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arXiv · 2407.07240

Vign\'eras orbifolds: isospectrality, regulators, and torsion homology

Alex Bartel·Aurel Page
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Abstract

We develop a new approach to the isospectrality of the orbifolds constructed by Vign\'eras. We give fine sufficient criteria for i-isospectrality in given degree i and for representation equivalence. These allow us to produce very small exotic examples of isospectral orbifolds: hyperbolic 3-orbifolds that are i-isospectral for all i but not representation equivalent, hyperbolic 3-orbifolds that are 0-isospectral but not 1-isospectral, and others. Using the same method, we also give sufficient criteria for rationality of regulator quotients Reg_i(Y_1)^2/Reg_i(Y_2)^2 for Vign\'eras orbifolds Y_1, Y_2, sometimes even when they are not isospectral. Moreover, we establish a link between the primes that enter in these regulator quotients and at which torsion homology of Y_1 and Y_2 can differ, and Galois representations.

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Alex Bartel, Aurel Page. 2024-07-09. Vign\'eras orbifolds: isospectrality, regulators, and torsion homology. https://arxiv.org/abs/2407.07240

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