Abstract
In this paper we compute the spectrum, in the sense of Berkovich, of an ultrametric linear differential equation with constant coefficients, defined over an affinoid domain of the analytic affine line \({{\mathbb {A}}}^{1,an}_{k}\). We show that it is a finite union of either closed disks or topological closures of open disks and that it satisfies a continuity property.
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Notes
In the language of Berkovich, this field can be naturally identified with the complete residue field of the point \(x_{0,r}\in {{\mathbb {A}}}^{1,an}_{k}\) (cf. Sect. 2.2).
Notice that it is relatively easy to show that any non trivial rank one connection over \({{\mathbb {C}}}(\!(S)\!)\) is set theoretically bijective. This follows from the classical index theorem of Malgrange [12]. However, the set theoretical inverse of the connection may not be automatically bounded. This is due to the fact that the base field \({{\mathbb {C}}}\) is trivially valued and the Banach open mapping theorem does not hold in general. However, it is possible to prove that any such set theoretical inverse is bounded (cf. [1]).
This can be motivated by the fact that the resolvent is an analytic function on the complement of the spectrum.
The kernel and the index of a differential equation over an analytic curve are independent of the choice of a derivation.
Note that, since all the structures of finite Banach A-module on M are equivalent, the spectrum does not depend on the choice of such a structure.
Which means that: \(M/{\text {Ker}}\varphi \) endowed with quotient topology is isomorphic to \({\text {Im}}\varphi \).
This topology is usually known as the exponential topology or Vietoris topology.
Note that, in the case where k is not trivially valued, we may assume that the \(\alpha _i\) are of type (1).
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Acknowledgements
The author wishes to express her gratitude to her advisors Andrea Pulita and Jérôme Poineau for their precious advice and suggestions, and for careful reading. She thanks Francesco Baldassarri, Frits Beukers, Antoine Ducros and Françoise Truc for useful occasional discussions and suggestions. She also thanks the referee for his suggestions and careful reading.
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Azzouz, T.A. Spectrum of a linear differential equation with constant coefficients. Math. Z. 296, 1613–1644 (2020). https://doi.org/10.1007/s00209-020-02482-z
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DOI: https://doi.org/10.1007/s00209-020-02482-z