This paper contains two parts toward studying abelian varieties from the classification point of view.In a series of papers[Doc.Math.,21,1607–1643(2016)],[Taiwan Residents J.Math.,20(4),723–741(2016)],etc.,the curre...This paper contains two parts toward studying abelian varieties from the classification point of view.In a series of papers[Doc.Math.,21,1607–1643(2016)],[Taiwan Residents J.Math.,20(4),723–741(2016)],etc.,the current authors and T.C.Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields.In this paper,we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q1/2.This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting.The purpose is to generalize the previous work by the second named author[Forum Math.,22(3),565–582(2010)]on abelian varieties with additional structures to similitude classes,which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.展开更多
基金the Natural Science Foundation of China(Grant No.11601395)supported by the MoST(Grant Nos.104-2115-M-001-001MY3 and 107-2115-M-001-001-MY2)。
文摘This paper contains two parts toward studying abelian varieties from the classification point of view.In a series of papers[Doc.Math.,21,1607–1643(2016)],[Taiwan Residents J.Math.,20(4),723–741(2016)],etc.,the current authors and T.C.Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields.In this paper,we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q1/2.This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting.The purpose is to generalize the previous work by the second named author[Forum Math.,22(3),565–582(2010)]on abelian varieties with additional structures to similitude classes,which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.
