Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



A non-Archimedean analogue of the Hodge-$\mathcal {D}$-conjecture for products of elliptic curves

Author: Ramesh Sreekantan
Journal: J. Algebraic Geom. 17 (2008), 781-798
Published electronically: March 4, 2008
MathSciNet review: 2424927
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Abstract: In this paper we show that the map \[ \partial :CH^2(E_1 \times E_2,1)\otimes \mathbb {Q} \longrightarrow PCH^1(\mathcal {X}_v)\] is surjective, where $E_1$ and $E_2$ are two non-isogenous semistable elliptic curves over a local field, $CH^2(E_1 \times E_2,1)$ is one of Bloch’s higher Chow groups and $PCH^1(\mathcal {X}_v)$ is a certain subquotient of a Chow group of the special fibre $\mathcal {X}_{v}$ of a semi-stable model $\mathcal {X}$ of $E_1 \times E_2$. On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-$\mathcal {D}$-conjecture of Beilinson - which is known to be true in this case by the work of Chen and Lewis (J. Algebraic Geom. 14 (2005), 213–240), and on the other, an analogue of the works of Speiß ($K$-Theory 17 (1999), 363–383), Mildenhall (Duke Math. J. 67 (1992), 387–406) and Flach (Invent. Math. 109 (1992), 307–327) in the case when the elliptic curves have split multiplicative reduction.

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Ramesh Sreekantan
Affiliation: School of Mathematics, Tata Insitute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai, 400 005 India
Address at time of publication: TIFR Centre for Applicable Mathematics, P.O. Bag No 03, Sharadanagar, Chikkabommasundara, Bangalore, 560 065 India

Received by editor(s): October 14, 2006
Received by editor(s) in revised form: January 22, 2007
Published electronically: March 4, 2008