Grothendieck’s pairing on component groups of Jacobians Siegfried Bosch (Münster)

Let A_{K} be an abelian variety over the field K of fractions of a discrete valuation ring R, and let A′_{K} be the dual abelian variety of A_{K}. Then A_{K} and A_{K′} have a natural integral structure over R, given by their Néron models A and A′. An interesting question is to investigate the duality between A_{K} and A′_{K} in terms of integral data, i. e., on the level of associated Néron models. Grothendieck has shown that the integral structure of this duality is encoded in a bilinear pairing

Φ_{A} × Φ_{A′} ⟶ ℚ/ℤ

on the component groups Φ_{A}, Φ_{A′} associated to the Néron models A, A′. In fact, the pairing represents the obstruction of extending the Poincaré bundle P on A_{K} × A′_{K} to a corresponding object on A × A′. Grothendieck conjectured that the pairing defines always a perfect duality between Φ_{A} and Φ_{A′}, and he was able to prove this in certain situations. The conjecture has been established in a number of additional cases, notably by work of Artin and Mazur (unpublished), Bégueri, and McCallum.

In the lecture we report about a new approach to Grothendieck’s pairing, which was developed in joint work with Dino Lorenzini. We start with an interpretation of the pairing in terms of certain vanishing orders of rational functions. Thereby we are able to relate Grothendieck’s pairing to Néron’s local height pairing. The latter has good functorial properties, in contrast to the behavior of Néron models and their component groups. For Jacobians, we thereby are able to transfer all ingredients of Grothedieck’s pairing to the underlying curve. Using Raynaud’s characterization of component groups of Jacobians, we arrive at an explicit formula for the pairing, involving only combinatorial data of the curve. The formula shows that, for Jacobians, Grothendieck’s pairing is always perfect, provided the residue field is perfect, and that there are counterexamples otherwise.