Let A be an abelian variety defined over a field k. In this paper we define a descending filtration {F r } r 0 of the group CH 0 (A) and prove that the successive quotientswhere K(k; A, . . . , A) is the Somekawa K-group attached to r-copies of the abelian variety A. In the special case when k is a finite extension of Q p and A has split multiplicative reduction, we compute the kernel of the map CH 0 (A)⊗Z[ 1 2 ] → Hom(Br(A), Q/Z)⊗Z[ 1 2 ], induced by the pairing CH 0 (A) × Br(A) → Q/Z.
Let K be a finite extension of Qp. Let A, B be abelian varieties over K with good reduction. For any integer m ≥ 1, we consider the Galois symbol K(K; A, B)/m →In this paper we give a geometric description of the image of this generalized Galois symbol by looking at the Tate duality pairingUnder this perfect pairing we compute the exact annihilator of the image of the Galois symbol in terms of an object of integral p-adic Hodge theory. In this way we generalize a result of Tate for H 1 . Moreover, our result has applications to zero cycles on abelian varieties.
For a reciprocity functor M we consider the local symbol complexwhere C is a smooth complete curve over an algebraically closed field k with generic point η C and ⊗ M is the product of Mackey functors. We prove that if M satisfies certain assumptions, then the homology of this complex is isomorphic to the K-group of reciprocity functors T (M, CH 0 (C) 0 )(Spec k).(n)Acknowledgement: I would like to express my great gratitude to my advisor, Professor Kazuya Kato, for his kind guidance throughout the process of obtaining the current result.
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