On Scattering Constants of Congruence Subgroups

  • Miguel Grados
  • Anna-Maria von PippichEmail author
Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 10)


Let Γ be a congruence subgroup of level N. The scattering constant of Γ at two cusps is given by the constant term at s = 1 in the Laurent expansion of the scattering function of Γ at these cusps. Scattering constants arise in Arakelov theory when establishing asymptotics for Arakelov invariants of the modular curve associated to Γ, as the level N tends to infinity. More precisely, in the known cases, scattering constants essentially contribute to the leading term of the asymptotics for the self-intersection of the relative dualizing sheaf. In this article, we prove an identity relating the scattering constants of Γ to certain scattering constants of the principal congruence subgroup \(\overline {\Gamma }(N)\). Providing an explicit formula for the latter, in case that N = 2 or N ≥ 3 is odd and square-free, we thereby present a systematic way of computing the scattering constants of Γ in these cases.



The authors acknowledge support from the International DFG Research Training Group Moduli and Automorphic Forms: Arithmetic and Geometric Aspects.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

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