arXiv subjects

Igor V. Nikolaev

Publications and source records attributed to Igor V. Nikolaev.

25 records · Page 2Linked to original sources

arXiv2024

Remark on height functions

Let $k$ be a number field and $V(k)$ an $n$-dimensional projective variety over $k$. We use the $K$-theory of a $C^*$-algebra $A_V$ associated to $V(k)$ to define a height of points of $V(k)$. The corresponding counting function is calculated and we show that it coincides with the known formulas for $n=1$. As an application, it is proved that the set $V(k)$ is finite, whenever the sum of the odd Betti numbers of $V(k)$ exceeds $n+1$. Our construction depends on the $n$-dimensional Minkowski question-mark function studied by Panti and others.

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arXiv2024

Shafarevich-Tate groups of abelian varieties

The Shafarevich-Tate group $W (\mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $\mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W (\mathscr{A})\cong Cl~(Λ)\oplus Cl~(Λ)$ or $W (\mathscr{A})\cong \left(\mathbf{Z}/2^k\mathbf{Z}\right) \oplus Cl_{~\mathbf{odd}}~(Λ)\oplus Cl_{~\mathbf{odd}}~(Λ)$, where $Cl~(Λ)$ is the ideal class group of a ring $Λ$ associated to the K-theory of the non-commutative tori and $2^k $ divides the order of $Cl~(Λ)$. The case of elliptic curves with complex multiplication is considered in detail.

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arXiv2024

Drinfeld modules as noncommutative tori

The Drinfeld module is a tool of the explicit class field theory for the function fields. We first observe a similarity of such modules with the noncommutative tori, and then use it to develop an explicit class field theory for the number fields. The case of the imaginary quadratic number fields is treated in detail.

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arXiv2023

Artin $L$-functions and noncommutative tori

Using the ideas of Deninger, we prove that the Artin $L$-functions coincide with such of the noncommutative tori. This result can be viewed as the Langlands reciprocity for noncommutative tori.

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arXiv2023

$L$-function of noncommutative tori

We introduce an analog of the $L$-function for noncommutative tori. It is proved that such a function coincides with the Hasse-Weil $L$-function of an elliptic curve with complex multiplication. As a corollary, one gets a localization formula for the noncommutative tori with real multiplication.

math.OA↗

arXiv2023

Remark on function field analogy

We study the analogy between number fields and function fields in one variable over finite fields. The main result is an isomorphism between the Hilbert class fields of class number one and a family of the function fields $\mathbf{F}_q(C)$ over a desingularized algebraic curve $C$. Our proof is based on the K-theory of the Serre $C^*$-algebras and birational geometry of the curve $C$. We apply the isomorphism to construct explicit generators of the Hilbert class fields coming from the torsion submodules of the Drinfeld module.

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arXiv2023

Elliptic curves and continued fractions

It is proved that the rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. As an illustration, we consider a family of elliptic curves with complex multiplication.

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