# zbMATH — the first resource for mathematics

A ‘Hardy-Littlewood’ approach to the norm form equation. (English) Zbl 0659.10051
Let $$L/\mathbb Q$$ be a number field, $${\mathcal O}$$ an order in $$L$$, and $$T: L\to\mathbb Q$$ the trace map. For $$\rho \in L^*$$ and $$s\in\mathbb C$$, define a Dirichlet series $T_{{\mathcal O}}(\rho,s)=\sum_{u\in {\mathcal O}^*,| T(\rho u)| >1}(\log | T(\rho u)|)^{-s}.$ Let $$r=r_ 1+r_ 2-1$$ be the usual quantity associated to $$L$$.
Theorem. The series for $$T_{{\mathcal O}}(\rho,s)$$ converges in the half-plane $$\text{Re}(s)>r$$. It has an analytic continuation to $$\text{Re}(s)>r-2$$, in which half-plane its only singularities are at $$s=r$$ and $$s=r-1$$.
The proof starts with the function $$U_{{\mathcal O}}(s)=\sum_{u\in {\mathcal O}^*,h(u)>0}h(u)^{-s}$$, where $$h(\cdot)$$ is the relative logarithmic height for $$L$$. This is stated to have similar properties to those desired for $$T_{{\mathcal O}}(\rho,s)$$, but the reader is referred to the author’s paper “Units in abelian group rings...” [Ill. J. Math. 33, No. 4, 542–553 (1989; Zbl 0897.11033)] for the proof. The author then gives explicit estimates relating log $$| T(\rho u)|$$ and $$h(u)$$. To prove these estimates he invokes two deep theorems, Evertse’s theorem on sums of $$n$$ $$S$$-units (which uses W. Schmidt’s subspace theorem) and a strong form of Baker’s theorem. The proof of the author’s main theorem is then a matter of breaking his sum up into pieces, pulling out a $$U_{{\mathcal O}}(s)$$ and a multiple of $$U_{{\mathcal O}}(s+1)$$, and using the deep estimates to show that the other terms are negligible. Then the properties of $$U_{{\mathcal O}}$$ imply the desired properties for $$T_{{\mathcal O}}(\rho,s)$$.
Finally, we remark that the set of solutions to a norm form equation is given (by a theorem of W. Schmidt) by a finite collection of sets of the form $$\{\rho u: u\in \mathcal O\}$$. This observation explains the title of the paper.

##### MSC:
 11M35 Hurwitz and Lerch zeta functions 11D57 Multiplicative and norm form equations 11J86 Linear forms in logarithms; Baker’s method 11J87 Schmidt Subspace Theorem and applications 11R42 Zeta functions and $$L$$-functions of number fields
Full Text:
##### References:
 [1] Schmidt, Diophantine Approximation 785 (1980) · Zbl 0421.10019 [2] DOI: 10.2307/1970824 · Zbl 0226.10024 · doi:10.2307/1970824 [3] Evertse, J. reine angew. Math 358 pp 6– (1985) [4] Baker, Transcendence Theory: Advances and Applications (1977) [5] Everest, J. reine angew. Math 375 pp 24– (1987) [6] DOI: 10.1112/jlms/s2-28.2.227 · Zbl 0521.12006 · doi:10.1112/jlms/s2-28.2.227 [7] Evertse, Compositio Math 53 pp 225– (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.