A non-Archimedean analogue of the Hodge-D-conjecture for products of elliptic curves

In this paper we show that the map % \partial:CH^2(E_1 \times E_2,1)\otimes \Q \longrightarrow PCH^1(\XX_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(\XX_v) is a certain subquotient of a Chow group of the special fibre \XX_{v} of a semi-stable model \XX of $E_1 \times E_2$. On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-\D-conjecture of Beilinson - which is known to be true in this case by the work of Chen and Lewis \cite{lech}, and on the other, an analogue of the works of Spei{\ss} \cite{spie}, Mildenhall \cite{mild} and Flach \cite{flac} in the case when the elliptic curves have split multiplicative reduction.Comment: 13 pages. To appear in the Journal of Algebraic Geometr

Higher order modular forms and mixed Hodge theory

In this paper we introduce a certain space of higher order modular forms of weight 0 and show that it has a Hodge structure coming from the geometry of the fundamental group of a modular curve. This generalizes the usual structure on classical weight 2 forms coming from the cohomology of the modular curve. Further we construct some higher order Poincare series to get higher order higher weight forms and using them we define a space of higher weight, higher order forms which has a mixed Hodge structure as well.Comment: 26 pages. To appear in Acta Arithmetica. New version corrects issues with text being truncate

Old and new motivic cycles on Abelian surfaces

Collino \cite{colo} discovered indecomposable motivic cycles in the group H^{2g-1}_{\mathcal M}(J(C),{\mathds Z}(g)). In an earlier paper we described the construction of some new motivic cycles which can be viewed as a generalization of Collino's cycle when $g=2$. In this paper we show that our new cycles are in fact related to Collino's cycles of higher genus. On one hand this suggests that new cycles are hard to find. On the other, it suggests that the tools developed to study Collino's cycle can be applied to our cycles.Comment: 13 page

Searches for proton decay and superheavy magnetic monopoles

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