Generalised Weber functions

Andreas Enge¹ ; François Morain²

¹ INRIA, LFANT Univ. Bordeaux, IMB CNRS, IMB, UMR 5251 33400 Talence, France
² INRIA Saclay–Île-de-France and LIX (CNRS/UMR 7161) École polytechnique 91128 Palaiseau Cedex, France

Acta Arithmetica, Tome 164 (2014) no. 4, pp. 309-341

Résumé

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.

Détail

DOI : 10.4064/aa164-4-1

Keywords: generalised weber function given mathfrak eta eta where eta dedekind function integer original function corresponds classify cases where power mathfrak evaluated quadratic integer generates ring class field associated order imaginary quadratic field compare heights invariants giving general formula degree modular equation relating mathfrak ultimate these invariants constructing reductions elliptic curves finite fields suitable cryptographic