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Yevgeny Zaytman

Publications and source records attributed to Yevgeny Zaytman.

9 recordsLinked to original sources
arXiv2025

Isogeny graphs of superspecial abelian varieties and Brandt matrices

Fix primes $p$ and $\ell$ with $\ell\neq p$. If $(A,λ)$ is a $g$-dimensional principally polarized abelian variety, an $(\ell)^g$-isogeny of $(A,λ)$ has kernel a maximal isotropic subgroup of the $\ell$-torsion of $A$; the image has a natural principal polarization. We define three isogeny graphs associated to such $(\ell)^g$-isogenies -- the big isogeny graph $\mathit{Gr}_{\!g}(\ell,p)$, the little isogeny graph $\mathit{gr}_{\!g}(\ell,p)$, and the enhanced isogeny graph $\widetilde{\mathit{gr}}_{\!g}(\ell, p)$. We prove that all three isogeny graphs are connected. One ingredient of the proof is strong approximation for the quaternionic unitary group, which has previously been applied to moduli of abelian varieties in charateristic $p$ by Chai, Ekedahl/Oort, and Chai/Oort. The adjacency matrices of the three isogeny graphs are given in terms of the Brandt matrices defined by Hashimoto, Ibukiyama, Ihara, and Shimizu. We study some basic properties of these Brandt matrices and recast the theory using the notion of Brandt graphs. We show that the isogeny graphs $\mathit{Gr}_{\!g}(\ell, p)$ and $\mathit{gr}_{\!g}(\ell, p)$ are in fact our Brandt graphs. We give the $\ell$-adic uniformization of $\mathit{gr}_{\!g}(\ell,p)$ and $\widetilde{\mathit{gr}}_{\!g}(\ell,p)$. The $(\ell+1)$-regular isogeny graph $\mathit{Gr}_1(\ell,p)$ for supersingular elliptic curves is well known to be Ramanujan. We calculate the Brandt matrices for a range of $g>1$, $\ell$, and $p$. These calculations give four examples with $g>1$ where the regular graph $\mathit{Gr}_{\!g}(\ell,p)$ has two vertices and is Ramanujan, and all other examples we computed with $g>1$ and two or more vertices were not Ramanujan. In particular, the $(\ell)^g$-isogeny graph is not in general Ramanujan for $g>1$.

math.NT↗
arXiv2024

Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves

Sarnak's conjecture in quantum computing concerns when the groups $\operatorname{PU}_2$ and $\operatorname{PSU}_2$ over cyclotomic rings $\mathbb{Z}[ζ_n, 1/2]$ with $ζ_n=e^{2πi/n}$, $4|n$, are generated by the Clifford-cyclotomic gate set. We previously settled this using Euler-Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group $G$ has $\operatorname{corank} G >0$ only if $G$ is not generated by torsion elements. In this paper we study the corank of these cyclotomic unitary groups in the families $n=2^s$ and $n=3\cdot 2^s$, $n\geq 8$, by letting them act on Bruhat-Tits trees. The quotients by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for $n=2^s$ and $n=3\cdot 2^s$ the corank groups doubly exponentially in $s$ as $s\rightarrow \infty$; it is $0$ precisely when $n=8,12, 16,24$ and indeed the cyclotomic unitary groups are generated by torsion elements (in fact by the Clifford-cyclotomic gates) for these $n$. We give explicit lower bounds for the corank in two different ways. The first is to bound the isotropy subgroups in the action on the tree by explicit cyclotomy. The second is to relate our graphs to Shimura curves over $F_n=\mathbb{Q}(ζ_n)^+$ via interchanging local invariants and applying a result of Selberg and Zograf. We show that the cyclotomy arguments give the stronger bounds. In a final section we execute a program of Sarnak to show that our results for the $n=2^s$ and $n=3\cdot 2^s$ families are sufficient to give a second proof of Sarnak's conjecture.

math.NT↗
arXiv2022

Isogeny complexes of superspecial abelian varieties

We consider the structures formed by isogenies of abelian varieties with polarizations that are not necessarily principal, specifically with the $[\ell]$-polarizations we have previously defined. Our primary interest is in superspecial abelian varieties, where the isogenies are related to quaternionic hermitian forms. We first consider isogeny graphs. We show that these $[\ell]$-isogeny graphs are a generalized Brandt graph and construct them entirely in terms of definite quaternion algebras. We prove that they are connected and give examples to show that the regular graphs obtained are sometimes Ramanujan and sometimes not. Isogenies of $[\ell]$-polarized abelian varieties can be closed under composition, with the consequence that such isogenies naturally form semi-simplicial complexes as introduced by Eilenberg and Zilber in 1950 (later also called $Δ$-complexes) -- the higher-dimensional analogues of multigraphs. We show that these isogeny complexes can be constructed from the arithmetic of hermitian forms over definite quaternion algebras and that they are quotients of the Bruhat-Tits building of the symplectic group by the action of a quaternionic unitary group. Working with quaternions these isogeny graphs and complexes are amenable to machine computation and we include many examples, concluding with a detailed examination of the $[2]$-isogeny complexes of superspecial abelian surfaces in characteristic $7$.

math.NT↗
arXiv2021

The Zariski closure of integral points on varieties parametrizing periodic continued fractions

Let $R$ be the ring of $S$-integers in a number field $K$. Let $\mathcal{B}=\{β, β^{\ast}\}$ be the multi-set of roots of a nonzero quadratic polynomial over $R$. There are varieties $V(\mathcal{B})_{N,k}$ defined over $R$ parametrizing periodic continued fractions $[b_1,\ldots , b_N,\overline{a_1,\ldots ,a_k}]$ for $β$ or $β^{\ast}$. We study the $R$-points on these varieties, finding contrasting behavior according to whether groups of units are infinite or not. If $R$ is the rational integers or the ring of integers in an imaginary quadratic field, we prove that the $R$-points of $V(\mathcal{B})_{N,k}$ are not Zariski dense. On the other hand, suppose that $β\not\in K\cup\{\infty\}$, $R^\times$ is infinite, and that there are infinitely many units in the (left) order $R_β$ of $βR+R\subseteq K(β)$ with norm to $K$ equal to $(-1)^k$. Then we prove that the $R$-points on $V(\mathcal{B})_{1,k}$ are Zariski dense for $k\geq 8$ and the $R$-points on $V(\mathcal{B})_{0,k}$ are Zariski dense for $k\geq 9$. We also prove that $V(\mathcal{B})_{1,k}$ and $V(\mathcal{B})_{0,k}$ are $K$-rational irreducible varieties for $k$ sufficiently large.

math.NT↗
arXiv2021

On the bounded generation of arithmetic ${\rm SL}_2$

Let $K$ be a number field and ${\mathcal O}$ be the ring of $S$-integers in $K$. Morgan, Rapinchuck, and Sury have proved that if the group of units ${\mathcal O}^{\times}$ is infinite, then every matrix in ${\rm SL}_2({\mathcal O})$ is a product of at most $9$ elementary matrices. We prove that under the additional hypothesis that $K$ has at least one real embedding or $S$ contains a finite place we can get a product of at most $8$ elementary matrices. If we assume a suitable Generalized Riemann Hypothesis, then every matrix in ${\rm SL}_2({\mathcal O})$ is the product of at most $5$ elementary matrices if $K$ has at least one real embedding, the product of at most $6$ elementary matrices if $S$ contains a finite place, and the product of at most $7$ elementary matrices in general.

math.NT↗
arXiv2020

Quotient graphs and amalgam presentations for unitary groups over cyclotomic rings

Suppose $4|n$, $n\geq 8$, $F=F_n=\mathbb{Q}(ζ_n+\barζ_n)$, and there is one prime $\mathfrak{p}=\mathfrak{p}_n$ above $2$ in $F_n$. We study amalgam presentations for $\operatorname{PU_{2}}(\mathbb{Z}[ζ_n, 1/2])$ and $\operatorname{PSU_{2}}(\mathbb{Z}[ζ_n, 1/2])$ with the Clifford-cyclotomic group in quantum computing as a subgroup. These amalgams arise from an action of these groups on the Bruhat-Tits tree $Δ=Δ_{\mathfrak{p}}$ for $\operatorname{SL_{2}}(F_\mathfrak{p})$ constructed via the Hamilton quaternions. We explicitly compute the finite quotient graphs and the resulting amalgams for $8\leq n\leq 48$, $n\neq 44$, as well as for $\operatorname{PU_{2}}(\mathbb{Z}[ζ_{60}, 1/2])$.

math.NT↗
arXiv2019

The Clifford-cyclotomic group and Euler-Poincaré characteristics

For an integer $n\geq 8$ divisible by $4$, let $R_n=\mathbb{Z}[ζ_n,1/2]$ and let $\operatorname{U}_2(R_n)$ be the group of $2\times 2$ unitary matrices with entries in $R_n$. Set $\operatorname{U}_2^ζ(R_n)=\{γ\in\operatorname{U}_2(R_n)\mid \detγ\in\langleζ_n\rangle\}$. Let $\mathcal{G}_n\subseteq \operatorname{U}_2^ζ(R_n)$ be the Clifford-cyclotomic group generated by a Hadamard matrix $H=\frac{1}{2}[\begin{smallmatrix} 1+i & 1+i\\1+i &-1-i\end{smallmatrix}]$ and the gate $T=[\begin{smallmatrix}1 & 0\\0 & ζ_n\end{smallmatrix}]$. We prove that $\mathcal{G}_n=\operatorname{U}_2^ζ(R_n)$ if and only if $n=8, 12, 16, 24$ and that $[\operatorname{U}_2^ζ(R_n):\mathcal{G}_n]=\infty$ if $\operatorname{U}_2^ζ(R_n)\neq \mathcal{G}_n$. We compute the Euler-Poincaré characteristics of the groups $\operatorname{SU}_2(R_n)$, $\operatorname{PSU}_2(R_n)$, $\operatorname{PU}_2(R_n)$, $\operatorname{PU}^ζ_2(R_n)$, and $\operatorname{SO}_3(R_n^+)$.

math.NT↗
arXiv2019

Integral points on varieties defined by matrix factorization into elementary matrices

Let ${\mathcal O}$ be the ring of $S$-integers in a number field $K$. For $A\in\rm{SL}_{2}(\mathcal{O})$ and $k\geq 1$, we define matrix-factorization varieties $V_k(A)$ over ${\mathcal O}$ which parametrize factoring $A$ into a product of $k$ elementary matrices; the equations defining $V_k(A)$ are written in terms of Euler's continuant polynomials. We show that the $V_k(A)$ are rational $(k-3)$-folds with an inductive fibration structure. We combine this geometric structure with arithmetic results to study the Zariski closure of the ${\mathcal O}$-points of $V_k(A)$. We prove that for $k\geq 4$ the ${\mathcal O}$-points on $V_k(A)$ are Zariski dense if $V_{k}(A)({\mathcal O})\neq\emptyset$ assuming the group of units ${\mathcal O}^{\times}$ is infinite. This shows that if $A$ can be written as a product of $k\geq 4$ elementary matrices, then this can be done in infinitely many ways in the strongest sense possible. This can then be combined with results on factoring into elementary matrices for ${\rm SL}_{2}({\mathcal O})$. One result is that for $k\geq 9$ the ${\mathcal O}$-points on $V_{k}(A)$ are Zariski dense if ${\mathcal O}^{\times}$ is infinite.

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