Hermite's Constant for Quadratic Number Fields

Modular Forms, a Computational Approach

“…Since Voronoǐ's pioneering work [Vor08], it has been the goal of many to extend his results from Q to a general algebraic number field F . Recently Coulangeon [Cou01], building on work of Icaza and Baeza [Ica97,BI97], has found a good notion of perfection for quadratic forms over number fields 12 . One of the key ideas in [Cou01] is that the correct notion of equivalence between Humbert forms involves not only the action of GL n (O F ), where O F is the ring of integers of F , but also the action of a certain continuous group U related to the units O × F .…”

On the computation of Hermite–Humbert constants: The algorithm of Cohn revisited

“…He also made conjectures for several other real quadratic fields. Those results for the first three number fields were also established in [2]. In [9] we presented an algorithm for calculating extreme Humbert forms in real quadratic number fields K with h K = 1.…”

Binary Hermitian forms over a cyclotomic field

“…Let O ⊂ F denote the ring of integers, and let· denote complex conjugation. Let k ⊂ F denote the real subfield k = Q( Let ι = (ι 1 , ι 2 ) denote the (non-complex conjugate) embeddings ι : F → C × C given by sending √ 5 to ( √ 5, − √ 5 ), or equivalently given by sending ζ to (ζ, ζ 3 ). Denote the nontrivial embedding by · .…”

“…Since the forms and their minimal vectors are given explicitly here, one could also extract invariants such as an additive analogue of the Hermite constant for Q(ζ 5 ) [3,20].…”