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

2022/07/21 by Vemulapalli, Sameera
#11H50 (Primary) 11H06 #11P21 #14H05 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2207.10522

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.

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