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

Benedict H. Gross; Chad Schoen

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 3, page 649-679
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Gross, Benedict H., and Schoen, Chad. "The modified diagonal cycle on the triple product of a pointed curve." Annales de l'institut Fourier 45.3 (1995): 649-679. <http://eudml.org/doc/75133>.

@article{Gross1995,
abstract = {Let $X$ be a curve over a field $k$ with a rational point $e$. We define a canonical cycle $\Delta _e\in Z^2(X^3)_\{\{\rm hom\}\}$. 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 $\langle \tau _*(\Delta _e),\tau ^\{\prime \}_*(\Delta _e)\rangle $ is well defined where $\tau $ and $\tau ^\{\prime \}$ are correspondences. The paper ends with a brief discussion of heights and $L$-functions in the case that $X$ is a modular curve.},
author = {Gross, Benedict H., Schoen, Chad},
journal = {Annales de l'institut Fourier},
keywords = {diagonal cycle; triple product of a pointed curve; regular models; semi-stable reduction; height pairing; -functions; modular curve},
language = {eng},
number = {3},
pages = {649-679},
publisher = {Association des Annales de l'Institut Fourier},
title = {The modified diagonal cycle on the triple product of a pointed curve},
url = {http://eudml.org/doc/75133},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Gross, Benedict H.
AU - Schoen, Chad
TI - The modified diagonal cycle on the triple product of a pointed curve
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 3
SP - 649
EP - 679
AB - Let $X$ be a curve over a field $k$ with a rational point $e$. We define a canonical cycle $\Delta _e\in Z^2(X^3)_{{\rm hom}}$. 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 $\langle \tau _*(\Delta _e),\tau ^{\prime }_*(\Delta _e)\rangle $ is well defined where $\tau $ and $\tau ^{\prime }$ are correspondences. The paper ends with a brief discussion of heights and $L$-functions in the case that $X$ is a modular curve.
LA - eng
KW - diagonal cycle; triple product of a pointed curve; regular models; semi-stable reduction; height pairing; -functions; modular curve
UR - http://eudml.org/doc/75133
ER -

References

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