Counting integral points on symmetric varieties with applications to arithmetic statistics
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916683964219392 |
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| author | Shankar, Arul Siad, Artane Swaminathan, Ashvin A. |
| author_facet | Shankar, Arul Siad, Artane Swaminathan, Ashvin A. |
| contents | In this article, we combine Bhargava's geometry-of-numbers methods with the dynamical point-counting methods of Eskin--McMullen and Benoist--Oh to develop a new technique for counting integral points on symmetric varieties lying within fundamental domains for coregular representations. As applications, we study the distribution of the $2$-torsion subgroup of the class group in thin families of cubic number fields, as well as the distribution of the $2$-Selmer groups in thin families of elliptic curves over $\mathbb{Q}$. For example, our results suggest that the existence of a generator of the ring of integers with small norm has an increasing effect on the average size of the $2$-torsion subgroup of the class group, relative to the Cohen--Lenstra predictions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01050 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Counting integral points on symmetric varieties with applications to arithmetic statistics Shankar, Arul Siad, Artane Swaminathan, Ashvin A. Number Theory Dynamical Systems In this article, we combine Bhargava's geometry-of-numbers methods with the dynamical point-counting methods of Eskin--McMullen and Benoist--Oh to develop a new technique for counting integral points on symmetric varieties lying within fundamental domains for coregular representations. As applications, we study the distribution of the $2$-torsion subgroup of the class group in thin families of cubic number fields, as well as the distribution of the $2$-Selmer groups in thin families of elliptic curves over $\mathbb{Q}$. For example, our results suggest that the existence of a generator of the ring of integers with small norm has an increasing effect on the average size of the $2$-torsion subgroup of the class group, relative to the Cohen--Lenstra predictions. |
| title | Counting integral points on symmetric varieties with applications to arithmetic statistics |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2304.01050 |