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Artane Siad

Publications and source records attributed to Artane Siad.

7 recordsLinked to original sources
arXiv2026

Monogenicity and 2-torsion in the class group of number fields of odd degree

We study the average $2$-torsion in the class group of monogenised fields of odd degree. Bhargava--Hanke--Shankar have recently shown that for a fixed signature, the average number of non-trivial $2$-torsion elements in the class group of monogenised cubic fields is exactly twice the value predicted by the Cohen--Lenstra--Martinet--Malle heuristic over the full $S_3$ family. For any odd degree $n \ge 3$ and signature, we prove that the average number of non-trivial $2$-torsion elements in the class group of monogenised fields is at most twice the value predicted by the Cohen--Lenstra--Martinet--Malle heuristic over the full $S_n$ family. Conditional on a tail estimate for $n \ge 5$, this establishes that the doubling phenomenon discovered by Bhargava--Hanke--Shankar persists across all odd degrees and signatures.

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arXiv2025

Hecke reciprocity and class groups

We compute the average size of $\mathrm{Cl}_F[2]$ in the family of cubic fields $F = \mathbb{Q}(\sqrt[3]{n})$. Specifically, as $F$ varies over the subfamily of wildly (resp. tamely) ramified fields $\mathbb{Q}(\sqrt[3]{n})$, the average size of $\mathrm{Cl}_F[2]$ is $3/2$ (resp. $2$). This tame/wild dichotomy is not accounted for by the class group heuristics in the literature. Analogously, when the extensions $F = K(\sqrt[3]{n})$ of $K = \mathbb{Q}(\sqrt{-3})$ are ordered by the norm of $n \in \mathcal{O}_K$, we show that the average size of $\mathrm{Cl}_F[2]$ is $3/2$, as is predicted by the Cohen--Martinet heuristics for $C_3$-extensions of $K$. Underlying our proofs is a reciprocity law for the relative class groups $\mathrm{Cl}_{F/K}[2]$ of odd degree extensions of number fields $F/K$. This leads us to propose class group heuristics for families of $K$-extensions with a fixed Galois $K$-group that explains the aberrant behavior in the family $\mathbb{Q}(\sqrt[3]{n})$ and predicts similar behavior in other special families. The other main ingredient is the work of Alpöge--Bhargava--Shnidman on the number of integral $G(\mathbb{Q})$-orbits in a $G$-invariant quadric with bounded invariants.

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arXiv2025

Counting integral points on symmetric varieties with applications to arithmetic statistics

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.

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arXiv2025

Entropy of Cohen-Lenstra measures: the $u$-aspect

Let ${\rm \mathbf{H}}(ν^{u}_{\rm CL})$ be the entropy of the Cohen-Lenstra measure on finite abelian $p$-groups associated to an integral unit-rank $0 \le u \in \mathbb{N}$. In this note, we show that $0 < {\rm \mathbf{H}}(ν^{u}_{\rm CL}) < \infty$ for all $u$, ${\rm \mathbf{H}}(ν^{u}_{\rm CL})$ is a strictly decreasing function of $u \ge 0$, and ${\rm \mathbf{H}}(ν^{u}_{\rm CL}) \xrightarrow{u \to \infty} 0$. In particular, this shows that the groupoid measure is an entropy maximizer in the class of Cohen-Lenstra measures of varying integral unit-rank on finite abelian $p$-groups.

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arXiv2024

Geometry-of-numbers methods in the cusp

In this article, we develop new methods for counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We illustrate these methods for a representation of cardinal interest in number theory, namely that of the split orthogonal group acting on the space of quadratic forms.

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arXiv2020

Monogenic fields with odd class number Part II: even degree

In 1801, Gauss proved that there were infinitely many quadratic fields with odd class number. We generalise this result by showing that there are infinitely many $S_n$-fields of any given even degree and signature that have odd class number. Also, we prove that there are infinitely many fields of any even degree at least $4$ and with at least one real embedding that have units of every signature. To do so, we bound the average number of $2$-torsion elements in the class group, narrow class group, and oriented class group of monogenised fields of even degree (and compute these averages precisely conditional on a tail estimate) using a parametrisation of Wood. These averages are the first $p$-torsion averages to be calculated for $p$ not coprime to the degree (in degree at least $3$), shedding light on the question of Cohen-Lenstra-Martinet-Malle type heuristics for class groups and narrow class groups at "bad" primes.

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