arXiv subjects

Ilya Zakharevich

Publications and source records attributed to Ilya Zakharevich.

10 recordsLinked to original sources

arXiv2024

Quadratic residue patterns, algebraic curves and a K3 surface

Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive $\ell$-tuples of quadratic residues, studying corresponding algebraic curves and their jacobians, which happen to be products of jacobians of hyperelliptic curves. In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove it. The core of this theorem is an unexpected relation between the number of points on the K3 surface and that on a CM elliptic curve.

math.AG↗

arXiv2023

Quadratic residue patterns and point counting on K3 surfaces

Quadratic residue patterns modulo a prime are studied since 19th century. We state the last unpublished result of Lydia Goncharova, reformulate it and prior results in terms of algebraic geometry, and prove it. The core of this theorem is an unexpected relation between the number of points on a K3 surface and that on a CM elliptic curve.

math.AG↗

arXiv2002

Geometic vertex operators

Vertex operators, being families of birational transformations of infinite-dimensional algebraic ``varieties'' M, act on appropriate line bundles on M. However, they act on (meromorphic) sections only as_partial operators_: they are defined on a subspace (in an appropriate_lattice of subspaces_ of Mer(M)), and send "smooth families" of vectors in such a subspace to smooth families. Axiomatizing this, we define_conformal fields_ as arbitrary families of partial operators in Mer(M) which satisfy both these properties. The ``variety'' M related to standard vertex operators is formed by rational functions of one variable z in Z=P^1, changing the variety Z one obtains different examples of M, and multidimensional analogues of vertex operators. One can cover Mer(M) by ``big smooth subsets''; these subsets are parameterized by appropriate projective bundles over Hilbert schemes of points on Z. We deduce conformal associativity relation for conformal fields, and conformal commutation relations for Laurent coefficients of commuting conformal fields from geometric properties of the Hilbert schemes. We start with providing examples of OPE in smooth families, i.e., smooth families a(s) and b(t) (of partial linear operators) such that a(s)b(t) has a pole when s=t. We also provide geometric description of boson-fermion correspondence, and relations of this correspondence to geometry of the set of meromorphic functions.

math.AG↗

arXiv2002

Curves of infinite genus I Riemann--Roch theorem for small degree

The most useful and interesting line bundles over algebraic curves of a very high genus have the ratio δof the degree to the genus close to half-integer values, usually δ\approx 0, δ\approx 1/2, or δ\approx 1; the numeric properties are very different in these three cases. This leads to three different theories for curves of infinite genus. For analytic curves of infinite genus, to get a theory parallel to algebraic geometry one needs to restrict attention to holomorphic sections satisfying some ``conditions on growth at infinity''. Each such condition effectively attaches an ``ideal point'' to the curve; this process is similar to compactification. The theory of holomorphic functions on curves with such ``ideal points'' is developed (the variant presented in the first part of the series is tuned to the case δ\approx 0). Conditions on the ``lengths of handles'' of the curve are found which ensure the geometry to be parallel to algebraic geometry. It turns out that these conditions give no restriction on the density of ideal points on the curve. In particular, such curves may have a dense set of ideal points; these curves have no smooth points at all, and have a purely fractal nature. (Such ``foam'' curves live near the ``periphery'' of the corresponding g = \infty moduli space; one needs to study these curves too, since they may be included in the support of natural measures arising on the moduli spaces.)

math.AG↗

arXiv2000

Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs

Veronese webs are rich geometric structures with deep relationships to various domains of mathematics. The PDEs which determine the Veronese web are overdetermined if dim >3, but in the case dim =3 they reduce to a special flavor of a non-linear wave equation. The symmetries embedded in the definition of a Veronese web reveal themselves as Bäcklund--Darboux transformations between these non-linear wave equations. On the other hand, the twistor transform identifies Veronese webs with moduli spaces of rational curves on certain complex surfaces. These moduli spaces can be described in terms of the non-linear Riemann problem. This reduces solutions of these non-linear wave equations to the non-linear Riemann problem. We examine these relationships in the particular case of 3-dimensional Veronese webs, simultaneously investigating how these notions relate to general notions of geometry of webs.

math-ph↗

arXiv2000

Webs, Lenard schemes, and the local geometry of bihamiltonian Toda and Lax structures

We introduce a criterion that a given bihamiltonian structure allows a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bihamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bihamiltonian structures. In a generic point the bihamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general results on geometry of webs. This demonstrates a possibility to apply a geometric language to problems on bihamiltonian integrable systems, such a possibility may be no less important than the particular results proven in this paper. Based on these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bihamiltonian Lie coalgebra. We also state general conjectures about geometry of more general ``homogeneous'' finite-dimensional bihamiltonian structures. The class of homogeneous structures is shown to coincide with the class of system integrable by Lenard scheme. The bihamiltonian structures which allow a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda lattice.

math.DG↗

arXiv2000

Kronecker webs, bihamiltonian structures, and the method of argument translation

We show that manifolds which parameterize values of first integrals of integrable finite-dimensional bihamiltonian systems carry a geometric structure which we call a {\em Kronecker web}. We describe two functors between Kronecker webs and integrable bihamiltonian structures, one is left inverse to another one. Conjecturally, these two functors are mutually inverse (for ``small'' open subsets). The above conjecture is proven provided the bihamiltonian structure allows an antiinvolution of a particular form. This implies the conjecture of \cite{GelZakh99Web} that on a dense open subset the bihamiltonian structure on ${\mathfrak g}^{*}$ is flat if ${\mathfrak g}$ is semisimple, or if ${\mathfrak g}={\mathfrak G}\ltimes \operatorname{ad}_{\mathfrak G}$ and ${\mathfrak G}$ is semisimple, and for some other Lie algebras of mappings.

math.SG↗

arXiv1997

Quasi-algebraic geometry of curves I. Riemann-Roch theorem and Jacobian

We discuss an analogue of Riemann-Roch theorem for curves with an infinite number of handles. We represent such a curve X by its Shottki model, which is an open subset U of CP^{1} with infinite union of circles as a boundary. An appropriate bundle on X is ω^{1/2} \otimes L, L being a bundle with (say) constants as gluing conditions on the circles. An admissible section of an appropriate bundle on X is a holomorphic half-form on U with given gluing conditions and H^{1/2}-smoothness condition. We study the restrictions on the mutual position of the circles and the gluing constants which guarantee the finite dimension of the space of appropriate sections of admissible bundles, and make the Riemann-Roch theorem hold. The resulting Jacobian variety is described as an infinite-dimension analogue of a torus.

alg-geom↗

arXiv1993

Poisson-Lie group of pseudodifferential symbols

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this ``universal'' Poisson--Lie group. Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in physical terminology) can be viewed as subspaces of the quotient (or Poisson reduction) of this Poisson--Lie group by the dressing action of the group of functions. Finally, we define an infinite set of functions in involution on the Poisson--Lie group that give the standard families of Hamiltonians when restricted to the submanifolds mentioned above. The Poisson structure and Hamiltonians on the whole group interpolate between the Poisson structures and Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical meaning of W_\infty as a limit of Poisson algebras W_εas εgoes to 0.

hep-th↗

arXiv1993

Poisson-Lie group of pseudodifferential symbols and fractional KP-KdV hierarchies

The Lie algebra of pseudodifferential symbols on the circle has a nontrivial central extension (by the ``logarithmic'' 2-cocycle) generalizing the Virasoro algebra. The corresponding extended subalgebra of integral operators generates the Lie group of classical symbols of all real (or complex) degrees. It turns out that this group has a natural Poisson-Lie structure whose restriction to differential operators of an arbitrary integer order coincides with the second Adler-Gelfand-Dickey structure. Moreover, for any real (or complex) αthere exists a hierarchy of completely integrable equations on the degree αpseudodifferential symbols, and this hierarchy for α=1 coincides with the KP one, and for an integer α=n>1$ and purely differential symbol gives the n-KdV-hierarchy.

hep-th↗