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

DOI:
https://doi.org/10.1090/S1056-3911-08-00477-3

Published electronically:
March 4, 2008

MathSciNet review:
2424927

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Abstract | References | Additional Information

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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Additional Information

**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

Email:
ramesh@math.tifr.res.in

Received by editor(s):
October 14, 2006

Received by editor(s) in revised form:
January 22, 2007

Published electronically:
March 4, 2008