A motivic Weil height machine for curves
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917127211974656 |
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| author | Betts, L. Alexander Dan-Cohen, Ishai |
| author_facet | Betts, L. Alexander Dan-Cohen, Ishai |
| contents | The rational points of a smooth curve $X$ over a number field $k$ map to the set of augmentations of the associated motivic algebra. An expectation, related to Kim's conjecture, is that for $X$ hyperbolic, the set of augmentations which come locally at each place of $k$ from a point is equal to the set of rational points. Our view is that this should provide a relative of the Grothendieck section conjecture which may be both more accessible, and more directly applicable, than the latter.
As a first step in this direction, we extend aspects of the ``Weil height machine'' to the set of such augmentations, and use this to prove a Manin--Dem'janenko-style finiteness result for motivic augmentations for particular curves. Along the way, we determine the structure of the cohomological motive of a $\mathbb{G}_m$-bundle over an algebraic variety as a highly structured algebra in the derived $\infty$-category of mixed motives with rational coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A motivic Weil height machine for curves Betts, L. Alexander Dan-Cohen, Ishai Algebraic Geometry Number Theory 14F42 (Primary) 11G50, 11D45 (Secondary) The rational points of a smooth curve $X$ over a number field $k$ map to the set of augmentations of the associated motivic algebra. An expectation, related to Kim's conjecture, is that for $X$ hyperbolic, the set of augmentations which come locally at each place of $k$ from a point is equal to the set of rational points. Our view is that this should provide a relative of the Grothendieck section conjecture which may be both more accessible, and more directly applicable, than the latter. As a first step in this direction, we extend aspects of the ``Weil height machine'' to the set of such augmentations, and use this to prove a Manin--Dem'janenko-style finiteness result for motivic augmentations for particular curves. Along the way, we determine the structure of the cohomological motive of a $\mathbb{G}_m$-bundle over an algebraic variety as a highly structured algebra in the derived $\infty$-category of mixed motives with rational coefficients. |
| title | A motivic Weil height machine for curves |
| topic | Algebraic Geometry Number Theory 14F42 (Primary) 11G50, 11D45 (Secondary) |
| url | https://arxiv.org/abs/2512.05284 |