}(3,\mathbb{R})/\mathrm{PSO}(3,\mathbb{R}))$. I will describe the Maass cusp forms of $L^2(\mathrm{PSL}(3,\mathbb{Z})\backslash\mathrm{PSL}(3,\mathbb{R}))$ 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 $\mathrm{SL}(3,\mathbb{Z})$ Maass forms to the non-spherical case, and I will discuss the structure of that proof, as well.