A new look at Jarvis’ distribution formula. (English) Zbl 1190.19003
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)], 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 K. Rolshausen and 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.
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.
Reviewer: Keith Johnson (Halifax)
MSC:
| 19F15 | Symbols and arithmetic (\(K\)-theoretic aspects) |
| 19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |
| 11G16 | Elliptic and modular units |
| 33E05 | Elliptic functions and integrals |
Keywords:
Siegel functions; Klein functions; distribution formula; elliptic curves; algebraic K-theoryReferences:
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