A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds
We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The p-th coefficients a(p) of the corresponding modular form can often be read off, at least conjecturally, from the truncated partial sums of the underlying hypergeom...
Gespeichert in:
| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Datum: | 2018 |
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2018
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209764 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds / W. Zudilin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-209764 |
|---|---|
| record_format |
dspace |
| spelling |
Zudilin, W. 2025-11-26T11:24:24Z 2018 A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds / W. Zudilin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 11F33; 11T24; 14G10; 14J32; 14J33; 33C20 arXiv: 1805.00544 https://nasplib.isofts.kiev.ua/handle/123456789/209764 https://doi.org/10.3842/SIGMA.2018.086 We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The p-th coefficients a(p) of the corresponding modular form can often be read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of p and from Weil's general bounds |a(p)| ≤ 2p⁽ᵐ⁻¹⁾/², where m is the weight of the form. Furthermore, the critical L-values of the modular form are predicted to be Q-proportional to the values of a related basis of solutions to the hypergeometric differential equation. Feedback from many colleagues has been extremely helpful in the preparation of this manuscript. I would like to thank Frits Beukers, Henri Cohen, Jesús Guillera, Günter Harder, Ling Long, Anton Mellit, Alexei Panchishkin, David Roberts, Emanuel Scheidegger, Duco van Straten, Alexander Varchenko, Fernando Rodriguez Villegas, John Voight, James Wan, Noriko Yui, and Don Zagier for their comments and responses to my questions. Special thanks are expressed to Vasily Golyshev for his clarification of the link between the critical L-values and the corresponding hypergeometrics, which underlie so-called gamma structures [10], and to Michael Somos for his powerful help in making some entries in Table 1 explicit. Finally, I am indebted to the anonymous referees for several helpful comments and corrections. This note grew out of the author’s talk at the BIRS Workshop “Modular Forms in String Theory” held in September 2016, and related discussions there. Later parts of this work were performed during the author’s visits to research institutions whose hospitality and scientific atmosphere were crucial to the success of the project. I thank the staff of the following institutes for providing such excellent conditions for research: BIRS (Banff, Canada, September 2016); MATRIX (Creswick, Australia, January 2017); ESI (Vienna, Austria, March 2017); MPIM (Bonn, Germany, December 2016 and July–August 2017); HIM (Bonn, Germany, March–April 2018). The author is partially supported by the Laboratory of Mirror Symmetry, NRU HSE, RF government grant, ag. no. 14.641.31.0001. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds |
| spellingShingle |
A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds Zudilin, W. |
| title_short |
A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds |
| title_full |
A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds |
| title_fullStr |
A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds |
| title_full_unstemmed |
A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds |
| title_sort |
hypergeometric version of the modularity of rigid calabi-yau manifolds |
| author |
Zudilin, W. |
| author_facet |
Zudilin, W. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The p-th coefficients a(p) of the corresponding modular form can often be read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of p and from Weil's general bounds |a(p)| ≤ 2p⁽ᵐ⁻¹⁾/², where m is the weight of the form. Furthermore, the critical L-values of the modular form are predicted to be Q-proportional to the values of a related basis of solutions to the hypergeometric differential equation.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209764 |
| citation_txt |
A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds / W. Zudilin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 37 назв. — англ. |
| work_keys_str_mv |
AT zudilinw ahypergeometricversionofthemodularityofrigidcalabiyaumanifolds AT zudilinw hypergeometricversionofthemodularityofrigidcalabiyaumanifolds |
| first_indexed |
2025-12-07T15:30:27Z |
| last_indexed |
2025-12-07T15:30:27Z |
| _version_ |
1850886104292524032 |