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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Remarque sur l'arithmétique des $ 2$-formes différentielles de deuxième espèce


Author: Boulahia Nejib
Journal: Proc. Amer. Math. Soc. 97 (1986), 389-392
MSC: Primary 14F20; Secondary 11G40, 14J20
MathSciNet review: 840615
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Abstract: We consider a complex smooth and proper surface $ X$ and a prime number $ l$. We know that the invariant $ {B_2} - \rho $ represents the dimension of the vector space of classes of $ 2$-forms of the second kind on $ X$. Grothendieck has observed that $ {B_2} - \rho $ 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 $ \omega $ of a $ 2$-form of the second kind an element

$\displaystyle {\omega _1}$

of $ {\omega}$-adic part of the Brauer group $ {H^2}({X_{{\text{et,}}}}\mathcal{O}_{{X_{{\text{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), $ {\omega _l}$ is then also interpreted as an element of $ {H^1}{({\Gamma _{{\text{et}}}},{i^*}J)_{(l)}}$ or, in the equivalent manner, as an element of the Tate-Schafarievitch group $ {\text{II}}{{\text{I}}^1}({\mathbf{C}}(\Gamma ),J)$ studied by Ogg, where $ J$ is the Jacobian of the generic fiber of $ X$ and $ {\mathbf{C}}(\Gamma )$ is the functions field of $ \Gamma $. Reciprocally each element of the $ l$-divisible part of $ {\text{II}}{{\text{I}}^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.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0840615-2
PII: S 0002-9939(1986)0840615-2
Article copyright: © Copyright 1986 American Mathematical Society