Higher Chow cycles on Abelian surfaces and a non-Archimedean analogue of the Hodge--conjecture

Abstract

We construct new indecomposable elements in the higher Chow group CH 2 (A, 1) of a principally polarized Abelian surface over a p-adic local field, which generalize an element constructed by Collino [Griffiths' infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393-415]. These elements are constructed using a generalization, due to Birkenhake and Wilhelm [Humbert surfaces and the Kummer plane, Trans. Amer. Math. Soc. 355 (2003), 1819-1841 (electronic)], of a classical construction of Humbert. They can be used to prove a non-Archimedean analogue of the Hodge-D-conjecture -namely, the surjectivity of the boundary map in the localization sequence -in the case where the Abelian surface has good and ordinary reduction.

Higher Chow cycles on a family of Kummer surfaces

“…Our construction uses the realization of the Kummer surfaces as the desingularized double coverings of P 1 × P 1 , and our construction of higher Chow cycles uses special types of (1, 1)-curves on P 1 × P 1 . This is similar to the construction of higher Chow cycles on abelian surfaces in [Sre14]. The fact that CH 2 ( X t , 1) ind 0 already follows from the result of Section 6 of [CDKL16] because our family is a base change of the family considered there.…”