scholarly journals Uniqueness of Polarization for the Autonomous 4-dimensional Painlevé-type Systems

2020 ◽  
Author(s):  
Akane Nakamura ◽  
Eric Rains
Keyword(s):  
Spectral Curve ◽  
Type Systems ◽  
Abelian Surface ◽  
Integral System ◽  
Genus 2 ◽  
Dimensional Integral

Abstract We prove that for any autonomous 4-dimensional integral system of Painlevé type, the Jacobian of the generic spectral curve has a unique polarization, and thus by Torelli’s theorem cannot be isomorphic as an unpolarized abelian surface to any other Jacobian. This enables us to identify the spectral curve and any irreducible genus 2 component of the boundary of an affine patch of the Liouville torus.

scholarly journals A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

2004 ◽  
Vol 47 (3) ◽  
pp. 398-406
Author(s):  
David McKinnon
Keyword(s):  
Number Field ◽  
Open Subset ◽  
Rational Points ◽  
Rational Curves ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Geometric Genus ◽  
Finite Set ◽  
Genus 2 ◽  
Kummer Surfaces

AbstractLet V be a K3 surface defined over a number field k. The Batyrev-Manin conjecture for V states that for every nonempty open subset U of V, there exists a finite set ZU of accumulating rational curves such that the density of rational points on U − ZU is strictly less than the density of rational points on ZU. Thus, the set of rational points of V conjecturally admits a stratification corresponding to the sets ZU for successively smaller sets U.In this paper, in the case that V is a Kummer surface, we prove that the Batyrev-Manin conjecture for V can be reduced to the Batyrev-Manin conjecture for V modulo the endomorphisms of V induced by multiplication by m on the associated abelian surface A. As an application, we use this to show that given some restrictions on A, the set of rational points of V which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Batyrev-Manin type.

scholarly journals ALBANESE VARIETIES OF CYCLIC COVERS OF THE PROJECTIVE PLANE AND ORBIFOLD PENCILS

2016 ◽  
Vol 227 ◽  
pp. 189-213
Author(s):  
E. ARTAL BARTOLO ◽  
J. I. COGOLLUDO-AGUSTÍN ◽  
A. LIBGOBER
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Plane Curves ◽  
Abelian Surface ◽  
Singular Curve ◽  
Cyclic Covers ◽  
Genus 2 ◽  
Multiple Fibers

The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, that is, a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is nontrivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve of genus 2 illustrating the extent to which these conditions are necessary.

CYCLIC COVERINGS OF ABELIAN VARIETIES AND THE GENERALIZED YANG–MILLS SYSTEM FOR A FIELD WITH GAUGE GROUP SU(2)

2008 ◽  
Vol 05 (06) ◽  
pp. 947-961
Author(s):  
A. LESFARI
Keyword(s):  
Dynamical System ◽  
Gauge Group ◽  
Integrable System ◽  
Double Cover ◽  
Abelian Surface ◽  
Invariant Surface ◽  
Yang Mills ◽  
Completely Integrable ◽  
Genus 2 ◽  
Constants Of Motion

In this paper, we consider a dynamical system related to the Yang–Mills system for a field with gauge group SU(2). We solve this system in terms of genus two hyperelliptic functions. The corresponding invariant surface defined by the two constants of motion can be completed as a cyclic double cover of an abelian surface (the jacobian of a genus 2 curve) and we show that this system is algebraic completely integrable in the generalized sense. Also we show that this system is part of an algebraic completely integrable system in five unknowns having three constants of motion.

Restriction of Stable Bundles on an Abelian Surface. The C2 = 2 Case

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Cristian Anghel
Keyword(s):  
Moduli Space ◽  
Stable Rank ◽  
Abelian Surface ◽  
Restriction Map ◽  
Stable Bundles ◽  
Genus 2 ◽  
Rank 2 ◽  
Dimension 2 ◽  
Rank 2 Bundles

Abstract In this note we describe the restriction map from the moduli space of stable rank 2 bundles with c2 = 2 on a jacobian X of dimension 2, to the moduli space of stable rank 2 bundles on the corresponding genus 2 curve C embedded in X.

scholarly journals Quasi-periodic solutions of the negative-order Jaulent–Miodek hierarchy

2019 ◽  
Vol 32 (03) ◽  
pp. 2050007 ◽  
Author(s):  
Jinbing Chen
Keyword(s):  
Periodic Solutions ◽  
Hamiltonian Structure ◽  
Spectral Curve ◽  
Type Systems ◽  
Finite Dimensional ◽  
Zero Curvature ◽  
Negative Order ◽  
Neumann Type ◽  
Jacobi Inversion ◽  
Negative Formula

A uniform construction of quasi-periodic solutions to the negative-order Jaulent–Miodek (nJM) hierarchy is presented by using a family of backward Neumann type systems. From the backward Lenard gradients, the nJM hierarchy is put into the zero-curvature setting and the bi-Hamiltonian structure displaying its integrability. The nonlinearization of Lax pair is generalized to the nJM hierarchy such that it can be reduced to a sequence of backward Neumann type systems, whose involutive solutions yield finite parametric solutions of the nJM hierarchy. The negative [Formula: see text]-order stationary JM equation is given to specify a finite-dimensional invariant subspace for the nJM flows. With a spectral curve determined by the Lax matrix, the nJM flows are linearized on the Jacobi variety of a Riemann surface. Finally, the Riemann–Jacobi inversion is applied to Abel–Jacobi solutions of the nJM flows, by which some quasi-periodic solutions are obtained for the nJM hierarchy.

scholarly journals Sato–Tate distributions and Galois endomorphism modules in genus 2

2012 ◽  
Vol 148 (5) ◽  
pp. 1390-1442 ◽  
Author(s):  
Francesc Fité ◽  
Kiran S. Kedlaya ◽  
Víctor Rotger ◽  
Andrew V. Sutherland
Keyword(s):  
Hyperelliptic Curves ◽  
Abelian Surface ◽  
Module Structure ◽  
Galois Module ◽  
Tate Group ◽  
Usp 4 ◽  
Galois Action ◽  
Tate Module ◽  
Genus 2

AbstractFor an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato–Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato–Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the ℝ-algebra generated by endomorphisms of $A_{{\overline {\mathbb Q}}}$ (the Galois type), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k, of which 34 can occur for k=ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics.

Religious Studies of Democratic Ukraine

2001 ◽  
pp. 3-12
Author(s):  
Anatolii M. Kolodnyi
Keyword(s):  
Religious Studies ◽  
Religious Experiences ◽  
Divine Nature ◽  
Deep Roots ◽  
Integral System ◽  
The World

Ukrainian religious studies have deep roots. We find the elements of it in the written descendants of the writings of Kievan Rus. From the prince's time, the universal way of vision, understanding and appreciation of the world for many Ukrainian thinkers becomes their own religious experiences. The main purpose of their works is not the desire to create a certain integral system of theological knowledge, but the desire to convey their personal religious-minded perception of the divine nature, harmony, beauty and perfection of God created the world.

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