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Nikolai S. Nadirashvili

Publications and source records attributed to Nikolai S. Nadirashvili.

4 recordsLinked to original sources
arXiv2023

Complete Description of Measures Corresponding to Abelian Varieties over Finite Fields

We study probability measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman- Vladuts theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. J.-P.Serre, using results of R.M.Robinson on conjugate algebraic integers, described the possible set of measures than can correspond to families of abelian varieties over a finite field. The problem whether all such measures actually occur was left open. Moreover, Serre supposed that not all such measures correspond to abelian varieties (for example, the Lebesgue measure on a segment). Here we settle Serre's problem proving that Serre conditions are sufficient, and thus describe completely the set of measures corresponding to abelian varieties.

math.AP↗
arXiv2018

An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane

We prove an isoperimetric inequality for the second non-zero eigenvalue of the Laplace-Beltrami operator on the real projective plane. For a metric of the unit area this eigenvalue is not greater than 20π. This value is attained in the limit by a sequence of metrics of area one on the projective plane. The limiting metric is singular and could be realized as a union of the projective plane and the sphere touching at a point, with standard metrics and the ratio of the areas 3:2. It is also proven that the multiplicity of the second non-zero eigenvalue on the projective plane is at most 6.

math.DG↗