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

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.