A Weil-étale version of the Birch and Swinnerton-Dyer formula over function fields (Q2009167)

The paper under review gives a reformulation of the Birch and Swinnerton-Dyer conjecture concerning the leading coefficient of the \(L\)-function of an abelian variety over a global function field. Under the assumption that the Tate-Shafarevich group is finite, the authors show that the leading coefficient of the \(L\)-function can be written as the alternating product of the orders of Weil-étale cohomology groups with coefficients in the Néron model of the abelian variety. Thus, the result fits into the general philosophy that special values of zeta and \(L\)-functions can be expressed as Euler characteristics of Weil-étale cohomology.