Decay of weighted cusp counts for congruence subgroups of SL₂ over number fields

2026/05/31 by Shengyuan Zhao
Mathematics · #math.NT #math.AG #math.GR

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Abstract

For congruence subgroups commensurable with SL₂ over number fields, we study cusp counts with arithmetic multiplicities. We prove that the ratio of the total weighted cusp count to the group index is bounded by a negative power of the norm of the congruence level, with an exponent that can be chosen explicitly in terms of the degree of the number field. This is a generalization of a theorem of Cox--Parry over rational numbers. The main input is an explicit local orbit estimate for arbitrary subgroups of exact level in SL₂(\mathcal O_K/\mathfrak p^e), uniform over all number fields of fixed degree. This estimate is covered by a general result of Finis--Lapid. We give a new explicit proof in our setting based on an analysis reminiscent of additive combinatorics.

Related

• Counting congruence subroups
• Estimates of cusp forms for certain cocompact arithmetic subgroups
• On the supremum of random cusp forms
• Computations directly on the cuspidal cohomology of congruence subgroups of SL(3, ℤ)
• A generalization of Mazur's theorem (Ogg's conjecture) for number fields