Mixed moments and the joint distribution of random groups

We study the joint distribution of random abelian and non-abelian groups. In the abelian case, we prove several universality results for the joint distribution of the multiple cokernels for random $p$-adic matrices. In the non-abelian case, we compute the joint distribution of random groups given by the quotients of the free profinite group by random relations. In both cases, we generalize the known results on the distribution of the cokernels of random $p$-adic matrices and random groups. Our proofs are based on the observation that mixed moments determine the joint distribution of random groups, which extends the works of Wood for abelian groups and Sawin for non-abelian groups.Comment: 27 page

Counting 33-dimensional algebraic tori over Q\mathbb{Q}

In this paper we count the number $N_3^{\text{tor}}(X)$ of $3$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor is bounded by $X$. We prove that $N_3^{\text{tor}}(X) \ll_{\varepsilon} X^{1 + \frac{\log 2 + \varepsilon}{\log \log X}}$, and this upper bound can be improved to $N_3^{\text{tor}}(X) \ll X (\log X)^4 \log \log X$ under the Cohen-Lenstra heuristics for $p=3$. We also prove that for $67$ out of $72$ conjugacy classes of finite nontrivial subgroups of $\operatorname{GL}_3(\mathbb{Z})$, Malle's conjecture for tori over $\mathbb{Q}$ holds up to a bounded power of $\log X$ under the Cohen-Lenstra heuristics for $p=3$ and Malle's conjecture for quartic $A_4$-fields.Comment: 33 pages, to appear in J. Number Theor

Joint distribution of the cokernels of random pp-adic matrices

In this paper, we study the joint distribution of the cokernels of random $p$-adic matrices. Let $p$ be a prime and $P_1(t), \cdots, P_l(t) \in \mathbb{Z}_p[t]$ be monic polynomials whose reductions modulo $p$ in $\mathbb{F}_p[t]$ are distinct and irreducible. We determine the limit of the joint distribution of the cokernels $\text{cok} (P_1(A)), \cdots, \text{cok}(P_l(A))$ for a random $n \times n$ matrix $A$ over $\mathbb{Z}_p$ with respect to Haar measure as $n \rightarrow \infty$. By applying the linearization of a random matrix model, we also provide a conjecture which generalizes this result. Finally, we provide a sufficient condition that the cokernels $\text{cok}(A)$ and $\text{cok}(A+B_n)$ become independent as $n \rightarrow \infty$, where $B_n$ is a fixed $n \times n$ matrix over $\mathbb{Z}_p$ for each $n$ and $A$ is a random $n \times n$ matrix over $\mathbb{Z}_p$.Comment: 18 page

On the lower bound of the number of abelian varieties over Fp\mathbb{F}_p

In this paper, we prove that the number $B(p,g)$ of isomorphism classes of abelian varieties over a prime field $\mathbb{F}_p$ of dimension $g$ has a lower bound $p^{\frac{1}{2} g^2 (1+o(1))}$ as $g \rightarrow \infty$. This is the first nontrivial result on the lower bound of $B(p,g)$. We also improve the upper bound $2^{34g^2} p^{\frac{69}{4} g^2 (1+o(1))}$ of $B(p,g)$ given by Lipnowski and Tsimerman (Duke Math. J. 167:3403-3453, 2018) to $p^{\frac{45}{4} g^2(1+o(1))}$.Comment: 20 pages, to appear in IMR