$\textit{K}$-Theory of Fermat Curves

Abstract

I investigate the $K_2$ groups of the quotients of Fermat curves given in projective coordinates by the equation $F_n:X^n+Y^n=Z^n$. On any quotient where the number of known elements is equal to the rank predicted by Beilinson’s Conjecture I verify numerically that the determinant of the matrix of regulator values agrees with the leading coefficient of the L-function up to a simple rational number. The main source of $K_2$ elements are the so-called “symbols with divisorial support at infinity” that were found by Ross in the 1990’s. These consist of symbols of the form {$\textit{f, g}$} where $\textit{f}$ and $\textit{g}$ have divisors whose points $\textit{P}$ all satisfy $\textit{XY Z(P)}$ = 0. The image of this subgroup under the regulator is computed and is found to be of rank predicted by Beilinson’s Conjecture on eleven nonisomorphic quotients of dimension greater than one. The $\textit{L}$-functions of these quotients are computed using Dokchitser’s ComputeL package and Beilinson’s Conjecture is verified numerically to a precision of 200 decimal digits. In chapter five, with careful analysis of a certain 2 × 2 determinant it is shown that a particular hyperelliptic quotient of all the Fermat curves has $K_2$ group of rank at least two. In the last chapter of the dissertation, a computational method is used in order to discover new elements of $K_2$. These elements are rigorously proven to be tame and allow for the full verification of Beilinson’s Conjecture on the Fermat curves $F_7$ and $F_9$. Also the method allows us to verify Beilinson’s Conjecture on certain hyperelliptic quotients of $F_8$ and $F_{10}$. Quotients where a similar method might be successful are also suggested.