Calabi-Yau Manifolds Over Finite Fields, II
We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of...
| Հիմնական հեղինակներ: | , , |
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| Ձևաչափ: | Journal article |
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2004
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| _version_ | 1826299006548967424 |
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| author | Candelas, P Ossa, X Rodriguez-Villegas, F |
| author_facet | Candelas, P Ossa, X Rodriguez-Villegas, F |
| author_sort | Candelas, P |
| collection | OXFORD |
| description | We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to `see' these curves in the geometry of the quintic. Having these zeta-functions to hand we are led to comment on their form in the light of mirror symmetry. That some residue of mirror symmetry survives into the zeta-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the zeta-functions are rational functions and the degrees of the numerators and denominators are exchanged between the zeta-functions for the manifold and its mirror. It is clear nevertheless that the zeta-function, as classically defined, makes an essential distinction between Kahler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a `quantum modification' of the zeta-function that restores the symmetry between the Kahler and complex structure parameters. We note that the zeta-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion. |
| first_indexed | 2024-03-07T04:55:20Z |
| format | Journal article |
| id | oxford-uuid:d66a11dc-db0b-420b-9ec5-23ea437b01e6 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T04:55:20Z |
| publishDate | 2004 |
| record_format | dspace |
| spelling | oxford-uuid:d66a11dc-db0b-420b-9ec5-23ea437b01e62022-03-27T08:33:16ZCalabi-Yau Manifolds Over Finite Fields, IIJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d66a11dc-db0b-420b-9ec5-23ea437b01e6Symplectic Elements at Oxford2004Candelas, POssa, XRodriguez-Villegas, FWe study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to `see' these curves in the geometry of the quintic. Having these zeta-functions to hand we are led to comment on their form in the light of mirror symmetry. That some residue of mirror symmetry survives into the zeta-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the zeta-functions are rational functions and the degrees of the numerators and denominators are exchanged between the zeta-functions for the manifold and its mirror. It is clear nevertheless that the zeta-function, as classically defined, makes an essential distinction between Kahler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a `quantum modification' of the zeta-function that restores the symmetry between the Kahler and complex structure parameters. We note that the zeta-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion. |
| spellingShingle | Candelas, P Ossa, X Rodriguez-Villegas, F Calabi-Yau Manifolds Over Finite Fields, II |
| title | Calabi-Yau Manifolds Over Finite Fields, II |
| title_full | Calabi-Yau Manifolds Over Finite Fields, II |
| title_fullStr | Calabi-Yau Manifolds Over Finite Fields, II |
| title_full_unstemmed | Calabi-Yau Manifolds Over Finite Fields, II |
| title_short | Calabi-Yau Manifolds Over Finite Fields, II |
| title_sort | calabi yau manifolds over finite fields ii |
| work_keys_str_mv | AT candelasp calabiyaumanifoldsoverfinitefieldsii AT ossax calabiyaumanifoldsoverfinitefieldsii AT rodriguezvillegasf calabiyaumanifoldsoverfinitefieldsii |