2013/12/31 by Yajun Zhou · 2 citations
Mathematics · #math.CA #math.NT
paper · pdf · doi:10.1007/s11139-014-9663-7
published as Ramanujan J. (2015) 38:227-329 · i+69 pages, 1 figure. Minor clarifications and notational changes. Numerical supplement (NumSuppAGF1.nb) downloadable from http://arxiv.org/format/1312.6352
arxiv created 2014/11/23 · arxiv updated 2015/10/23
In the framework of Kontsevich-Zagier periods, we derive integral representations for weight-k automorphic Green's functions invariant under modular transformations in \varGamma0(N) (N∈\mathbb Z≥1 ), provided that there are no cusp forms on the respective Hecke congruence groups with an even integer weight k≥4. These Kontsevich-Zagier integral representations for automorphic Green's functions give explicit formulae for certain Eichler-Shimura maps connecting Eichler cohomology to Maaß cusp forms. We construct integral representations for weight-4 Gross-Zagier renormalized Green's functions (automorphic self-energy) from limit scenarios of the respective Kontsevich-Zagier integrals. We reduce the weight-4 automorphic self-energy on X0(4)(\mathbb C)=\varGamma0(4)∖\mathfrak H^* to an explicit form, which supports an algebraicity conjecture of Gross and Zagier.