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On a Cohen–Lenstra heuristic for Jacobians of random graphs

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Abstract

In this paper, we make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are based on a Cohen–Lenstra-type heuristic saying that a finite abelian group with duality pairing appears with frequency inversely proportional to the size of the group times the size of the group of automorphisms that preserve the pairing. We conjecture that the Jacobian of a random graph is cyclic with probability a little over .7935. We determine the values of several other statistics on Jacobians of random graphs that would follow from our conjectures. In support of the conjectures, we prove that random symmetric matrices over \({\mathbb {Z}}_p\), distributed according to Haar measure, have cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.

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Acknowledgments

The authors thank Matt Baker, Wei Ho, Matt Kahle, and the referees. The fourth author was supported in part by NSF grant DMS-1068689 and NSF CAREER grant DMS-1149054. The fifth author was supported by an American Institute of Mathematics Five-Year Fellowship and National Science Foundation grants DMS-1147782 and DMS-1301690.

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Authors and Affiliations

  1. Yale University, New Haven, CT, USA

    Julien Clancy, Nathan Kaplan, Timothy Leake & Sam Payne

  2. Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI, 53705, USA

    Melanie Matchett Wood

  3. American Institute of Mathematics, 600 East Brokaw Road, San Jose, CA, 95112, USA

    Melanie Matchett Wood

Authors
  1. Julien Clancy
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  2. Nathan Kaplan
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  3. Timothy Leake
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  4. Sam Payne
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  5. Melanie Matchett Wood
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Correspondence to Sam Payne.

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Clancy, J., Kaplan, N., Leake, T. et al. On a Cohen–Lenstra heuristic for Jacobians of random graphs. J Algebr Comb 42, 701–723 (2015). https://doi.org/10.1007/s10801-015-0598-x

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  • DOI: https://doi.org/10.1007/s10801-015-0598-x

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