Spectrum of a linear differential equation with constant coefficients
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Mathematische Zeitschrift
Spectrum of a linear differential equation with constant coefficients Tinhinane A. Azzouz1,2 Received: 14 May 2018 / Accepted: 17 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 A1,an . We show that it is a finite union of either closed disks or topological closures k of open disks and that it satisfies a continuity property. Keywords Berkovich spaces · p-adic differential equations · Spectral theory
1 Introduction Differential equations constitute an important tool for the investigation of algebraic and analytic varieties, over the complex and the p-adic numbers. Notably, de Rham cohomology is one of the most powerful way to obtain algebraic and analytic informations. Besides, ultrametric phenomena appear naturally when studying formal Taylor solutions of the equation around singular and regular points. As soon as the theory of ultrametric differential equations became a central topic of investigation around 1960, after the work of Dwork, Robba, et al., the following intersting phenomena appeared. In the ultrametric setting, the solutions of a linear differential equation may fail to converge as expected, even if the coefficients of the equation are entire functions. For example, over the ground field Q p of p-adic numbers, the exponential power series n exp(T ) = n≥0 Tn! which is solution of the equation y = y has radius of convergence 1
equal to | p| ( p−1) even though the equation shows no singularities. However, the behaviour of the radius of convergence is well controlled, and its knowledge permits to obtain several information about the equation. Namely it controls the finite dimensionality of the de Rham cohomology. For more details, we refer the reader to the recent work of Poineau and Pulita [14,16,17] and [15].
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Tinhinane A. Azzouz [email protected]
1
Univ. Grenoble Alpes, CNRS, Institut Fourier, 38000 Grenoble, France
2
Present Address: Univ. Alger 1, Faculté des Sciences, Département Mathématiques et Informatique, Alger, Algeria
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T. A. Azzouz
The starting point of this paper is an interesting relation between this radius and the notion of spectrum (in the sense of Berkovich). Before discussing more in detail our results, we shall quickly explain this relation. Consider a ring A together with a derivation d : A → A. A differential module on (A, d) is a finite free A-module M together with a Z-linear map ∇:M→M satisfying for all m ∈ M and all f ∈ A the relation ∇( f m) = d( f )m + f ∇(m). If an isomorphism M ∼ = Aν is given, then ∇ coincides with an operator of the form d + G : Aν → Aν where d acts on Aν component by component and G is a square matrix with coefficients in A. If A is moreover a Banach algebra with respect to a given norm . and Aν is endowed with the max norm, then we can endow the algebra of continuous
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