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arXiv · 2206.12442

Families of $\phi$-congruence subgroups of the modular group

Angelica Babei·Andrew Fiori·Cameron Franc
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Abstract

We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed $\phi$-congruence subgroups, are obtained by reducing homomorphisms $\phi$ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi-unipotent group, and on the other hand from maps into symplectic groups of degree four. In the quasi-unipotent case we also provide a detailed discussion of the corresponding modular forms, using the fact that the tower of curves in this case contains the tower of isogenies over the elliptic curve $y^2=x^3-1728$ defined by the commutator subgroup of the modular group.

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Angelica Babei, Andrew Fiori, Cameron Franc. 2022-06-24. Families of $\phi$-congruence subgroups of the modular group. https://arxiv.org/abs/2206.12442

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