|
Эта публикация цитируется в 4 научных статьях (всего в 4 статьях)
Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations
Sujay K. Ashoka, Dileep P. Jatkarb, Madhusudhan Ramanc a Institute of Mathematical Sciences, Homi Bhabha National Institute (HBNI),
IV Cross Road, C. I. T. Campus, Taramani, Chennai 600 113, India
b Harish-Chandra Research Institute, Homi Bhabha National Institute (HBNI),
Chhatnag Road, Jhunsi, Allahabad 211 019, India
c Department of Theoretical Physics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Navy Nagar, Colaba, Mumbai 400 005, India
Аннотация:
We study various relations governing quasi-automorphic forms associated to discrete subgroups of ${\rm SL}(2,\mathbb{R}) $ called Hecke groups. We show that the Eisenstein series associated to a Hecke group ${\rm H}(m)$ satisfy a set of $m$ coupled linear differential equations, which are natural analogues of the well-known Ramanujan identities for quasi-modular forms of ${\rm SL}(2,\mathbb{Z})$. Each Hecke group is then associated to a (hyper-)elliptic curve, whose coefficients are determined by an anomaly equation. For the $m=3$ and $4$ cases, the Ramanujan identities admit a natural geometric interpretation as a Gauss–Manin connection on the parameter space of the elliptic curve. The Ramanujan identities also allow us to associate a nonlinear differential equation of order $ m $ to each Hecke group. These equations are higher-order analogues of the Chazy equation, and we show that they are solved by the quasi-automorphic Eisenstein series $E_2^{(m)}$ associated to ${\rm H}(m) $ and its orbit under the Hecke group. We conclude by demonstrating that these nonlinear equations possess the Painlevé property.
Ключевые слова:
Hecke groups, Chazy equations, Painlevé analysis.
Поступила: 6 мая 2019 г.; в окончательном варианте 29 декабря 2019 г.; опубликована 1 января 2020 г.
Образец цитирования:
Sujay K. Ashok, Dileep P. Jatkar, Madhusudhan Raman, “Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations”, SIGMA, 16 (2020), 001, 26 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma1538 https://www.mathnet.ru/rus/sigma/v16/p1
|
Статистика просмотров: |
Страница аннотации: | 118 | PDF полного текста: | 27 | Список литературы: | 16 |
|