Counting congruence subgroups
Copyright policy )Counting congruence subgroups
Let $��$ denote the modular group $SL(2,\Bbb Z)$ and $C_n(��)$ the number of congruence subgroups of $��$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log C_n(��)}{(\log n)^2/\log\log n} = \frac{3-2\sqrt{2}}{4}.$ We also present a very general conjecture giving an asymptotic estimate for $C_n(��)$ for general arithmetic groups. The lower bound of the conjecture is proved modulo the generalized Riemann hypothesis for Artin-Hecke L-functions, and in many cases is also proved unconditionally.
30 pages
Unimodular groups, congruence subgroups (group-theoretic aspects), numbers of subgroups, Subgroup theorems; subgroup growth, algebraic number fields, modular group, Structure of modular groups and generalizations; arithmetic groups, rings of integers, Group Theory (math.GR), Discrete subgroups of Lie groups, arithmetic groups, semisimple simply-connected connected algebraic groups, Riemann surfaces, congruence subgroups, FOS: Mathematics, Asymptotic results on counting functions for algebraic and topological structures, Mathematics - Group Theory
Unimodular groups, congruence subgroups (group-theoretic aspects), numbers of subgroups, Subgroup theorems; subgroup growth, algebraic number fields, modular group, Structure of modular groups and generalizations; arithmetic groups, rings of integers, Group Theory (math.GR), Discrete subgroups of Lie groups, arithmetic groups, semisimple simply-connected connected algebraic groups, Riemann surfaces, congruence subgroups, FOS: Mathematics, Asymptotic results on counting functions for algebraic and topological structures, Mathematics - Group Theory
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