Realization of Groups with Pairing as Jacobians of Finite Graphs
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Author(s)
Gaudet, Louis
Jensen, David
Ranganathan, Dhruv
Wawrykow, Nicholas
Weisman, Theodore
Date Issued
November 2, 2018
Publisher
Springer International Publishing
Version
Author's final manuscript
Abstract
Abstract
We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur as the Jacobian of a simple graph. There exist infinite families of finite abelian groups that do not occur as the Jacobians of simple graphs.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00026-018-0406-0
