Joint PU/IAS Number Theory

In the 1940s Mahler initiated the program of determining the bass note spectrum $$\mathrm{Spec}(P):=\left\{\inf_{\underline{x} \in\Lambda \setminus \underline{0}}\left\vert P(\underline{x})\right\vert, \Lambda \subset \mathbb{R}^k \text{ a unimodular lattice} \right\} $$ for some homogeneous form $P$. Understanding this spectrum is central in the geometry of numbers and offers a unified framework for many unsolved problems in number theory and homogeneous dynamics. However, not many cases are known. When $P$ is a binary quadratic form, $\mathrm{Spec}(P)$ is well studied and is classically related to the Markoff spectrum. For binary forms of higher degrees much less is known. Mordell determined the extremal lattices for the spectra of binary cubic forms and further conjectured the existence of spectral gaps, a conjecture later disproved by Davenport. In this talk, we resolve this spectral problem for all binary cubic forms and more generally for all binary forms $P$ of degree $n\geq3$, by showing that quite contrary to these conjectures the spectrum $\mathrm{Spec}(P)$ is always an interval.