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Berkovich (Invent. Math. 125(2):367–390, ), Fujiwara (Duke Math. J. 80(1):15–57, ) and Huber (J. Algebraic Geom. 7(2):359–403, ) have proved that the fiber of the vanishing cycles at a point of the special fiber depends only on the formal completion at this point. We refine this result and prove the invariance under formal completion of the perverse monodromy filtration on the fiber of vanishing cycles. This result is used in an essential way by Boyer (Invent. Math. doi:, ). [ABSTRACT FROM AUTHOR]

Abstract: This article consists of two independent, but related, parts. The first one proves the vanishing of the Chow group of classes of zero-cycles of degree zero modulo rational equivalence for a cubic hypersurface of dimension ⩾10 on a p-adic or field (and, in fact, the R-triviality of such a hypersurface). This is done without the assumption of good reduction (or even smoothness). The second part goes in the other direction and gives an explicit example of a smooth cubic hypersurface of dimension 3 (necessarily of bad reduction) on a field such as (or ) whose Chow group of classes of zero-cycles of degree zero modulo rational equivalence does not vanish. [Copyright &y& Elsevier]

Abstract: Let Dumont has conjectured the identity , which generalizes, in particular, the classical results of Lagrange, Gauß, Jacobi and Kronecker on the sums of two, three and four squares. We give a combinatorial proof of Dumont''s conjecture. To cite this article: B. Lass, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]

Abstract: In this Note, we give a geometric proof of Kolchin''s Theorem, using resolution of singularities. We give a counter-example in positive characteristic. To cite this article: J. Nicaise, J. Sebag, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]

Thanks to results of A. Beauville (Acta Math. 164 (1990) 211–235), we give an explicit description of translation-invariant vector fields on affine Jacobians of spectral curves, which gives a mathematical support for the work of F.A. Smirnov and V. Zeitlin (preprint math-ph/0203037). In the hyperelliptic case, this description is due to D. Mumford (Tata Lectures on Theta, Vol. II, Birkha¨user, Boston, 1983). To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003). [Copyright &y& Elsevier]