The modified diagonal cycle on the triple product of a pointed curve
Gross, Benedict H. ; Schoen, Chad
Annales de l'Institut Fourier, Tome 45 (1995), p. 649-679 / Harvested from Numdam
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Soit X une courbe sur un corps k et soit e∈X(k). Nous définissons un cycle canonique Δ e ∈Z 2 (X 3 ) hom . Supposons que k est un corps de nombres et que X a un modèle semi-stable sur les entiers de k dont les composantes irréductibles des fibres sont lisses. Nous construisons un modèle régulier de X 3 et vérifions que l’accouplement de Beilinson-Bloch, 〈τ * (Δ e ),τ * ′ (Δ e )〉, est bien défini, où τ et τ ′ sont les correspondances. Si X est une courbe modulaire et τ et τ ′ sont les opérateurs de Hecke convenables, on conjecture une formule liant la dérivée d’une fonction L avec l’accouplement de Beilinson-Bloch.

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.

@article{AIF_1995__45_3_649_0,
     author = {Gross, Benedict H. and Schoen, Chad},
     title = {The modified diagonal cycle on the triple product of a pointed curve},
     journal = {Annales de l'Institut Fourier},
     volume = {45},
     year = {1995},
     pages = {649-679},
     doi = {10.5802/aif.1469},
     mrnumber = {96e:14008},
     zbl = {0822.14015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1995__45_3_649_0}
}

Gross, Benedict H.; Schoen, Chad. The modified diagonal cycle on the triple product of a pointed curve. Annales de l'Institut Fourier, Tome 45 (1995) pp. 649-679. doi : 10.5802/aif.1469. http://gdmltest.u-ga.fr/item/AIF_1995__45_3_649_0/

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