Quadrics in arithmetic statistics
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866914970708475904 |
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| author | Alpöge, Levent |
| author_facet | Alpöge, Levent |
| contents | We (re)introduce the circle method into arithmetic statistics.
More specifically, we combine the circle method with Bhargava's counting technique in order to give a general method that allows one to treat arithmetic statistical problems in which one is trying to count orbits on a subvariety of affine space defined by the vanishing of a quadratic invariant.
We explain this method by way of example by computing the average size of $2$-Selmer groups in the families $y^2 = x^3 + B$ and $y^2 = x^3 + B^2$.
In the course of the argument we introduce a smoothed form of Bhargava's aforementioned method, as well as a trick with which we formally deduce that the above averages are $3$ from knowledge of the averages over "unconstrained" families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_03947 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quadrics in arithmetic statistics Alpöge, Levent Number Theory Algebraic Geometry We (re)introduce the circle method into arithmetic statistics. More specifically, we combine the circle method with Bhargava's counting technique in order to give a general method that allows one to treat arithmetic statistical problems in which one is trying to count orbits on a subvariety of affine space defined by the vanishing of a quadratic invariant. We explain this method by way of example by computing the average size of $2$-Selmer groups in the families $y^2 = x^3 + B$ and $y^2 = x^3 + B^2$. In the course of the argument we introduce a smoothed form of Bhargava's aforementioned method, as well as a trick with which we formally deduce that the above averages are $3$ from knowledge of the averages over "unconstrained" families. |
| title | Quadrics in arithmetic statistics |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2110.03947 |