p-Adic Properties of Hauptmoduln with Applications to Moonshine

• Опубліковано в: : Symmetry, Integrability and Geometry: Methods and Applications
• Дата: 2019
• Автори: Chen, R.C., Marks, S., Tyler, M.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2019
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/210189
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: p-Adic Properties of Hauptmoduln with Applications to Moonshine / R.C. Chen, S. Marks, M. Tyler // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 42 назв. — англ.

The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln, including the j-function, coincide precisely with the graded characters of the monster module, an infinite-dimensional graded representation of the monster group. On the other hand, Lehner and Atkin proved that the coefficients of the j-function satisfy congruences modulo pⁿ for p ∈ {2, 3, 5, 7, 11}, which led to the theory of p-adic modular forms. We combine these two aspects of the j-function to give a general theory of congruences modulo powers of primes satisfied by the Hauptmoduln appearing in monstrous moonshine. We prove that many of these Hauptmoduln satisfy such congruences, and we exhibit a relationship between these congruences and the group structure of the monster. We also find a distinguished class of subgroups of the monster with graded characters satisfying such congruences.