G-functions, motives, and unlikely intersections -- old and new
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912191982075904 |
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| author | André, Yves |
| author_facet | André, Yves |
| contents | In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more ``motivic" symmetries; and how Bombieri's principle of global relations can be used to control the height of such parameters (which was also one of the origins of the André-Oort conjecture). At the end, we sketch the recent revival of the G-function method in the context of unlikely intersections and the Zilber-Pink conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | G-functions, motives, and unlikely intersections -- old and new André, Yves Number Theory Algebraic Geometry 11G, 11J, 14K In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more ``motivic" symmetries; and how Bombieri's principle of global relations can be used to control the height of such parameters (which was also one of the origins of the André-Oort conjecture). At the end, we sketch the recent revival of the G-function method in the context of unlikely intersections and the Zilber-Pink conjecture. |
| title | G-functions, motives, and unlikely intersections -- old and new |
| topic | Number Theory Algebraic Geometry 11G, 11J, 14K |
| url | https://arxiv.org/abs/2501.09867 |