### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 060, 33 pages      arXiv:1710.01071      https://doi.org/10.3842/SIGMA.2018.060
Contribution to the Special Issue on Moonshine and String Theory

### $(2+)$-Replication and the Baby Monster

Chris Cummins a and Rodrigo Matias b
a) Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd Ouest, Montréal, H3G 1M8, Québec, Canada
b) Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Portugal

Received October 04, 2017, in final form May 31, 2018; Published online June 16, 2018

Abstract
The definitions of replicable and completely replicable functions are intimately related to the Hecke operators for the modular group. We define the notions of ''$(2+)$-replicable'' and ''completely $(2+)$-replicable'' functions by considering the Hecke operators for $\Gamma_0(2)^+$. We prove that the McKay-Thompson series for $2\cdot\mathbb{B}$, as computed by Höhn, are completely $(2+)$-replicable.

Key words: moonshine; baby monster; replication.

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References

1. Alexander D., Cummins C., McKay J., Simons C., Completely replicable functions, in Groups, Combinatorics & Geometry (Durham, 1990), London Math. Soc. Lecture Note Ser., Vol. 165, Cambridge University Press, Cambridge, 1992, 87-98.
2. Borcherds R.E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. USA 83 (1986), 3068-3071.
3. Borcherds R.E., Generalized Kac-Moody algebras, J. Algebra 115 (1988), 501-512.
4. Borcherds R.E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405-444.
5. Carnahan S., Generalized moonshine I: genus-zero functions, Algebra Number Theory 4 (2010), 649-679, arXiv:0812.3440.
6. Carnahan S., Generalized moonshine II: Borcherds products, Duke Math. J. 161 (2012), 893-950, arXiv:0908.4223.
7. Carnahan S., Generalized moonshine IV: monstrous Lie algebras, arXiv:1208.6254.
8. Conway J.H., Norton S.P., Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308-339.
9. Cummins C.J., Congruence subgroups of groups commensurable with ${\rm PSL}(2,{\mathbb Z})$ of genus 0 and 1, Experiment. Math. 13 (2004), 361-382.
10. Cummins C.J., Gannon T., Modular equations and the genus zero property of moonshine functions, Invent. Math. 129 (1997), 413-443.
11. Cummins C.J., Norton S.P., Rational Hauptmoduls are replicable, Canad. J. Math. 47 (1995), 1201-1218.
12. Diamond F., Shurman J., A first course in modular forms, Graduate Texts in Mathematics, Vol. 228, Springer-Verlag, New York, 2005.
13. Ferenbaugh C.R., On the modular functions involved in ''Monstrous Moonshine'', Ph.D. Thesis, Princeton University, 1992.
14. Ford D., McKay J., Norton S., More on replicable functions, Comm. Algebra 22 (1994), 5175-5193.
15. Frenkel I.B., Lepowsky J., Meurman A., A natural representation of the Fischer-Griess Monster with the modular function $J$ as character, Proc. Nat. Acad. Sci. USA 81 (1984), 3256-3260.
16. Frenkel I.B., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, Inc., Boston, MA, 1988.
17. GAP - Groups, Algorithms, and Programming, Version 4.8.7, 2017, http://www.gap-system.org/.
18. Höhn G., Generalized moonshine for the baby monster, Preprint, 2003.
19. Kozlov D.N., On functions satisfying modular equations for infinitely many primes, Canad. J. Math. 51 (1999), 1020-1034.
20. Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progress in Mathematics, Vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004.
21. Martin Y., On modular invariance of completely replicable functions, in Moonshine, the Monster, and Related Topics (South Hadley, MA, 1994), Contemp. Math., Vol. 193, Amer. Math. Soc., Providence, RI, 1996, 263-286.
22. Norton S.P., More on moonshine, in Computational Group Theory (Durham, 1982), Academic Press, London, 1984, 185-193.
23. Norton S.P., Generalized moonshine, in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., Vol. 47, Amer. Math. Soc., Providence, RI, 1987, 208-210.
24. Norton S.P., Moonshine-type functions and the CRM correspondence, in Groups and symmetries, CRM Proc. Lecture Notes, Vol. 47, Amer. Math. Soc., Providence, RI, 2009, 327-342.
25. Shimura G., Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, Vol. 11, Princeton University Press, Princeton, NJ, 1994.