We investigate action of a subgroup G1 of the Picard group on finite sets using coset diagrams. We show that its actions on the sets of 3, 4, 5, 6, 8, and 12 elements yield building blocks of Coset diagrams and that t...We investigate action of a subgroup G1 of the Picard group on finite sets using coset diagrams. We show that its actions on the sets of 3, 4, 5, 6, 8, and 12 elements yield building blocks of Coset diagrams and that these blocks can be connected together so that a diagram of n vertices can be obtained. We show that various combinations of these blocks represent alternating and symmetric groups of various degrees. We show also that the action of G1 on a set of n vertices is transitive.展开更多
We focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group.As an ...We focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group.As an application, we give criteria of the integral solvability of the diophantine equation p = x2+ ny2 over a class of imaginary quadratic fields where p is a prime element.展开更多
In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group F = PSL（2, Z[i]） on Q（i, √3）. Graphical interpretation of ...In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group F = PSL（2, Z[i]） on Q（i, √3）. Graphical interpretation of amalgamation of the components of F is also given. Some elements a＋b√3/c of Q（i, √3） and their conjugates a-b√3/c a c over Q（i） have different signs in the orbits of the biquadratic field Q（i, √3） when acted upon by F. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Fa, and they form a closed path which is the only closed path in the orbit Гa. We also devise a procedure to obtain ambiguous numbers of the form a-b√3/c, where b is a positive integer.展开更多
For a commutative ring R, its related characterizations are given by investigating the structure and properties of H0R. Furthermore, by virtue of H0structure, some important characterizations of CPF properties and con...For a commutative ring R, its related characterizations are given by investigating the structure and properties of H0R. Furthermore, by virtue of H0structure, some important characterizations of CPF properties and connected properties on K0R are obtained from related rings.展开更多
Which algebraic groups are Picard varieties?BRION Michel Abstract We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projectiv...Which algebraic groups are Picard varieties?BRION Michel Abstract We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety having only finitely many non-normal points.In contrast,no Witt group of dimension st least 3 over a perfect field of prime characteristic is isogenous to a Picard variety obtained by this construction.Keywords algebraic group,Picard variety。展开更多
文摘We investigate action of a subgroup G1 of the Picard group on finite sets using coset diagrams. We show that its actions on the sets of 3, 4, 5, 6, 8, and 12 elements yield building blocks of Coset diagrams and that these blocks can be connected together so that a diagram of n vertices can be obtained. We show that various combinations of these blocks represent alternating and symmetric groups of various degrees. We show also that the action of G1 on a set of n vertices is transitive.
基金supported by National Natural Science Foundation of China(Grant No.11471314)the National Basic Research Program of China(973 Project)(Grant No.2011CB302401)the National Center for Mathematics and Interdisciplinary Sciences,Chinese Academy of Sciences
文摘We focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group.As an application, we give criteria of the integral solvability of the diophantine equation p = x2+ ny2 over a class of imaginary quadratic fields where p is a prime element.
文摘In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group F = PSL（2, Z[i]） on Q（i, √3）. Graphical interpretation of amalgamation of the components of F is also given. Some elements a＋b√3/c of Q（i, √3） and their conjugates a-b√3/c a c over Q（i） have different signs in the orbits of the biquadratic field Q（i, √3） when acted upon by F. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Fa, and they form a closed path which is the only closed path in the orbit Гa. We also devise a procedure to obtain ambiguous numbers of the form a-b√3/c, where b is a positive integer.
文摘For a commutative ring R, its related characterizations are given by investigating the structure and properties of H0R. Furthermore, by virtue of H0structure, some important characterizations of CPF properties and connected properties on K0R are obtained from related rings.
文摘Which algebraic groups are Picard varieties?BRION Michel Abstract We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety having only finitely many non-normal points.In contrast,no Witt group of dimension st least 3 over a perfect field of prime characteristic is isogenous to a Picard variety obtained by this construction.Keywords algebraic group,Picard variety。
