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
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

$\displaystyle \partial:CH^2(E_1 \times E_2,1)\otimes \mathbb{Q} \longrightarrow PCH^1(\mathcal{X}_v)$

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

DOI: https://doi.org/10.1090/S1056-3911-08-00477-3
Received by editor(s): October 14, 2006
Received by editor(s) in revised form: January 22, 2007
Published electronically: March 4, 2008

American Mathematical Society