Artin's $L$-functions and Gassmann Equivalence

Nagata, Kiyoshi

doi:10.3836/tjm/1270150724

Cited by 6 publications

(6 citation statements)

References 1 publication

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“…♦The next Proposition 3.2.4 provides a spectral criterion that is equivalent to weak conjugacy, and is an immediate corollary of Proposition 3.1.1. It is analogous to a number theoretical result of Nagata[59]. Proposition 3.2.4.…”

supporting

confidence: 64%

“…Proposition E. If we have a diagram (2), then H 1 and H 2 are weakly conjugate if and only if the multiplicity of the zero eigenvalue in σ M i pRes G H i Ind G H j 1q is independent of i, j " 1, 2. This result, reformulated in Proposition 3.2.4, is proven by an adaptation of a number theoretical argument of Nagata [59]. The crucial differential geometric ingredient is the spectral characterisation of the multiplicity of the trivial representation in any given representation (Lemma 2.6.3).…”

Section: Introductionmentioning

confidence: 84%

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Twisted isospectrality, homological wideness and isometry

Cornelissen¹,

Peyerimhoff²

2021

Preprint

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Given a manifold (or, more generally, a developable orbifold) M0 and two closed Riemannian manifolds M1 and M2 with a finite covering map to M0, we give a spectral characterisation of when they are equivalent Riemannian covers (in particular, isometric), assuming a representation-theoretic condition of homological wideness: if M is a common finite cover of M1 and M2 and G is the covering group of M over M0, the condition involves the action of G on the first homology group of M (it holds, for example, when there exists a rational homology class on M whose orbit under G consists of |G| linearly independent homology classes). We prove that, under this condition, Riemannian covering equivalence is the same as isospectrality of finitely many twisted Laplacians on the manifolds, acting on sections of flat bundles corresponding to specific representations of the fundamental groups of the manifolds involved. Using the same methods, we provide spectral criteria for weak conjugacy and strong isospectrality. In the negative curvature case, we formulate an analogue of our result for the length spectrum. The proofs are inspired by number-theoretical analogues. We study examples where the representation theoretic condition does and does not hold. For example, when M1 and M2 are commensurable non-arithmetic closed Riemann surfaces of negative Euler characteristic, there is always such an M0, and the condition of homological wideness always holds.

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supporting

confidence: 64%

Section: Introductionmentioning

confidence: 84%

Twisted isospectrality, homological wideness and isometry

Cornelissen¹,

Peyerimhoff²

2021

Preprint

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“…This is meromorphic function of complex variable s. For the sake of brevity we will denote them by L F (ρ). In 1986 K.Nagata published the paper [9] from which a careful reader could extract the following result: Theorem 2. Let K, K ′ denote two finite separable geometric extensions of F q (x).…”

Section: Introductionmentioning

confidence: 99%

On Artin L-functions and Gassmann Equivalence for Global Function Fields

Solomatin

2016

Preprint

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In this paper we present an approach to study arithmetical properties of global function fields by working with Artin L-functions. In particular we recall and then extend a criteria of two function fields to be arithmetically equivalent in terms of Artin L-functions of representations associated to the common normal closure of those fields. We provide few examples of such non-isomorphic fields and also discuss an algorithm to construct many such examples by using torsion points on elliptic curves. Finally, we will show how to apply our results in order to distinguish two global fields by a finite list of Artin L-functions.Acknowledgements: I would like to thank both my advisors, namely professor Bart de Smit and professor Karim Belabas, for their useful advices during the project.

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“…The square of the amplitude of each frequency component was used as the power of the signal 23 ) . MDF is a representative value of the frequency that divides the area of the EMG power spectrum in the extracted muscle radioform into two equal areas 24 ) , and it is an index of the overall muscle fatigue of the EMG spectrum waveform. In addition, MDF transitions to the lower frequency band over time when muscle fatigue appears, both during maximal and submaximal exertion of muscle strength 23 , 24 ) .…”

Section: Discussionmentioning

confidence: 99%

“…MDF is a representative value of the frequency that divides the area of the EMG power spectrum in the extracted muscle radioform into two equal areas 24 ) , and it is an index of the overall muscle fatigue of the EMG spectrum waveform. In addition, MDF transitions to the lower frequency band over time when muscle fatigue appears, both during maximal and submaximal exertion of muscle strength 23 , 24 ) . Regarding muscle fatigue by electromyogram frequency analysis, since “movement of the EMG power spectrum to a lower frequency (wave slowing)” is defined as muscle fatigue 24 ) , decrease in MDF was defined as muscle fatigue.…”

Section: Discussionmentioning

confidence: 99%