On the L-Function of Some Kummer Curves with Complex Multiplication

Florin Nicolae

MSC 2000

11G40 $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture

11G15 Complex multiplication and moduli of abelian varieties

Abstract
Let l be a prime number. For $n \ge 1$ the L-function of the Kummer Curve $C_n :y^l = x(x^{l^n} - 1)$ over the cyclotomic field $\Bbb{Q} (\zeta_{l^{n+1}}, \zeta_{l^{n+1}} := {\rm exp} \frac{2\pi i}{l^{n+1}}$, is a product of $2g_n = l^n (l-1)$ Hecke L-fucntions with Jacobi sums Grössencharaktere. The jacobian variety of $C_n$ over the field of complex numbers is a simple abelian variety with complex multiplication ring of endomorphisms isomorphic to the ring of integers $\Bbb{Z} [\zeta_{l^{n+1}}]$.