The modified diagonal cycle on the triple product of a pointed curve
Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 649-679.

Soit X une courbe sur un corps k et soit eX(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.

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     title = {The modified diagonal cycle on the triple product of a pointed curve},
     journal = {Annales de l'Institut Fourier},
     pages = {649--679},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
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     year = {1995},
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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) no. 3, pp. 649-679. doi : 10.5802/aif.1469. https://aif.centre-mersenne.org/articles/10.5802/aif.1469/

[Al Kl] A. Altmann and S. Kleiman, Introduction to Grothendieck Duality Theory, Lect. Notes in Math. 146, Springer-Verlag, New York (1970) | MR | Zbl

[Be1] A. Beilinson, Higher regulators and values of L-functions, J. Soviet Math., 30 (1985), 2036-2070. | Zbl

[Be2] A. Beilinson, Height pairing between algebraic cycles. In : Current trends in arithmetic algebraic geometry, Contemp. Math., vol. 67 (1987), 1-24. | MR | Zbl

[Bl] S. Bloch, Height pairing for algebraic cycles, J. Pure Appl. Algebra, 34 (1984), 119-145. | MR | Zbl

[Ce] G. Ceresa, C is not algebraically equivalent to C- in its Jacobian, Annals of Math., 117 (1983), 285-291. | MR | Zbl

[Co vG] E. Colombo, and B. Van Geemen, Note on curves on a Jacobian, Compositio Math., 88 (1993), 333-353. | EuDML | Numdam | MR | Zbl

[De1] P. Deligne, Théorie de Hodge III, Publ. Math. IHES, 44 (1975), 5-77. | EuDML | Numdam | Zbl

[De2] P. Deligne, La conjecture de Weil II, Publ. Math. IHES, 52 (1981), 273-308.

[De3] P. Deligne, Formes modulaires et représentations l-adiques, Sém. Bourbaki, Exp. 355, Springer Lecture Notes, 179 (1969), 139-172. | EuDML | Numdam | Zbl

[DeMu] P. Deligne, and D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. IHES, 36 (1969), 75-110. | EuDML | Numdam | MR | Zbl

[DeRa] P. Deligne, and M. Rapoport, Les schémas de modules de courbes elliptiques, in : Modular forms of one variable II, Springer Lecture Notes, 349 (1973), 143-316. | MR | Zbl

[Des] M. Deschamps, Réduction semi-stable, in : Sém. sur les pinceaux de courbes de genre au moins deux, Astérisque, 86 (1981). | Zbl

[Fu] W. Fulton, Intersection theory, Springer Ergebnisse 3 Folge, Band 2 (1984). | MR | Zbl

[GiSo] H. Gillet, and C. Soulé, Intersection theory using Adams operations, Inv. Math., 90 (1987), 243-278. | EuDML | MR | Zbl

[GKu] B. Gross, and S. Kudla, Heights and the central critical values of triple product L-functions, Compositio Math., 81 (1992), 143-209. | EuDML | Numdam | MR | Zbl

[Ha] R. Hartshorne, Algebraic Geometry, Springer, 1977. | MR | Zbl

[Mi] J.S. Milne, Étale cohomology, Princeton Univ., Press, 1980. | MR | Zbl

[Mu] D. Mumford, Rational equivalence of zero cycles on surfaces, J. Math., Kyoto Univ., 9 (1969), 195-204. | MR | Zbl

[Né] A. Néron, Quasi-fonctions et hauteurs sur les variétés abéliennes, Annals of Math., 82 (1965), 249-331. | MR | Zbl

[Sch] A. Scholl, Motives for modular forms, Inv. Math., 100 (1990), 419-430. | EuDML | MR | Zbl

[Si] J. Silverman, The arithmetic of elliptic curves, Springer Graduate Texts in Math., 106 (1986). | MR | Zbl

[Ta1] J. Tate, Endomorphisms of abelian varieties over finite fields., Inv. Math., 2 (1966), 134-144. | EuDML | MR | Zbl

[Ta2] J. Tate, Conjectures on algebraic cycles in l-adic cohomology. In : Motives, Proc. of Symposia in Pure Math., 55, Part 1 (1994), 71-83. | MR | Zbl

[Ve] J-L. Verdier, Classe d'homologie associée à un cycle, Astérisque, 36-37 (1976), 101-151. | Zbl

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