scholarly journals Classification of Prime Knots in the Thickened Surface of Genus 2 Having Diagrams with at Most 4 Crossings

2020 ◽  
Vol 7 (1) ◽  
pp. 32-46 ◽  
Author(s):  
A.A. Akimova ◽  
Keyword(s):  
Genus 2 ◽  
Thickened Surface

Classification of real moduli spaces over genus 2 curves

1995 ◽  
Vol 57 (2) ◽  
pp. 207-215 ◽  
Author(s):  
Shuguang Wang
Keyword(s):  
Moduli Spaces ◽  
Genus 2 ◽  
Genus 2 Curves

scholarly journals Classification of periodic transformations of an orientable surface of genus two

2021 ◽  
Vol 23 (2) ◽  
pp. 147-158
Author(s):  
Denis A. Baranov ◽  
Olga V. Pochinka
Keyword(s):  
Sufficient Conditions ◽  
Orientable Surface ◽  
Topological Conjugacy ◽  
Modular Surface ◽  
Periodic Surface ◽  
Genus 2 ◽  
Genus Two ◽  
Necessary And Sufficient ◽  
Genus One

Abstract. In this paper, we find all admissible topological conjugacy classes of periodic transformations of a two-dimensional surface of genus two. It is proved that there are exactly seventeen pairwise topologically non-conjugate orientation-preserving periodic pretzel transformations. The implementation of all classes by lifting the full characteristics of mappings from a modular surface to a surface of genus two is also presented. The classification results are based on Nielsen’s theory of periodic surface transformations, according to which the topological conjugacy class of any such homeomorphism is completely determined by its characteristic. The complete characteristic carries information about the genus of the modular surface, the ramified bearing surface, the periods of the ramification points and the turns around them. The necessary and sufficient conditions for the admissibility of the complete characteristic are described by Nielsen and for any surface they give a finite number of admissible collections. For surfaces of a small genus, one can compile a complete list of admissible characteristics, which was done by the authors of the work for a surface of genus 2.

scholarly journals The classification of quasi-regular polyhedra of genus 2

1992 ◽  
Vol 7 (4) ◽  
pp. 347-357 ◽  
Author(s):  
Reinhard Franz ◽  
Daniel Huson
Keyword(s):  
Regular Polyhedra ◽  
Genus 2

ON HYPERELLIPTIC C∞–LEFSCHETZ FIBRATIONS OF FOUR-MANIFOLDS

1999 ◽  
Vol 01 (02) ◽  
pp. 255-280 ◽  
Author(s):  
BERND SIEBERT ◽  
GANG TIAN
Keyword(s):  
Ruled Surfaces ◽  
Lefschetz Fibrations ◽  
Genus 2 ◽  
Four Manifolds

We show that hyperelliptic symplectic Lefschetz fibrations are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. This reduces the classification of genus 2 fibrations to the classification of certain symplectic submanifolds in rational ruled surfaces.

scholarly journals EXTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE

2019 ◽  
Vol 71 (1) ◽  
pp. 175-196
Author(s):  
Kenta Funayoshi ◽  
Yuya Koda
Keyword(s):  
Orientable Surface ◽  
Closed Orientable Surface ◽  
Genus 2

Abstract An automorphism $f$ of a closed orientable surface $\Sigma $ is said to be extendable over the 3-sphere $S^3$ if $f$ extends to an automorphism of the pair $(S^3, \Sigma )$ with respect to some embedding $\Sigma \hookrightarrow S^3$. We prove that if an automorphism of a genus-2 surface $\Sigma $ is extendable over $S^3$, then $f$ extends to an automorphism of the pair $(S^3, \Sigma )$ with respect to an embedding $\Sigma \hookrightarrow S^3$ such that $\Sigma $ bounds genus-2 handlebodies on both sides. The classification of essential annuli in the exterior of genus-2 handlebodies embedded in $S^3$ due to Ozawa, and the second author plays a key role.

scholarly journals A non-solvable extension of ℚ unramified outside 7

2012 ◽  
Vol 148 (3) ◽  
pp. 669-674 ◽  
Author(s):  
Luis V. Dieulefait
Keyword(s):  
Galois Group ◽  
Cusp Form ◽  
Fourier Coefficients ◽  
Galois Representation ◽  
Maximal Subgroups ◽  
Genus 2 ◽  
Solvable Extension ◽  
Level 1 ◽  
Siegel Cusp Form

AbstractWe consider a mod 7 Galois representation attached to a genus 2 Siegel cusp form of level 1 and weight 28 and using some of its Fourier coefficients and eigenvalues computed by N. Skoruppa and the classification of maximal subgroups of PGSp(4,p) we show that its image is as large as possible. This gives a realization of PGSp(4,7) as a Galois group over ℚ and the corresponding number field provides a non-solvable extension of ℚ which ramifies only at 7.

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.

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