Triangle Groups: Automorphic Forms and Nonlinear Differential Equations

Ashok, Sujay K.; Jatkar, Dileep P.; Raman, Madhusudhan

doi:10.3842/SIGMA.2020.102

Triangle Groups: Automorphic Forms and Nonlinear Differential Equations

Sujay K. Ashok
(
- IMSc, Chennai
)
,
Dileep P. Jatkar
(
- Harish-Chandra Res. Inst.
)
,
Madhusudhan Raman
(
- Tata Inst.
)

Apr 13, 2020

13 pages

Published in:

SIGMA 16 (2020) 102

Published: Oct 11, 2020

e-Print:

2004.06035 [nlin.SI]

DOI:

10.3842/SIGMA.2020.102 (publication)

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Abstract: (SIGMA)

We study the relations governing the ring of quasiautomorphic forms associated to triangle groups with a single cusp, thereby extending our earlier results on Hecke groups. The Eisenstein series associated to these triangle groups are shown to satisfy Ramanujan-like identities. These identities in turn allow us to associate a nonlinear differential equation to each triangle group. We show that they are solved by the quasiautomorphic weight-2 Eisenstein series associated to the triangle group and its orbit under the group action. We conclude by discussing the Painlevé property of these nonlinear differential equations.

triangle groups
Chazy equations
Painlevé analysis
differential equations: nonlinear
orbit

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