Majumder, Priyanka (2022)
Bounds for Canonical Green's Functions of Cofinite Fuchsian Groups at Cusps.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00020399
Dissertation, Erstveröffentlichung, Verlagsversion

Kurzbeschreibung (Abstract)

In this thesis, we study Green’s functions on modular curves with respect to the canonical metric which come from Arakelov’s theory. Let Γ be a Fuchsian subgroup of PSL2(R) and H be the upper half-plane. Then the quotient space X = Γ\H is conformally equivalent to a non-compact Riemann surface. The canonical Green’s function gcan (z, w), is a function of z, w ∈ X (z ≠ w), which is uniquely characterized by the differential equation d z d cz gcan (z, w) + δw (z) = μcan (z), where δw (z) is the Dirac delta distribution, and gcan (z, w) satisfies the normalization condition ∫x gcan (z, w) μcan (z) = 0 with w ∈ X. In this thesis we reprove some known asymptotic bounds for the canonical Greens function associated to Γ0(N ), Γ1(N ), and Γ(N ). The asymptotic bound for the canonical Green’s function associated with Γ0(N ) (with square-free N ) was first proved by A. Abbes, P. Michel, and E. Ullmo. In this thesis we reproved their result using a different approach namely, we use hyperbolic heat kernels. We express the difference between the hyperbolic Green’s function and the canonical Green’s function, ghyp (z, w) − gcan (z, w), in terms of integrals involving the hyperbolic heat kernel Khyp (t; z, w) (t ∈ R >0 ; z, w ∈ X). Then we prove that at two inequivalent cusps of a cofinite Fuchsian subgroup the canonical Green’s function can be bounded in terms of the scattering constants, the Kronecker limit functions, and the Selberg zeta constant, etc. Then, we consider some examples of cofinite Fuchsian subgroups, Γ0(N ), Γ1(N ), and Γ(N ), which are the most important congruence subgroups. Furthermore, using the hyperbolic heat kernel approach we are also able to prove some new results. In the case of Γ0(N ) we are able to remove the square-free condition on N . We also prove an asymptotic bound for the canonical Green’s function associated with a general congruence subgroup. Note that, this approach with the hyperbolic heat kernel has been introduced by J. Jorgenson, J. Kramer and further extended by A. Aryasomyajula.

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