Tambara functors are the analogue of commutative rings in equivariant algebra. A little over a decade ago, Nakaoka defined the Zariski spectrum of prime ideals of a Tambara functor, and Nakaoka and Calle-Ginnett computed the analogue of $Spec(\mathbb{Z})$ over finite cyclic groups. In this talk, I will survey some new techniques for studying the spectra of Tambara functors. The main application is a description of the Tambara affine line, the equivariant analogue of $Spec(\mathbb{Z}[x])$, over cyclic groups of prime order. This is joint work with David Chan, David Mehrle, Ben Spitz, and Danika Van Niel.
