2014
DOI: 10.1112/s0010437x13007690
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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 cl… Show more

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“…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.…”
Section: K Satosupporting
confidence: 55%
“…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.…”
Section: K Satosupporting
confidence: 55%