\(K_ 2\) of a Fermat quotient and the value of its \(L\)-function
Let \(C : x^5 + y^5 = 1\) be the Fermat curve of exponent one. One has an action of \(\mathbb{Z}/5 \mathbb{Z}\) on \(C\) by \((x,y) \mapsto (\zeta^k x, \zeta^{-k} y)\), where \(k \in \mathbb{Z}/5 \mathbb{Z}\) and \(\zeta\) is a fixed (nontrivial) fifth root of unity. Consider the quotient curve \(X\) of \(C\) by this action, and write \(\pi : C \to X\) for the projection. Then \(X\) is a nonsingular curve of genus 2 and, as an affine curve, it can be given by the equation \(X : w^5 = u(1 - u)\), where \(w = xy\) and \(u = x^5\). It is known that \(K_2 X\) is generated by symbols \(\{\;,\;\}\). Write \(\text{reg}_X : K_2 X \to H^1 (X (\mathbb{C}), 2 \pi i \mathbb{R})^+\) for the regulator map, where the superscript + denotes the invariant subspace under complex conjugation of both \(X(\mathbb{C})\) and \(2 \pi i \mathbb{R}\). It has (real) dimension two, the genus of \(X\). Beilinson's conjecture states: (i) \(\text{reg}_X (K_2 X)\) is a lattice in \(H^1 (X (\mathbb{C}), 2 \pi i \mathbb{R})^+\); (ii) \(\text{det(reg}_X (K_2 X)_\mathbb{Q})\) is equal to the first non-zero coefficient of \(L(H^1 (X), s)\) at \(s = 0\) up to a non-zero rational number. Here \(\text{ord}_{s = 0} L(H^1(X),s) = 2\). \textit{R. Ross} [C. R. Acad. Sci., Paris, Sér. I 312, No. 1, 1-5 (1991; Zbl 0744.14006)] has shown that the element \(\alpha = \{1 - w,u\} \in K_2 X\) has nontritival image under the regulator map. Here an explicit element \(\beta \in K_2 X\), linearly independent of \(\alpha\), is presented and it is shown, actually by an astute numerical integration procedure, that the determinant of the matrix of the integrals of \(\pi^* \text{reg} \alpha\) and \(\pi^* \text{reg} \beta\) along two homologically independent 1-cycles on \(C\) does not vanish. One takes \(\beta = \pi_* \{x + y, {1 - x \over y}\}\). On the other hand, by a careful treatment of theta series, a computation of \(L(H^1 (X), 2)\) (which by the functional equation is related to \(L^{(2)} (H^1 (X), 0))\) is explained. Putting everything together leads to numerical evidence for Beilinson's conjecture.
numerical evidence for Beilinson's conjecture
scientific article; zbMATH DE number 847545
scientific article