Bounds on Successive Minima of Orders in Number Fields and Scrollar Invariants of Curves

TL;DR

This paper investigates the constraints on successive minima of orders in number fields and scrollar invariants of curves, providing explicit classifications for many cases and revealing their multiplicative nature.

Contribution

It characterizes all multiplicative constraints on successive minima of orders in number fields and determines these constraints explicitly for many degrees, inspired by Lenstra's conjecture.

Findings

All nontrivial multiplicative constraints on successive minima are derived from multiplication.

Explicit determination of all such constraints for degrees less than 18.

Analogous results established for scrollar invariants of curves.

Abstract

Orders and fractional ideals in number fields provide interesting examples of lattices. We ask: what lattices arise from orders in number fields? We prove that all nontrivial multiplicative constraints on successive minima of orders come from multiplication. Moreover, inspired by a conjecture of Lenstra, for infinitely many positive integers $n$ (including all $n < 18$), we explicitly determine all multiplicative constraints on successive minima of orders in degree $n$ number fields. We also prove analogous results for scrollar invariants of curves.