Congruences between Bernoulli-Hurwitz numbers in the supersingular case
Let \(K\) be a complex quadratic field with integers \(\mathcal O_K\). Let \(H\) be the Hilbert class field of \(K\) and \(E\) an elliptic curve over \(H\) with complex multiplication by \(\mathcal O_K\). Let \(\wp (z)\) be the Weierstrass \(\wp\)-function of \(E\). Let \(n\) be an even positive integer which is also a multiple of the number of roots of unity in \(K\). Then, one defines the Bernoulli-Hurwitz numbers, \(\text{BH}(n)\in H\), by \[ \wp (z)=\frac{1}{z^2}+\sum_{n\geq 2}\frac{\text{BH}(n+2)}{n+2}\frac{z^n}{n}. \] Let \(p\) be a prime \(\geq 3\) which remains inert in \(K\) and suppose that the model of \(E\) attached to \(\wp (z)\) has good reduction above \(p\). Upon multiplying \(\text{BH}(n)\) by certain factors, N. Katz was able to establish congruences of Kubota-Leopoldt type. In the present paper, the author gives a refinement of these congruences.
scientific article; zbMATH DE number 4156519