A Survey and a Complement of Fundamental Hermite Constants

In this note, we give an account of further development of generalized Hermite constants after [W1]. In [W5], we introduced the fundamental Hermite constant γ(G,Q, k) of a pair (G,Q) of a connected reductive group G and a maximal parabolic subgroup Q of G both defined over a global field k. Though we use adelic language, the definition of γ(G,Q, k) is given as a natural generalization of the definition of the original Hermite constant γn. It was proved in [W5], among other things, that some properties of Hermite–Rankin’s constant, e.g., Rankin’s inequality, can be generalized to fundamental Hermite constants. We will give a survey of these results in the first two sectons of this note. In Section 1, we recall Hermite– Rankin’s constant and its generalization due to Thunder [T2]. Section 2 is a summary of our papers [W5] and [W6], in which we define the fundamental Hermite constant γ(G,Q, k) and state properties of γ(G,Q, k). In Example 1, we show that Thunder’s generalization is none other than the fundamental Hermite constant of GLn defined over an algebraic number field. Section 3 is a complement of properties of fundamental Hermite constants, in which we will study a behavior of fundamental Hermite constants under central k-isogenies.