A new look at Jarvis' distribution formula
In two papers [Bull. Lond. Math. Soc. 32, No. 2, 146--154 (2000; Zbl 1024.11042), Manuscr. Math. 103, No. 3, 329--337 (2000; Zbl 1024.11041)], \textit{F. Jarvis} proved that if \(L\) and \(\Lambda\) are lattices in \({\mathbb C}\), \(\psi: E = {\mathbb C}/L \to {\mathbb C}/\Lambda=F\) an isogeny between the corresponding elliptic curves, \(D\) a divisor satisfying certain conditions and \(\phi_{\Lambda}\), \(\phi_L\) the Siegel functions associated to the lattices \(\Lambda\) and \(L\) then the following distribution formula holds: \[ \phi_\Lambda(\psi(D)) = \prod_{S \in \ker(\psi)} \phi_L(D \oplus S) \] This result played an essential role in the construction of elements in the second \(K\)-group of an elliptic curve given by \textit{K. Rolshausen} and \textit{N. Schappacher} in [J. Reine Angew. Math. 495, 61--77 (1998; Zbl 0887.11027)]. In the paper under review the author gives a proof of a general distribution formula for Siegel functions of divisors on \({\mathbb C}\) which implies Jarvis' result when one descends to elliptic curves. The main ingredient in the proof is the distribution formula for Klein functions.
scientific article; zbMATH DE number 5554699