Vign\'eras orbifolds: isospectrality, regulators, and torsion homology
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 Regi(Y1)2/Regi(Y2)2 for Vign\'eras orbifolds Y1, Y2, 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 Y1 and Y2 can differ, and Galois representations.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.