## Relative regulators of number fields.(English)Zbl 0945.11022

Let $$L$$ be an algebraic number field. R. Zimmert [Invent. Math. 62, 367-380 (1980; Zbl 0456.12003)] found positive constants $$c_0,c_1$$ such that the regulator $$\text{Reg}(L)$$ satisfies $\text{Reg}(L)\geq c_oc_1^{[L:\mathbb Q]}.$ A.-M. Bergé and J. Martinet [Sémin. Théor. Nombres, Paris/Fr. 1987-88, Prog. Math. 81, 23-50 (1990; Zbl 0699.12014)] defined a relative regulator associated to an extension $$L/K$$, which is essentially the ratio $$\text{Reg}(L)/\text{Reg}(K)$$. They asked whether a result analogous to that of Zimmert holds for the relative regulator. The present authors derive such a result, i.e., they prove the following
Main Theorem: There exist positive absolute constants $$d_0,d_1$$ such that for any extension $$L/K$$ of number fields we have $\text{Reg}(L)/\text{Reg}(K)\geq\big(d_0d_1^{[L:K]}\big)^{[K:\mathbb Q]}.$
Reviewer: V.Ennola (Turku)

### MSC:

 11R27 Units and factorization 11R47 Other analytic theory

### Keywords:

relative regulator; theta series

### Citations:

Zbl 0456.12003; Zbl 0699.12014
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