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The modified diagonal cycle on the triple product of a pointed curve. (English) Zbl 0822.14015

Summary: Let \(X\) be a curve over a field \(k\) with a rational point \(e\). We define \(\Delta _ e\in Z^ 2(X^ 3)_{\text{hom}}\), a canonical cycle. 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 \(\langle \tau _ *(\Delta _ e),\tau '_ *(\Delta _ e)\rangle \) is well defined where \(\tau \) and \(\tau '\) are correspondences. The paper ends with a brief discussion of heights and \(L\)-functions in the case that \(X\) is a modular curve.

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14G05 Rational points
14H25 Arithmetic ground fields for curves
14G35 Modular and Shimura varieties
14C25 Algebraic cycles
11G18 Arithmetic aspects of modular and Shimura varieties
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