Journal of Algebraic Combinatorics

, Volume 42, Issue 3, pp 701–723 | Cite as

On a Cohen–Lenstra heuristic for Jacobians of random graphs

  • Julien Clancy
  • Nathan Kaplan
  • Timothy Leake
  • Sam PayneEmail author
  • Melanie Matchett Wood


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.


Random graphs Cohen–Lenstra heuristics Cokernels of random matrices Sandpile groups Graph Jacobians 



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.


  1. 1.
    Bhargava, M., Kane, D., Lenstra, H., Poonen, B., Rains, E.: Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves, preprint. arXiv:1304.3971 (2013)
  2. 2.
    Bannai, E., Munemasa, A.: Duality maps of finite abelian groups and their applications to spin models. J. Algebr. Comb. 8(3), 223–233 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cassels, J.: Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung. J. Reine Angew. Math. 211, 95–112 (1962)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Cohen, H., Lenstra, H.: Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983). In: Lecture Notes in Mathematics, vol. 1068, pp. 33–62. Springer, Berlin (1984)Google Scholar
  5. 5.
    Cohen, H., Lenstra Jr, H.: Heuristics on class groups, Number theory (New York, 1982). In: Lecture Notes in Mathematics, vol. 1052, pp. 26–36. Springer, Berlin (1984)Google Scholar
  6. 6.
    Clancy, J., Kaplan, N., Leake, T., Payne, S., Wood, M.: On a Cohen–Lenstra heuristic for Jacobians of random graphs. arXiv:1402.5129
  7. 7.
    Clancy, J., Leake, T., Payne, S.: A note on Jacobians, Tutte polynomials, and two-variable zeta functions of graphs. Experiment. Math. 24, 1–7 (2015)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Conway, J.H., Sloane, N.J.A.: Sphere packings, lattices and groups, third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer, New York (1999). With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. VenkovGoogle Scholar
  9. 9.
    Delaunay, C.: Heuristics on Tate-Shafarevitch groups of elliptic curves defined over \({\mathbb{Q}}\). Exp. Math. 10(2), 191–196 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Delaunay, C.: Heuristics on class groups and on Tate-Shafarevich groups: the magic of the Cohen–Lenstra heuristics, Ranks of elliptic curves and random matrix theory. In: London Mathematical Society Lecture Note Series, vol. 341, pp. 323–340. Cambridge Univ. Press, Cambridge (2007)Google Scholar
  11. 11.
    Fulman, J.: Hall-Littlewood polynomials and Cohen–Lenstra heuristics for Jacobians of random graphs, to appear in Ann. Comb. arXiv:1403.0473, (2014)
  12. 12.
    Friedman, E., Washington, L.: On the distribution of divisor class groups of curves over a finite field, Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, pp. 227–239 (1989)Google Scholar
  13. 13.
    Gaudet, L., Jensen, D., Ranganathan, D., Wawrykow, N., Weisman, T.: Realization of groups with pairing as Jacobians of finite graphs, preprint. arXiv:1410.5144 (2014)
  14. 14.
    Lorenzini, D.: Arithmetical graphs. Math. Ann. 285(3), 481–501 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    MacWilliams, J.: Orthogonal matrices over finite fields. Am. Math. Mon. 76, 152–164 (1969)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Miranda, R.: Nondegenerate symmetric bilinear forms on finite abelian 2-groups. Trans. Am. Math. Soc. 284(2), 535–542 (1984)zbMATHGoogle Scholar
  17. 17.
    Shokrieh, F.: The monodromy pairing and discrete logarithm on the Jacobian of finite graphs. J. Math. Cryptol. 4(1), 43–56 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Stanley, R.: Enumerative combinatorics. Cambridge Studies in Advanced Mathematics, 49, vol. 1, 2nd edn. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  19. 19.
    Wagner, D.: The critical group of a directed graph, preprint. arXiv:math/0010241 (2000)
  20. 20.
    Wall, C.T.C.: Quadratic forms on finite groups, and related topics. Topology 2, 281–298 (1964)CrossRefGoogle Scholar
  21. 21.
    Wood, M.: The distribution of sandpile groups of random graphs, preprint. arXiv:1402.5149 (2014)

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Julien Clancy
    • 1
  • Nathan Kaplan
    • 1
  • Timothy Leake
    • 1
  • Sam Payne
    • 1
    Email author
  • Melanie Matchett Wood
    • 2
    • 3
  1. 1.Yale UniversityNew HavenUSA
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  3. 3.American Institute of MathematicsSan JoseUSA

Personalised recommendations