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Milnor \(K\)-groups and zero-cycles on products of curves over \(p\)-adic fields. (English) Zbl 0985.14003
The authors investigate the Chow groups of products of curves over a \(p\)-adic field \(k\). They show that for smooth, projective, geometrically connected curves, \(X_i\), with \(X_i(k)\neq 0\) whose Jacobians, \(J_i\), have a mixture of good ordinary and split multiplicative reduction (which the authors call semi-ordinary) the Chow groups have the property that \(\text{CH}_0(X_i\times\dots \times X_d)/m\) is finite for any \(m\). The method of proof is to express \(\text{CH}_0(X_i\times\dots \times X_d)\) in terms of the Milnor \(K\)-groups introduced by M. Somekawa [\(K\)-theory 4, 105–119 (1990; Zbl 0721.14003)]. These can then be bounded using the usual Milnor \(K\)-groups of \(k\) following B. Kahn [C. R. Acad. Sci. Paris, Sér. I 314, 1039–1042 (1992; Zbl 0785.14027)].

MSC:
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14G20 Local ground fields in algebraic geometry
14C15 (Equivariant) Chow groups and rings; motives
11G55 Polylogarithms and relations with \(K\)-theory
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
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