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

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 \[ {\omega _1}\] 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.