arXiv subjects
Publications and source records attributed to Igor Nikolaev.
We study geometry of the ring of integers $O_K$ of a number field $K$. Namely, it is proved that the inclusion $\mathbf{Z}\subset O_K$ defines a covering of the Riemann sphere $\mathbf{C}P^1$ ramified over the points $\{0,1,\infty\}$. Our approach is based on the notion of a Serre $C^*$-algebra. As an application, a new proof of the Belyi Theorem is given.
We recover the Jones polynomials of knots and links from the K-theory of a cluster C*-algebra of the sphere with two cusps. In particular, an interplay between the Chebyshev and Jones polynomials is studied.
We prove an analog of Belyi's theorem for the algebraic surfaces. Namely, any non-singular algebraic surface can be defined over a number field if and only it covers the complex projective plane with ramification at three knotted two-dimensional spheres.
We recover the Dehornoy order on the braid group $B_{2g+n}$ from the tracial state on a cluster $C^*$-algebra $\mathbb{A}(S_{g,n})$ associated to the surface $S_{g,n}$ of genus $g$ with $n$ boundary components. It is proved that the space of left-ordering of the fundamental group $\pi_1(S_{g,n})$ is a totally disconnected dense subspace of the projective Teichm\"uller space $\mathbb{P}T_{g,n}\cong \mathbf{R}^{6g-7+2n}$. In particular, each left-ordering of $\pi_1(S_{g,n})$ defines the orbit of a Riemann surface $S_{g,n}$ under the geodesic flow on the space $T_{g,n}$.
We study non-commutative algebraic geometry of Artin, Serre and Tate in terms of the operator algebras. Namely, we define the Serre $C^*$-algebra $\mathcal{A}_X$ of a projective variety $X$ as the norm-closure of a representation of the twisted homogeneous coordinate ring of $X$ by the linear operators on a Hilbert space $\mathcal{H}$. It is proved that $X$ is homeomorphic to the space of all irreducible representations of the crossed product of $\mathcal{A}_X$ by an automorphism of $\mathcal{A}_X$. The case of rational elliptic curves $X$ is considered in detail.
This note is the follow up to a paper by M. Waldschmidt.
We study dynamics of the Lattès maps in the complex plane in terms of the Cuntz-Krieger algebras associated to the endomorphisms of the non-commutative tori. In particular, it is shown that iterations of the Lattès maps can be reduced to the dynamics of the subshifts of finite type. Using such a reduction, we calculate the zeta function of the Lattès maps.
We prove Faltings Finiteness Theorem using Rieffel's classification of the noncommutative tori.
We construct a functor from the smooth 4-dimensional manifolds to the hyper-algebraic number fields, i.e. fields with non-commutative multiplication. It is proved that that the simply connected 4-manifolds correspond to the abelian extensions. We recover the Rokhlin and Donaldson's Theorems from the Galois theory of the non-commutative fields.
We recast elliptic surfaces over the projective line in terms of the non-commutative tori and one-parameter families of the periodic continued fractions. The correspondence is used to study the Picard numbers, the ranks and the minimal models of such surfaces. As an example, we calculate the Picard numbers of elliptic surfaces with fibers having complex multiplication.
It is proved that the Wedderburn Theorem on finite division rings implies that all knots and links in the smooth 4-dimensional manifolds are trivial.
We construct a functor from the category of elliptic curves to a category of noncommutative tori. Our proof is based on an isomorphism between the Sklyanin algebras and dense sub-algebras of the noncommutative tori.
This is a credit mini-course in French prepared for a Summer School at the University of Sherbrooke. The course consists of three one-and-half hour lectures and three credit exercises for a class of advanced graduate students.
It is proved that the profinite completion of the mapping class group Mod (g,n) of a surface of genus g with n boundary components is isomorphic to such of the arithmetic group GL(6g-6+2n, Z). We establish a relation between the normal subgroups of Mod (g,n) and the absolute Galois group G(K) of a number field K. Using the Tits alternative, we prove the Shafarevich Conjecture saying that the group G(Q^ab) of the maximal abelian extension of the field of rationals is isomorphic to a free profinite group.
We introduce a $C^*$-algebra $\mathscr{A}_V$ of a variety $V$ over the number field $K$ and a $C^*$-algebra $\mathscr{A}_G$ of a reductive group $G$ over the ring of adeles of $K$. Using Pimsner's Theorem we construct an embedding $\mathscr{A}_V\hookrightarrow \mathscr{A}_G$, where $V$ is a $G$-coherent variety, e.g. the Shimura variety of $G$. The embedding is an analog of the Langlands reciprocity for $C^*$-algebras. It follows from the $K$-theory of the inclusion $\mathscr{A}_V\subset\mathscr{A}_G$ that the Hasse-Weil $L$-function of $V$ is a product of the automorphic $L$-functions corresponding to irreducible representations of the group $G$.
The arithmetic complexity counts the number of algebraically independent entries in the periodic continued fraction $\theta=[b_1,\dots, b_N, \overline{a_1,\dots,a_k}]$. If $\mathscr{A}_{\theta}$ is a noncommutative torus corresponding to the rational elliptic curve $\mathscr{E}(K)$, then the rank of $\mathscr{E}(K)$ is given by a simple formula $r(\mathscr{E}(K))= c(\mathscr{A}_{\theta})-1$, where $c(\mathscr{A}_{\theta})$ is the arithmetic complexity of $\theta$. We prove that $c(\mathscr{A}_{\theta})$ is equal to the dimension of the Brock-Elkies-Jordan variety of $\theta$ introduced in [1]. Following Zagier and Lemmermeyer, we evaluate the Shafarevich-Tate group of $\mathscr{E}(K)$.
A dynamical analog of the prime ideals for simple non-commutative rings is introduced. We prove a factorization theorem for the dynamical ideals. The result is used to classify the surface knots and links in the smooth 4-dimensional manifolds.
We compute the genus of a rational quadratic form in terms of the K-theory of a C*-algebra attached to the adelic orthogonal group of the form. As a corollary, one gets a higher composition law for the rational quadratic forms. As an illustration, we consider the Gauss composition of the binary quadratic forms.