Linear relations of p-adic periods of 1-motives (thesis)
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913623232741376 |
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| author | Mohajer, Mohammadreza |
| author_facet | Mohajer, Mohammadreza |
| contents | In this thesis, we aim to develop p-adic analogs of known results for classical periods, focusing specifically on 1-motives. We establish an integration theory for 1-motives with good reductions, which generalizes the Colmez-Fontaine-Messing p-adic integration for abelian varieties with good reductions. We also compare the integration pairing with other pairings such as those induced by crystalline theory. Additionally, we introduce a formalism for periods and formulate p-adic period conjectures related to p-adic periods arising from this integration pairing. Broadly, our p-adic period conjecture operates at different depths, with each depth revealing distinct relations among the p-adic periods. Notably, the classical period conjecture (Kontsevich-Zanier conjecture over $\bar{\mathbb{Q}}$) for 1-periods fits within our framework, and, according to the classical subgroup theorem of Huber-Wüstholz for 1-motives, the conjecture for classical periods of 1-motives holds true at depth 1. Finally, we identify three $\mathbb{Q}$-structures arising from $\bar{\mathbb{Q}}$-rational points of the formal p-divisible group associated with a 1-motive $M$ with a good reduction at $p$, and we prove p-adic period conjectures at depths 2 and 1, relative to periods induced by the p-adic integration of $M$ and these $\mathbb{Q}$-structures. Our proof involves a p-adic version of the subgroup theorem that we obtain for 1-motives with good reductions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03118 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear relations of p-adic periods of 1-motives (thesis) Mohajer, Mohammadreza Number Theory Algebraic Geometry 14C30, 11S80, 14F30, 32G20, 14Kxx, 14K20, 14L05, 11G25, 14F42, 14C15 In this thesis, we aim to develop p-adic analogs of known results for classical periods, focusing specifically on 1-motives. We establish an integration theory for 1-motives with good reductions, which generalizes the Colmez-Fontaine-Messing p-adic integration for abelian varieties with good reductions. We also compare the integration pairing with other pairings such as those induced by crystalline theory. Additionally, we introduce a formalism for periods and formulate p-adic period conjectures related to p-adic periods arising from this integration pairing. Broadly, our p-adic period conjecture operates at different depths, with each depth revealing distinct relations among the p-adic periods. Notably, the classical period conjecture (Kontsevich-Zanier conjecture over $\bar{\mathbb{Q}}$) for 1-periods fits within our framework, and, according to the classical subgroup theorem of Huber-Wüstholz for 1-motives, the conjecture for classical periods of 1-motives holds true at depth 1. Finally, we identify three $\mathbb{Q}$-structures arising from $\bar{\mathbb{Q}}$-rational points of the formal p-divisible group associated with a 1-motive $M$ with a good reduction at $p$, and we prove p-adic period conjectures at depths 2 and 1, relative to periods induced by the p-adic integration of $M$ and these $\mathbb{Q}$-structures. Our proof involves a p-adic version of the subgroup theorem that we obtain for 1-motives with good reductions. |
| title | Linear relations of p-adic periods of 1-motives (thesis) |
| topic | Number Theory Algebraic Geometry 14C30, 11S80, 14F30, 32G20, 14Kxx, 14K20, 14L05, 11G25, 14F42, 14C15 |
| url | https://arxiv.org/abs/2411.03118 |