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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
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##### References:
  Schmidt, Diophantine Approximation 785 (1980) · Zbl 0421.10019  DOI: 10.2307/1970824 · Zbl 0226.10024 · doi:10.2307/1970824  Evertse, J. reine angew. Math 358 pp 6– (1985)  Baker, Transcendence Theory: Advances and Applications (1977)  Everest, J. reine angew. Math 375 pp 24– (1987)  DOI: 10.1112/jlms/s2-28.2.227 · Zbl 0521.12006 · doi:10.1112/jlms/s2-28.2.227  Evertse, Compositio Math 53 pp 225– (1984)
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