Triangle configurations, and Beilinson's conjecture for of the product of a curve with itself

Abstract

The aim of this thesis is to look into Beilinson's conjecture on the rank of the integral part of certain algebraic -groups of varieties over number fields, as applied to where is a (smooth projective) curve. In particular, it examines whether non-zero integral elements can be obtained from linear combinations of certain special types of elements which I refer to as ``triangle'' configurations. Most of the thesis examines the special case where is an elliptic curve.

The main result is that whenever any rational linear combination of such triangle configurations lies in the integral part of , then its image under the Beilinson regulator map is the same as that of a ``decomposable'' integral element, which is to say, one consisting only of constant functions along various curves. Hence, if Beilinson's conjecture is correct and the regulator is injective on the integral part, then no previously unknown integral elements can be produced from these triangle constructions.

I will also examine the same question for some slighly more general elements of , and will show that (subject to one conjecture, which seems highly likely to be true, although I have been unable to prove it rigorously) the same result holds, provided that we restrict ourselves to an individual ``triangle'', as opposed to arbitrary linear combinations. This will follow from conditions on such a triangle which are both necessary (always) and sufficient (at least for certain special types of elliptic curve) for integrality.