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Igor V. Nikolaev

Publications and source records attributed to Igor V. Nikolaev.

At least 19 recordsLinked to original sources
arXiv2026

Number fields as curves over $\mathbf{F}_1$

We study function fields in one variable over the field with one element $\mathbf{F}_1$. It is proved that the Galois extensions of $\mathbf{Q}$ are isomorphic to the curves over $\mathbf{F}_1$ being understood as the Deitmar schemes. Specifically, one gets explicit formulas linking the genus and the number of cusps of an algebraic curve over the extension $\mathbf{F}_{1^m}$ of order $m\ge 1$ of the field $\mathbf{F}_1$ and the $m$-th roots of unity of the corresponding number field. It follows that elliptic curves with one cusp over $\mathbf{F}_1$ are either the cyclotomic fields or the maximal abelian unramified extensions of the quadratic number fields. Our proof depends on representaion of the Drinfeld modules by the bounded linear operators on a Hilbert space and the crossed product structure of the Cuntz-Krieger algebras.

math.NT↗
arXiv2026

K-theory of Etesi C*-algebras

We study the $C^*$-algebra $\mathbb{E}_{\mathscr{M}}$ of a smooth 4-dimensional manifold $\mathscr{M}$ introduced by Gábor Etesi. It is proved that the $\mathbb{E}_{\mathscr{M}}$ is a stationary AF-algebra. We calculate the topological and smooth invariants of $\mathscr{M}$ in terms of the K-theory of the $C^*$-algebra $\mathbb{E}_{\mathscr{M}}$. Using Gompf's Stable Diffeomorphism Theorem, it is shown that all smoothings of $\mathscr{M}$ form a torsion abelian group. The latter is isomorphic to the Brauer group of a number field associated to the K-theory of $\mathbb{E}_{\mathscr{M}}$.

math.GT↗
arXiv2026

Geometry of $\mathbf{F}_1$ and Cuntz-Krieger algebras

We study a natural map between projective varieties $V(\mathbf{F}_{1})$ over the field with one element and the Cuntz-Krieger algebras $O_A$. Using the $K$-theory of $O_A$, we calculate the Frobenius action and cardinality of the set $V(\mathbf{F}_{1^r})$. It is proved that the zeta function of $V(\mathbf{F}_{1})$ satisfies all Weil's Conjectures except for an analog of the Riemann hypothesis. We use the crossed product structure of $O_A$ to establish a morphism of the schemes $\operatorname{Spec} ~(\mathbf{Z})\to \operatorname{Spec} ~(\mathbf{F}_{1})\simeq \{\operatorname{pt}\}$.

math.NT↗
arXiv2026

Local factors and Cuntz-Pimsner algebras

We recast the local factors of the Hasse-Weil zeta function at infinity in terms of the Cuntz-Pimsner algebras. The nature of such factors is an open problem studied by Deninger and Serre.

math.NT↗
arXiv2026

Arakelov geometry of Cuntz-Pimsner algebras

We use $K$-theory of the $C^*$-algebras to study the Arakelov geometry, i.e. a compactification of the arithmetic schemes $V\to Spec ~\mathbf{Z}$. In particular, it is proved that the Picard group of $V$ is isomorphic to the $K_0$-group of a Cuntz-Pimsner algebra associated to $V$. We apply the result to the finiteness problem for the algebraic varieties over number fields.

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arXiv2026

Class number zeta function of imaginary quadratic fields

We introduce a zeta function counting imaginary quadratic number fields by their class numbers. It is proved that such a function is rational depending only on the eight roots of unity of degrees $1$ and $2$. As a corollary, one gets a lower bound $2p$ for the number of imaginary quadratic fields of the prime class number $p$. Our method is based on the study of periodic points of a dynamical system arising in the representation theory of the Drinfeld modules by the bounded linear operators on a Hilbert space.

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arXiv2026

Quantum arithmetic of Drinfeld modules

We study the quantum invariants of projective varieties over the number fields. Namely, explicit formulas for a functor $\mathscr{Q}$ on such varieties are proved. The case of abelian varieties with complex multiplication is treated in detail.

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arXiv2025

Lambert $W$-function and Gauss class number one conjecture

We study fixed points of a function arising in a representation theory of the Drinfeld modules by the bounded linear operators on a Hilbert space. We prove that such points correspond to number fields of the class number one. As an application, one gets a solution to the Gauss conjecture for the real quadratic fields of class number one.

math.NT↗
arXiv2025

Galois cohomology revisited

We recast the Galois cohomology of the variety $V$ over a number field $k$ in terms of the K-theory of a $C^*$-algebra $\mathscr{A}_V$ connected to $V$. It is proved that $V$ is isomorphic to $V'$ over $k$ (algebraic closure of $k$, resp.) if and only if $\mathscr{A}_V$ is isomorphic (Morita equivalent, resp.) to $\mathscr{A}_{V'}$. In particular, the Morita equivalent $C^*$-algebras $\mathscr{A}_V$ parametrize twists of the variety $V$. The case of rational elliptic curves is considered in detail.

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arXiv2025

Jones Index Theorem revisited

We prove the Jones Index Theorem using the K-theory of a cluster $C^*$-algebra of the Riemann sphere with two boundary components.

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arXiv2025

Birational geometry of quaternions

The Hilbert class field of the quaternion algebra $B$ is an algebra $\mathscr{H}(B)$ such that every two-sided ideal of $B$ is principal in $\mathscr{H}(B)$. We study the avatars of $B$ and $\mathscr{H}(B)$, i.e. algebraic surfaces attached to the quaternion algebras. It is proved that the avatar of $\mathscr{H}(B)$ is obtained from the avatar of $B$ by a birational map. We apply this result to the function field analogy.

math.NT↗
arXiv2025

Quantum dynamics of elliptic curves

We calculate K-theory of a crossed product $C^*$-algebra of the noncommutative torus with real multiplication by elliptic curve $\mathscr{E}(K)$ over a number field $K$. This result is used to evaluate the rank and the Shafarevich-Tate group of $\mathscr{E}(K)$.

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arXiv2025

K-theory of Minkowski question-mark function

It is proved that the Minkowski question-mark function comes from the K-theory of Cuntz-Pimsner algebras. We apply this result to calculate the action of Frobenius endomorphism at the infinite prime. Such a problem was raised by Serre and Deninger in the theory of local factors of zeta functions of projective varieties.

math.KT↗
arXiv2025

Trace cohomology revisited

We use a cohomology theory coming from the canonical trace on a C*-algebra of the projective variety to prove an analog of the Riemann Hypothesis for the Kuga-Sato varieties over finite fields.

math.AG↗
arXiv2024

Quantum arithmetic

We formalize the quantum arithmetic, i.e. a relationship between number theory and operator algebras. Namely, it is proved that rational projective varieties are dual to the $C^*$-algebras with real multiplication. Our construction fits all axioms of the quantum arithmetic conjectured by Manin and others. Applications to elliptic curves, Shafarevich-Tate groups of abelian varieties and height functions are reviewed.

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