Exotic arithmetic structure on the first Hurwitz triplet

Abstract

We find that the first Hurwitz triplet possesses two distinct arithmetic structures. As Shimura curves $X_1$, $X_2$, $X_3$, whose levels are with norm 13. As non-congruence modular curves $Y_1$, $Y_2$, $Y_3$, whose levels are 7. Both of them are defined over $\mathbb{Q}(\cos \frac{2\pi }{7})$. However, for the third non-congruence modular curve $Y_3$, there exist an "exotic" duality between the associated non-congruence modular forms and the Hilbert modular forms, both of them are related to $\mathbb{Q}(e^{\frac{2\pi i}{13}})$! Our results have relations and applications to modular equations of degree fourteen (including Jacobian modular equation and "exotic" modular equation), "triality" of the representation of $PSL(2,13)$, Haagerup subfactor, geometry of the exceptional Lie group $G_2$, and even the Monster finite simple group $\mathbb{M}$!

57 pages

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