arXiv subjects

Alexey G. Gorinov

Publications and source records attributed to Alexey G. Gorinov.

5 recordsLinked to original sources

arXiv2026

Combinatorics of double cosets and fundamental domains for the subgroups of the modular group

As noticed by R.~Kulkarni, the conjugacy classes of subgroups of the modular group correspond bijectively to bipartite cuboid graphs. We'll explain how to recover the graph corresponding to a subgroup $G$ of $\mathrm{PSL}_2(\mathbb{Z})$ from the combinatorics of the right action of $\mathrm{PSL}_2(\mathbb{Z})$ on the right cosets $G\backslash\mathrm{PSL}_2(\mathbb{Z})$. This gives a method of constructing nice fundamental domains (which Kulkarni calls "special polygons") for the action of $G$ on the upper half plane. For the classical congruence subgroups $Γ_0(N)$, $Γ_1(N)$, $Γ(N)$ etc. the number of operations the method requires is the index times something that grows not faster than a polynomial in $\log N$. This is roughly the square root of the number of operations required by the naive procedure. We give algorithms to locate an element of the upper half-plane on the fundamental domain and to write a given element of $G$ as a product of independent generators. We also (re)prove a few related results about the automorphism groups of modular curves. For example, we give a simple proof that the automorphism group of $X(N)$ is $\mathrm{SL}_2(\mathbb{Z}/N)/\{\pm I\}$.

math.NT↗

arXiv2026

Affine models for Noetherian schemes

Let $S$ be a base scheme, assumed separated and Noetherian. We define \emph{adequate classes} of morphisms of $S$-schemes by formalizing certain properties of homotopy equivalences of complex algebraic varieties. Other examples of adequate classes include morphisms of varieties over an algebraically closed field of positive characteristic which induce an isomorphism of étale cohomology, and weak equivalences in the cdh topology. An \emph{affine model} for a Noetherian scheme $X$ over $S$ is an affine scheme $M_X$ equipped with an adequate morphism $M_X\to X$. In this paper we construct affine models for arbitrary separated schemes of finite type over $S$. Our construction can be viewed as a generalization of the Jouanolou trick. As an application, we construct a mixed Hodge structure on the Leray spectral sequence of an arbitrary proper morphism $f:X\to Y$ of complex algebraic varieties, generalizing an argument by Donu Arapura which assumed $Y$ quasi-projective and $f$ projective.

math.AG↗

arXiv2026

Binomial rings, and integral homology of complements of compact toric arrangements

An \emph{affine subtorus} of the compact torus $T=(S^1)^n$ is a translated copy of a Lie subgroup. Given a finite collection $T_1,\ldots, T_k$ of such subtori, and a prime $p$, we describe an explicit chain complex that calculates the group $H_*(T-\bigcup_{i=1}^k T_i,\mathbb{Z}_{(p)})$. %The complex is determined by the integral homology maps induced by the inclusions $T_J\subset T_I$ where $I\subset J\subset\{1,\ldots, k\}$ and $T_I$ denotes $\bigcap_{i\in I} T_i$. Our main tool is the binomial models for spaces constructed by T.~Ekedahl. We use these results to express the groups $H_*(T-\bigcup_{i=1}^k T_i,\mathbb{Z})$. We also show that the Mayer-Vietoris spectral sequence that converges to the homology of $T-\bigcup_{i=1}^k T_i$ collapses at the second page rationally, and also integrally under some assumptions on the arrangement $T_1,\ldots, T_k$, with all extension problems being trivial in the latter case.

math.AT↗

arXiv2013

Rational cohomology of the moduli spaces of pointed genus 1 curves

We give a combinatorial description of the rational cohomology of the moduli spaces of pointed genus 1 curves with $n$ marked points and level $N$ structures. More precisely, we explicitly describe the $E_2$ term of the Leray spectral sequence of the forgetful mapping $\EuScript{M}_{1,n}(N)\to\EuScript{M}_{1,1}(N)$ and show that the result is isomorphic to the rational cohomology of $\EuScript{M}_{1,n}(N)$ as a rational mixed Hodge structure equipped with an action of the symmetric group $\mathfrak{S}_n$. The classical moduli space $\EuScript{M}_{1,n}$ is the particular case N=1.

math.AG↗

arXiv2003

Cesaro asymptotics for the orders of SL_k(Z_n)$ and GL_k(Z_n) as n -> infinity

Given an integer k>0, our main result states that the sequence of orders of the groups SL_k(\Z_n) (respectively, of the groups GL_k(Z_n)) is Cesaro equivalent as n -> infinity to the sequence C_1(k) n^{k^2-1} (respectively, C_2(k)n^{k^2}), where the coefficients C_1(k) and C_2(k) depend only on k; we give explicit formulas for C_1(k) and C_2(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function is Cesaro equivalent to n * 6/pi^2. We present some experimental facts related to the main result.

math.CO↗