We compute all irregular primes less than 163 577 856. For all of these primes we verify that the Kummer–Vandiver conjecture holds and that the λ-invariant is equal to the index of irregularity.

We study invariants of Calabi-Yau varieties in positive characteristic, especially the height of the Artin-Mazur formal group. We illustrate these results by Calabi-Yau varieties of Fermat and Kummer type. 2000 Mathematics Subject Classification: 14J32, 14L05, 14F

We introduce a notion of highly Kummer-faithful fields and study its relationship with the fields. also give some examples For example, if $k$ is number field finite degree over \mathbb{Q}, $g$ an integer $> 0$ $\mathbf{m}=(m_p)_p$ family non-negative integers, where $p$ ranges all prime numbers, then extension $k_{g,\mathbf{m}}$ obtained by adjoining to coordinates elements $p^{m_p}$-torsion s...

In this paper we deal with a degenerate version of the trisecant conjecture (see Welters [W1]). Let (X , [Θ]) be an indecomposable principally polarized abelian variety and let Θ be a symmetric representative of the polarization. We shall denote by θ a non-zero section of the sheaf O X (Θ). The linear system |2Θ| is base-point-free and it is independent of the choice of Θ. The image of the morp...