Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions
A rising philosophy in the theory of automorphic representations in number theory is that higher central derivatives of L-functions of automorphic forms should correspond to the intersection numbers of special cycles on moduli spaces. A classic early result along this philosophy was achieved by Gros...
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| Format: | Thesis |
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Massachusetts Institute of Technology
2022
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| Online Access: | https://hdl.handle.net/1721.1/139475 |
| _version_ | 1826209976461295616 |
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| author | Chen, Yongyi |
| author2 | Wei Zhang |
| author_facet | Wei Zhang Chen, Yongyi |
| author_sort | Chen, Yongyi |
| collection | MIT |
| description | A rising philosophy in the theory of automorphic representations in number theory is that higher central derivatives of L-functions of automorphic forms should correspond to the intersection numbers of special cycles on moduli spaces. A classic early result along this philosophy was achieved by Gross and Zagier, who proved that the derivative of the L-function of an elliptic curve is equal, up to a constant, to the Néron-Tate height pairing of a special point called a Heegner point on the elliptic curve.
A more recent result was proven in the function field case by Yun and Zhang which showed that higher derivatives of the base change L-function of an unramified automorphic representation over PGL₂ over a function field are equal, up to a constant, to the self-intersection number, inside the moduli stack of PGL₂-shtukas, of the moduli stack of shtukas for an anisotropic torus.
We prove in the function field case that the higher derivatives of the square of the L-function of unramified automorphic representations over PGL₂ are equal, up to a constant, to the self-intersection number, inside the moduli stack of PGL₂-shtukas, of the moduli stack of shtukas for the split torus. |
| first_indexed | 2024-09-23T14:38:06Z |
| format | Thesis |
| id | mit-1721.1/139475 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T14:38:06Z |
| publishDate | 2022 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1394752022-01-15T03:16:25Z Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions Chen, Yongyi Wei Zhang Massachusetts Institute of Technology. Department of Mathematics A rising philosophy in the theory of automorphic representations in number theory is that higher central derivatives of L-functions of automorphic forms should correspond to the intersection numbers of special cycles on moduli spaces. A classic early result along this philosophy was achieved by Gross and Zagier, who proved that the derivative of the L-function of an elliptic curve is equal, up to a constant, to the Néron-Tate height pairing of a special point called a Heegner point on the elliptic curve. A more recent result was proven in the function field case by Yun and Zhang which showed that higher derivatives of the base change L-function of an unramified automorphic representation over PGL₂ over a function field are equal, up to a constant, to the self-intersection number, inside the moduli stack of PGL₂-shtukas, of the moduli stack of shtukas for an anisotropic torus. We prove in the function field case that the higher derivatives of the square of the L-function of unramified automorphic representations over PGL₂ are equal, up to a constant, to the self-intersection number, inside the moduli stack of PGL₂-shtukas, of the moduli stack of shtukas for the split torus. Ph.D. 2022-01-14T15:13:36Z 2022-01-14T15:13:36Z 2021-06 2021-05-25T12:46:41.476Z Thesis https://hdl.handle.net/1721.1/139475 0000-0003-3019-4187 In Copyright - Educational Use Permitted Copyright MIT http://rightsstatements.org/page/InC-EDU/1.0/ application/pdf Massachusetts Institute of Technology |
| spellingShingle | Chen, Yongyi Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions |
| title | Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions |
| title_full | Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions |
| title_fullStr | Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions |
| title_full_unstemmed | Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions |
| title_short | Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions |
| title_sort | self intersection of manin drinfeld cycles and taylor expansion of l functions |
| url | https://hdl.handle.net/1721.1/139475 |
| work_keys_str_mv | AT chenyongyi selfintersectionofmanindrinfeldcyclesandtaylorexpansionoflfunctions |