Searcharxiv⌕ Search

arXiv subjects

Ivan Horozov

Publications and source records attributed to Ivan Horozov.

At least 19 recordsLinked to original sources
arXiv2026

Euler Characteristics of a Family of Congruence Subgroups of $GL_m(\Z)$

The congruence subgroups $\Gamma_1(m,p)$ that we consider here are subgroups of $GL_m(\Z)$ that fix the vector $(0,\dots,0,1) \mod p$, where $p\geq 5$ is a prime. We present a method and many computations of homological Euler characteristics of $GL_m(\Z)$ and $\Gamma_1(m,p)$ with coefficients in any highest weight representation $V$. By homological Euler characteristics we mean the alternating dimensions of cohomology of the group with coefficient in $V$. We compute the homological Euler characteristics for $\Gamma_1(2,p)$, and $\Gamma_1(3,p)$ with coefficients in any finite dimensional highest weight representation. Also we compute the homological Euler characteristics for of $\Gamma_1(4,p)$ and $\Gamma_1(5,p)$ with coefficients in the trivial and the determinant representations. We give application to cohomology of $\Gamma_1(3,p)$ with trivial and with determinant representation. We also give an alternative method for computing the cohomology of $GL_4(\Z)$ compared to \cite{GL4}. The methods in this paper are a continuation of result from \cite{Thesis, EulerChar}.

math.NT↗
arXiv2022

Distribution of primes represented by polynomials and Multiple Dedekind zeta functions

n this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for any irreducible polynomial $f$. Here, we formulate a conjecture for all irreducible polynomials. We also consider conjectures for distribution of pairs of primes. It show unexpected relation to multiple Dedekind zeta function - at $s=2$ for one prime and at $(s_1,s_2)=(2,2)$ for pairs of primes. We tested the conjecture for pairs of primes for several quadratic fields. The conjecture for pairs of primes and multiple Dedekind zeta function over the Gaussian integers provide error less than a tenth of a percent. We also tested conjectures that compare sets of primes in a pair of different quadratic fields. Numerically, such quotients can be expressed in terms of regulators and class numbers. Some of the data, together with the code, is available on GitHub, (see \cite{Zouberou}).

math.NT↗
arXiv2022

Cohomology of SL(3,Z) with coefficients in the standard representation

This paper is a natural continuation of a joint paper with Bajpai, Harder and Moya Giusti \cite{BHHM}, even though it began as an answer to Goncharov's question. It that paper, we had complete description for all representations except for odd symmetric powers and their dual ones. For those representations we were left with two options: certain one dimensional module is a ghost space or not. Here we find the $H^2(SL_3(\Z),V_3)$ has ghost classes. It means that it is generated by a class from the cohomology of the Borel subgroup. With the techniques developed here, we show that the $d_2$ map of the spectral sequence for the boundary cohomology of $GL_4(\Z)$ is non-trivial if and only if there is a ghost class in $GL_3(\Z)$ (see Propositions 11 and 12.) We use a result of Elbaz-Vincent, Gangl and Soule to show that a spectral sequence related to $GL_4(\Z)$ does not degenerate at $E_2$-level. Then $d_2$ is non-trivial. Therefore, we obtain that $H^2(SL_3(\Z),V_3))$ is a ghost space, where $V_3$ is the standard representation.

math.NT↗
arXiv2021

Euler characteristic and cohomology of $\mathrm{Sp}_4(\mathbb{Z})$ with nontrivial coefficients

In this article, the cohomology of the arithmetic group $\mathrm{Sp}_4(\mathbb{Z})$ with coefficients in any finite dimensional highest weight representation $\mathcal{M}_λ$ have been studied. Euler characteristic with coefficients in $\mathcal{M}_λ$ have been carried out in detail. Combining the results obtained on Euler characteristic and the work of Harder on Eisenstein cohomology, the description of the cuspidal cohomology has been achieved. At the end, we employ our study to compute the dimensions for the cohomology spaces $H^{\bullet}(\mathrm{Sp}_4(\mathbb{Z}), \mathcal{M}_λ)$.

math.NT↗
arXiv2020

Boundary and Eisenstein Cohomology of $\mathrm{SL}_3(\mathbb{Z})$

In this article, several cohomology spaces associated to the arithmetic groups $\mathrm{SL}_3(\mathbb{Z})$ and $\mathrm{GL}_3(\mathbb{Z})$ with coefficients in any highest weight representation $\mathcal{M}_λ$ have been computed, where $λ$ denotes their highest weight. Consequently, we obtain detailed information of their Eisenstein cohomology with coefficients in $\mathcal{M}_λ$. When $\mathcal{M}_λ$ is not self dual, the Eisenstein cohomology coincides with the cohomology of the underlying arithmetic group with coefficients in $\mathcal{M}_λ$. In particular, for such a large class of representations we can explicitly describe the cohomology of these two arithmetic groups. We accomplish this by studying the cohomology of the boundary of the Borel-Serre compactification and their Euler characteristic with coefficients in $\mathcal{M}_λ$. At the end, we employ our study to discuss the existence of ghost classes.

math.NT↗
arXiv2018

Periods of Mixed Tate Motives over Real Quadratic Number Rings

Recently, the author defined multiple Dedekind zeta values \cite{MDZF} associated to a number $K$ field and a cone $C$. In this paper we construct explicitly non-trivial examples of mixed Tate motives over the ring of integers in $K$, for a real quadratic number field $K$ and a particular cone C. The period of such a motive is a multiple Dedekind zeta values at $(s_1,s_2)=(1,2)$, associated to the pair $(K;C)$, times a nonzero element of $K$.

math.AG↗
arXiv2018

Multiple Dedekind Zeta Values are Periods of Mixed Tate Motives

Recently, the author defined multiple Dedekind zeta values [5] associated to a number K field and a cone C. These objects are number theoretic analogues of multiple zeta values. In this paper we prove that every multiple Dedekind zeta value over any number field K is a period of a mixed Tate motive. Moreover, if K is a totally real number field, then we can choose a cone C so that every multiple Dedekind zeta associated to the pair (K;C) is unramified over the ring of algebraic integers in K. In [7], the author proves similar statements in the special case of a real quadratic fields for a particular type of a multiple Dedekind zeta values. The mixed motives are defined over K in terms of a the Deligne-Mumford compactification of the moduli space of curves of genus zero with n marked points.

math.AG↗
arXiv2018

Noncommutative modular symbols and Eisenstein series

We form real-analytic Eisenstein series twisted by Manin's noncommutative modular symbols. After developing their basic properties, these series are shown to have meromorphic continuations to the entire complex plane and satisfy functional equations in some cases. This theory neatly contains and generalizes earlier work in the literature on the properties of Eisenstein series twisted by classical modular symbols.

math.NT↗
arXiv2017

Iterated Integrals and higher order invariants

We deduce from the work of Chen, that the restriction morphism from closed free iterated integrals to closed iterated integrals on loops is onto. We use this to show that the module of higher order invariants of smooth functions is generated by free closed iterated integrals.

math.DG↗
arXiv2015

Shuffle Product for Multiple Dedekind Zeta Values over Imaginary Quadratic Fields

Multiple Dedekind zeta values were recently defined by the second author. In a separate paper, the second author constructed double shuffle relations in some cases as a response to questions asked by Richard Hain and Alexander Goncharov. In this paper, we develop the technique for obtaining more shuffle relations and produce many examples of shuffle products over an imaginary quadratic field. We also define the notion of self shuffle of a (multiple) Dedekind zeta value and use it at many instances. We define a refinement of the multiple Dedekind zeta values. Our key examples are self shuffles of the Dedekind zeta at 2 and at 3, the shuffle product of the Dedekind zeta of 2 times itself, and the shuffle product of the Dedekind zeta at 2 times the Dedekind zeta at 3. We obtain one unexpected result that the self shuffle of multiple Dedekind zeta at (1,2) minus the self shuffle of the twisted (with a permutation) multiple Dedekind zeta value at (1,2) is a very simple expression in terms of the refined multiple Dedekind zeta values.

math.NT↗
arXiv2014

Multiple Zeta Values and Ideles

In this paper we give two idelic representations of the multiple zeta values - one using iterated integrals over the finite ideles and the other using iterated integrals over the idele class group. Each of the representations leads to a shuffle relation. Thus, we recover in a unified way the two types of shuffle relations of multiple zeta values via the iterated integrals over finite ideles and via iterated integrals over the idele class group.

math.NT↗
arXiv2014

Non-commutative Hilbert modular symbols

The main goal of this paper is to construct non-commutative Hilbert modular symbols. However, we also construct commutative Hilbert modular symbols. Both the commutative and the non-commutative Hilbert modular symbols are generalizations of Manin's classical and non-commutative modular symbols. We prove that many cases of (non-)commutative Hilbert modular symbols are periods in the sense on Kontsevich-Zagier. Hecke operators act naturally on them. Manin defines the non-commutative modilar symbol in terms of iterated path integrals. In order to define non-commutative Hilbert modular symbols, we use a generalization of iterated path integrals to higher dimensions, which we call iterated integrals on membranes. Manin examines similarities between non-commutative modular symbol and multiple zeta values both in terms of infinite series and in terms of iterated path integrals. Here we examine similarities in the formulas for non-commutative Hilbert modular symbol and multiple Dedekind zeta values, recently defined by the author, both in terms of infinite series and in terms of iterated integrals on membranes.

math.NT↗
arXiv2013

Double Shuffle Relations for Multiple Dedekind Zeta Values

This paper contains examples of shuffle relations among multiple Dedekind zeta values. Dedekind zeta values were defined by the author in his paper "Multiple Dedekind zeta functions". Here we concentrate on the cases of real or imaginary quadratic fields with up to double iteration. We give examples of integral shuffle relation in terms of iterated integrals over membranes and of infinite sum shuffle relation, sometimes called stuffle relation. Using both types of shuffles for the product $ζ_{K,C}(2)ζ_{K,C}(2)$, we find relations among multiple Dedekind zeta values for real and for imaginary quadratic fields.

math.NT↗
arXiv2013

On the Contou-Carrere Symbol for Surfaces

This is a preliminary report on the Contou-Carrere symbol for surfaces. It consists of two parts. In the first part, we recall technical results needed to define the symbol. The second part is where we compute all components of the Coutou-Carrere symbol for surfaces, using iterated integrals over membranes.

math.AG↗
arXiv2013

Reciprocity Laws on Algebraic Surfaces via Iterated Integrals

This paper presents a proof of reciprocity laws for the Parshin symbol and for two new local symbols, defined here, which we call 4-function local symbols. The reciprocity laws for the Parshin symbol are proven using a new method - via iterated integrals. The usefulness of this method is shown by two facts - first, by establishing new local symbols - the 4-function local symbols and their reciprocity laws and, second, by providing refinements of the Parshin symbol in terms of bi-local symbols, each of which satisfies a reciprocity law. The K-theoretic variant of the first 4-function local symbol is defined in the Appendix. It differs by a sign from the one defined via iterated integrals. Both the sign and the K-theoretic variant of the 4-function local symbol satisfy reciprocity laws.

math.AG↗
arXiv2013

Multiple Dedekind Zeta Functions

In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation, can be written as infinite sum, and have analytic continuation. We compute explicitly the value of a multiple residue of certain MDZF over a quadratic number field at the point (1,1,1,1). Based on such computations, we state two conjectures about MDZV.

math.NT↗
arXiv2012

Parallel Transport on Higher Loop Spaces

We construct a parallel transport on higher loop spaces of a manifold in term of a higher dimensional generalization of iterated path integrals. Under mild assumptions, we define a de Rham complex on higher loop spaces and we recover a known result of Hain of a de Rham structure on higher homotopy groups of a manifold. The key ingredient is a new definition of iterated integrals on membranes, which also have applications in number theory, algebraic geometry and mathematical physics.

math.AT↗