Explicit Jordan decompositions for ideal lattices in CM fields

Abstract

Let $E$ be a CM number field, $F$ a totally real subfield (e.g. $\mathbb{Q}$), and let $\mathfrak{a}$ be a non-zero fractional ideal of $E$. Endowed with the Hermitian trace form $h_{E/F}(x,y)={\mathrm{Tr}}_{E/F}(x\overline{y})$, the ideal $\mathfrak{a}$ defines an ideal lattice over ${\mathcal{O}}_F$. In this paper, we give explicit formulas for the Jordan decomposition of this lattice at a prime ideal $\mathfrak{p}\subset {\mathcal{O}}_F$, in terms of the prime ideal factorization of $\mathfrak{a}$ in $E$. Following the approach of Erez, Morales, and Perlis, we reduce the computation to the local behavior at the primes above $\mathfrak{p}$. Our results provide local invariants for the isometry relation between ideal lattices, with potential applications to the study of structured lattices arising in arithmetic and cryptography.

30 pages