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Generalised Weber functions

Andreas EngeFrançois Morain — 2014

Acta Arithmetica

A generalised Weber function is given by N ( z ) = η ( z / N ) / η ( z ) , where η(z) is the Dedekind function and N is any integer; the original function corresponds to N=2. We classify the cases where some power 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 N ( z ) and j(z). Our ultimate goal is the use of these invariants in constructing...

Constructing elliptic curves over finite fields using double eta-quotients

Andreas EngeReinhard Schertz — 2004

Journal de Théorie des Nombres de Bordeaux

We examine a class of modular functions for Γ 0 ( N ) whose values generate ring class fields of imaginary quadratic orders. This fact leads to a new algorithm for constructing elliptic curves with complex multiplication. The difficulties arising when the genus of X 0 ( N ) is not zero are overcome by computing certain modular polynomials. Being a product of four η -functions, the proposed modular functions can be viewed as a natural generalisation of the functions examined by Weber and usually employed...

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