On the syntomic regulator for products of elliptic curves

Abstract

We consider the syntomic regulator on the integral motivic cohomology of a smooth proper surface over a p-adic field and apply a recent formula of Besser that uses p-adic integration theory, in particular his theory of triple indices on Coleman integrals, to the case of a self-product of an elliptic curve. The method is suitable to separate decomposable from indecomposable elements in the (integral) motivic cohomology. As an interesting example, we construct an element that, though not given in decomposable form, becomes decomposable after taking p-adic completion.