Lower order terms in the shape of cubic fields

Abstract: The ring of integers of a degree n number field may be viewed as an n-dimensional lattice within the canonical embedding. Spectrally expanding the space of lattices, we study the distribution of lattice shapes of rings of integers when cubic fields are ordered by discriminant by studying the Weyl sums testing the lattice shape against the real analytic Eisenstein series and Maass cusp forms. In the case of Eisenstein series we identify a lower order main term of order $X^{11/12}$ when fields of discriminant of order $X$ are counted with a smooth weight. \\ Joint work with Eun Hye Lee. Recent work of Lee and Ramin Tagloo-Bighash promises to extend these ideas to integral orbits in general prehomogeneous vector spaces.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)