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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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