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

Abstract

In this paper we show that the mapis surjective, where E 1 and E 2 are two non-isogenous semistable elliptic curves over a local field, CH 2 (E 1 × E 2 , 1) is one of Bloch's higher Chow groups and P CH 1 (X v ) is a certain subquotient of a Chow group of the special fibre X v of a semi-stable model X of E 1 × E 2 . On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-Dconjecture of Beilinson -which is known to be true in this case by the work of Chen and Lewis (J. the case when the elliptic curves have split multiplicative reduction.

Higher Chow cycles on Abelian surfaces and a non-Archimedean analogue of the Hodge--conjecture

“…This theorem can be extended to work in the case where A has semi-stable reduction. A special case was studied in [Sre08].…”

“…In fact, one can formulate this conjecture more generally for primes p of semi-stable reduction [Sre08], but in this paper we will only deal with primes of good reduction. We can also consider this conjecture for X over a global field itself -looking at the boundary map for all primes -but that is a much harder question.…”

“…• m = 2, n = 1 and X = E 1 × E 2 where E 1 and E 2 are elliptic curves over a local field K p with bad semi-stable reduction at p. In [Sre08] we generalized Spiess's work to the case of semi-stable reduction. Here we had to use a semi-stable model X of E 1 × E 2 .…”

K1of products of Drinfeld modular curves and special values ofL-functions

“…In [Sre08], we formulated a function field analogue of the Beilinson conjectures. In particular we defined a group which, at a finite place, plays the role of the real Deligne cohomology.…”

“…In [Sre08] we defined a regulator map r D,ν from the motivic cohomology to the ν-adic Deligne cohomology and, in analogy with the Beilinson conjectures, conjectured that the image is a full lattice. Finally, in some cases, we made a conjecture on the special value of the L-function.…”