Kontsevich-Zagier integrals for automorphic Green’s functions. I. (English) Zbl 1379.11057
Summary: In the framework of Kontsevich-Zagier periods, we derive integral representations for weight-\(k\) automorphic Green’s functions invariant under modular transformations in \(\varGamma_0(N)\) \((N\in\mathbb Z_{\geq 1})\), provided that there are no cusp forms on the respective Hecke congruence groups with an even integer weight \(k\geq 4\). 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 \(X_0(4)(\mathbb C)=\varGamma_0(4)\smallsetminus\mathfrak H^\ast\) to an explicit form, which supports an algebraicity conjecture of B. H. Gross and D. B. Zagier [Invent. Math. 84, 225–320 (1986; Zbl 0608.14019)].
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F03 | Modular and automorphic functions |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
| 11Y70 | Values of arithmetic functions; tables |
| 33B15 | Gamma, beta and polygamma functions |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
Keywords:
Kontsevich-Zagier periods; automorphic Green’s functions; Ramanujan’s alternative base theory; Jacobi elliptic functions; Eichler-Shimura maps; Gross-Zagier renormalizationCitations:
Zbl 0608.14019References:
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