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.
Similar content being viewed by others
References
Bach, E., Sorenson, J.: Explicit bounds for primes in residue classes. Math. Comp. 65(216), 1717–1735 (1996)
Baker, M., Norine, S.: Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007)
Baker, M., Shokrieh, F.: Chip-firing games, potential theory on graphs, and spanning trees. J. Combin. Theory Ser. A 120(1), 164–182 (2013)
Bosch, S., Lorenzini, D.: Grothendieck’s pairing on component groups of Jacobians. Invent. Math. 148(2), 353–396 (2002)
Clancy, J., Kaplan, N., Leake, T., Payne, S., Wood, M.M.: On a Cohen-Lenstra heuristic for Jacobians of random graphs. J. Algebraic Combin. 42(3), 701–723 (2015)
Clancy, J., Leake, T., Payne, S.: A note on Jacobians, Tutte polynomials, and two-variable zeta functions of graphs. Exp. Math. 24(1), 1–7 (2015)
Cori, R., Rossin, D.: On the sandpile group of dual graphs. European J. Combin. 21(4), 447–459 (2000)
Davenport, H.: Multiplicative Number Theory. Third edition. Revised and with a preface by Hugh L. Montgomery. Graduate Texts in Mathematics, Vol. 74. Springer-Verlag, New York (2000)
Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613–1616 (1990)
Lorenzini, D.: Arithmetical properties of Laplacians of graphs. Linear and Multilinear Algebra 47(4), 281–306 (2000)
Lorenzini, D.J.: Arithmetical graphs. Math. Ann. 285(3), 481–501 (1989)
Miranda, R.: Nondegenerate symmetric bilinear forms on finite abelian 2-groups. Trans. Amer. Math. Soc. 284(2), 535–542 (1984)
Pollack, P.: Prime splitting in abelian number fields and linear combinations of Dirichlet characters. Int. J. Number Theory 10(4), 885–903 (2014)
Shokrieh, F.: The monodromy pairing and discrete logarithm on the Jacobian of finite graphs. J. Math. Cryptol. 4(1), 43–56 (2010)
Wall, C.: Quadratic forms on finite groups, and related topics. Topology 2(4), 281–298 (1963)
Wood, M.: The distribution of sandpile groups of random graphs. J. Amer. Math. Soc. 30(4), 915–958 (2017)
Acknowledgements
This project was completed as part of the 2014 Summer Undergraduate Mathematics Research at Yale (SUMRY) program, where the second and third authors were supported as mentors and the first, fourth, and fifth authors were supported as participants. It is a pleasure to thank all involved in the program for creating a vibrant research community. We benefited from conversations with Dan Corey, Andrew Deveau, Jenna Kainic, Nathan Kaplan, Susie Kimport, Dan Mitropolsky, and Anup Rao. We thank Sam Payne for suggesting the problem. We are also especially grateful to Paul Pollack, whose ideas significantly strengthened the results of this paper. Finally, we thank the referees for their careful reading and insightful comments. The authors were supported by NSF grant CAREER DMS-1149054 (PI: Sam Payne).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gaudet, L., Jensen, D., Ranganathan, D. et al. Realization of Groups with Pairing as Jacobians of Finite Graphs. Ann. Comb. 22, 781–801 (2018). https://doi.org/10.1007/s00026-018-0406-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-018-0406-0