This study examines recurrence relations for modular traces in arithmetic groups, indicating full determination at prime power levels.
Recently, recurrence relations satisfied by the modular traces of the Hauptmodul for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>N</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {Γ }₀(N) of genus zero have been established. However, these relations do not determine all values of the modular traces. In this study, we examine additional recursions and demonstrate that the values of the modular traces are fully determined for prime power levels.
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Kang et al. (2025) studied this question.
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