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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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[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)
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