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Abstract Let $$S$$ be a scheme and let $$\pi : \mathcal{G} \to S$$ be a $${\mathbb{G}}_{m,S}$$-gerbe corresponding to a torsion class $$[\mathcal{G}]$$ in the cohomological Brauer group $${\operatorname{Br}}^{\prime}(S)$$ of $$S$$. We show that the cohomological Brauer group $${\operatorname{Br}}^{\prime}(\mathcal{G})$$ of $$\mathcal{G}$$ is isomorphic to the quotient of $${\operatorname{Br}}^{\prime}(S)$$ by the subgroup generated by the class $$[\mathcal{G}]$$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes.
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This content will become publicly available on April 10, 2026
Counting integral points on symmetric varieties with applications to arithmetic statistics
Abstract 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 . 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.
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- Award ID(s):
- 1926686
- PAR ID:
- 10616120
- Publisher / Repository:
- London Mathematical Society
- Date Published:
- Journal Name:
- Proceedings of the London Mathematical Society
- Volume:
- 130
- Issue:
- 4
- ISSN:
- 0024-6115
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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