Height pairing between algebraic cycles
Height pairing between algebraic cycles
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代数循环之间的高度配对
DOI：
10.1007/bfb0078364
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
A. Beilinson
中科院分区：
文献类型：
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作者：
A. Beilinson
(X)«（）. Birth和Swinnerton-Dyer的猜想声称，在s= 1时，函数L（H1（X），s）具有rk阶的零CH 1（X）O，其首系数等于Neron-Tate正则高度parring的行列式乘以周期矩阵行列式直到某个有理倍数（在下文中我们不需要它的精确值）。对于其它的LL函数，Swinnerton-Dyer证明了函数L（H2 i-1（X），s）在s= i（=临界带的中间）处有阶为rkCHi（X）0的零点.本文的目的是定义CHi（X）0和CHdimX+ li（X）0之间的正则高度配对，它与i= 1时的Neron-Tate高度配对一致，并且其行列式乘以周期矩阵的行列式应该在理论上等于L（H2 i-1（X），s）在s= i*）的前导的有理多项式（我无法想象的性质）。这种配对也应该出现在Riemann-Roch型定理中，如Arakelov-Faltings（见[15]）。事实上，我们的高度配对是在CH'IX}的某个子群上定义的;在非常合理的（=证据等级远高于B-SwD）局部假设下，这个子群应该与整个CH'（X）o重合。
(X)«(). The conjecture of Birth and SWinnerton-Dyer claims that at s= 1 the function L (H1 (X), s) has zero of order rk CH1 (X) Owith the leading coefficient equal to the determinant of Neron-Tate canonical height parring multiplied by the period matrix determinant up to some rational multiple (we do not need its exact value in what follows). As for the other LLfunctions, Swinnerton-Dyer conjectured[20J that the function L (H2i-l (X), s) has at s= i (= the middle of the cri tical strip) the zero of order rk CHi (X) o. The aim of this note is to define the canonical height pairing between CHi (X) O and CHdimX+ li (X) o that coincides with the Neron-Tate one for i= l and whose determinant multiplied by the period matrix determinant should conjecturally be equal up to a rational mUltiple (of the nature I cannot imagine) to the leading of L (H2i-1 (X), s) at s= i*). This pairing should also occure in Riemann-Roch type theorems a la Arakelov-Faltings (see [15]).*) e In fact our height pairing is defined on a certain subgroup of CH'IX}; under the very plausible (:= of rank of evidence far higher then B-SwD) local conjectures this sUbgroups should coincide with the whole CH'(X) o.