Grothendieck's pairing on Neron component groups
The group of connected components of the special fiber of the Neron model of an abelian variety is called the Neron component group. In SGA7, Grothendieck constructed a canonical pairing between the Neron component groups of an abelian variety and its dual over a local field with perfect residue field. He conjectured that the pairing is perfect. In this talk, we prove his conjecture. A key tool is the category of fields viewed as a Grothendieck site. The cohomology of an abelian variety can be regarded as a sheaf on this new site. This allows us to treat Neron models completely functorial in the derived category of sheaves. From the known case of semistable abelian varieties, we deduce the perfectness in full generality by Galois descent.