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Triangle configurations, and Beilinson's conjecture for $K_{1}^{(2)}$ of the product of a curve with itself

ZIGMOND, ROBIN,JAMES (2010) Triangle configurations, and Beilinson's conjecture for $K_{1}^{(2)}$ of the product of a curve with itself. Doctoral thesis, Durham University.



The aim of this thesis is to look into Beilinson's conjecture on the rank of the integral part of certain algebraic $K$-groups of varieties over number fields, as applied to $K_{1}^{(2)}(C\times C)$ where $C$ 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 $C$ is an elliptic curve.

The main result is that whenever any rational linear combination of such triangle configurations lies in the integral part of $K_{1}^{(2)}(E\times E)$, 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 $K_{1}^{(2)}(E\times E)$, 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.

Item Type:Thesis (Doctoral)
Award:Doctor of Philosophy
Faculty and Department:Faculty of Science > Mathematical Sciences, Department of
Thesis Date:2010
Copyright:Copyright of this thesis is held by the author
Deposited On:24 May 2010 10:51

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