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Journal Article

Remarque Sur L'Arithmétique Des 2-Formes Différentielles de Deuxième Espèce

Boulahia Nejib
Proceedings of the American Mathematical Society
Vol. 97, No. 3 (Jul., 1986), pp. 389-392
DOI: 10.2307/2046223
Stable URL: http://www.jstor.org/stable/2046223
Page Count: 4
We consider a complex smooth and proper surface X and a prime number l. We know that the invariant B2 - ρ represents the dimension of the vector space of classes of 2-forms of the second kind on X. Grothendieck has observed that B2 - ρ is an arithmetic invariant, and this leads naturally to interpret a 2-form of the second kind from an arithmetic point of view. By using Grothendieck's techniques we associate to each class ω of a 2-form of the second kind an element ωl of l-adic part of the Brauer group $H^2(X_{\operatorname{et}}, \mathscr{O}^\ast_{X_{\operatorname{et}}})$ of X. In the case where X is a fibered space on a curve, if one subjects this fibration to some given conditions (Artin), ωl is then also interpreted as an element of $H^1(\Gamma_{\operatorname{et}}, i^\ast J)_{(l)}$ or, in the equivalent manner, as an element of the Tate-Schafarievitch group $\operatorname{III}^1(\mathbf{C}(\Gamma), J)$ studied by Ogg, where J is the Jacobian of the generic fiber of X and C(Γ) is the functions field of Γ. Reciprocally each element of the l-divisible part of $\operatorname{III}^1(\mathbf{C}(\Gamma), J)_{(l)}$ comes from a 2-form of the second kind. This correspondence permits us to deduce some consequences on 2-forms of the second kind.