Generalised Weber functions
Tom 164 / 2014
Acta Arithmetica 164 (2014), 309-341 MSC: 11G15, 14K22. DOI: 10.4064/aa164-4-1
A generalised Weber function is given by $\mathfrak w_N(z) = \eta (z/N)/\eta (z)$, where $\eta (z)$ is the Dedekind function and $N$ is any integer; the original function corresponds to $N=2$. We classify the cases where some power $\mathfrak w_N^e$ evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating $\mathfrak w_N(z)$ and $j(z)$. Our ultimate goal is the use of these invariants in constructing reductions of elliptic curves over finite fields suitable for cryptographic use.