On supersingular curves and abelian varieties.

*(English)*Zbl 0641.14007The author presents a number of results on supersingular curves and abelian varieties. First he classifies \(\ell\)-adic and rational polarizations, in particular principal polarizations on a product of supersingular elliptic curves. To this end he introduces mass functions on groupoids and proves some formulas for the mass of certain sets of polarizations.

In the second part the author shows that a necessary condition for the Jacobian of a curve of genus \(g\) over a field of characteristic \(p\) to be a product of supersingular elliptic curves is 2g\(\leq p(p-1)\). He also finds a sufficient condition for this to happen (involving group representations), and gives some examples.

In the last section families of genus two curves with supersingular generic fibre are characterized. This is applied to show that the total space of a family of curves over \({\mathbb{P}}^ 1\) is of general type if its Jacobian has good reduction everywhere, the only exception being a product of two supersingular elliptic curves in characteristic 2.

In the second part the author shows that a necessary condition for the Jacobian of a curve of genus \(g\) over a field of characteristic \(p\) to be a product of supersingular elliptic curves is 2g\(\leq p(p-1)\). He also finds a sufficient condition for this to happen (involving group representations), and gives some examples.

In the last section families of genus two curves with supersingular generic fibre are characterized. This is applied to show that the total space of a family of curves over \({\mathbb{P}}^ 1\) is of general type if its Jacobian has good reduction everywhere, the only exception being a product of two supersingular elliptic curves in characteristic 2.

Reviewer: F.Herrlich

##### MSC:

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14H40 | Jacobians, Prym varieties |

14K15 | Arithmetic ground fields for abelian varieties |

14H25 | Arithmetic ground fields for curves |