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The modified diagonal cycle on the triple product of a pointed curve

Benedict H. GrossChad Schoen — 1995

Annales de l'institut Fourier

Let X be a curve over a field k with a rational point e . We define a canonical cycle Δ e Z 2 ( X 3 ) hom . Suppose that k is a number field and that X has semi-stable reduction over the integers of k with fiber components non-singular. We construct a regular model of X 3 and show that the height pairing τ * ( Δ e ) , τ * ' ( Δ e ) is well defined where τ and τ ' are correspondences. The paper ends with a brief discussion of heights and L -functions in the case that X is a modular curve.

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