A finer Tate duality theorem for local Galois symbols

Abstract

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.

Galois symbol maps for abelian varieties over a $p$-adic field

“…which is called the Albanese kernel is also written by the Somekawa K-group as 14]). From the same computation as in [21], Theorem 4.1, we recover [3], Corollary 8.9.…”

Algebraic geometry in mixed characteristic

“…When V is crystalline, Theorem 2.18 may be viewed as an integral and cochain-level variant of this statement. Such integral refinements have been formulated previously in special cases (e.g., [102]); it would be interesting to compare them to Theorem 2.18.…”

Weak Approximation for $0$-cycles on a product of elliptic curves

“…, where E k is the Néron model of E k over Spec(O k ). Since E k has good ordinary reduction, it follows by [Gaz18,Proposition 8.8] that these are exactly the homomorphisms…”