Superspecial abelian varieties over finite prime fields
An abelian variety defined over a field \(K\) of characteristic \(p\) is called superspecial if it is isomorphic over the algebraic closure \(\overline{K}\) to a product of supersingular elliptic curves. Fix a supersingular elliptic curve \(E_0/{\mathbb F}_p\) (with an extra condition if \(p = 2\) or \(3\)).NEWLINENEWLINEIn the paper under consideration, the number of \(g\)-dimensional superspecial abelian varieties \(A\) over \({\mathbb F}_p\) is determined such that there is an \({\mathbb F}_p\)-isogeny from \(E_0^g\) to \(A\). The result, Theorem 1.1, relates the number in question with the class number of the imaginary quadratic field \({\mathbb Q}(\sqrt{-p})\), and thereby generalizes a classical result of Deuring for the case \(g=1\),NEWLINENEWLINEThe proof is by algebraic means and uses the module theory of a certain not necessarily maximal order in a semisimple commutative \({\mathbb Q}\)-algebra.NEWLINENEWLINEBesides the proof, the article presents a condensed but highly valuable discussion of similar problems: relations with weighted/unweighted class numbers, geometric and arithmetic mass formulas, supersingular instead of superspecial abelian varieties, with or without polarizations, Drinfeld modules, etc.
scientific article; zbMATH DE number 6094128
scientific article