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Class invariants for quartic CM fields. (English) Zbl 1172.11018
The article is concerned with explicit bounds on the primes appearing in the denominators of class invariants of primitive quartic CM fields. The construction of such bounds have significance in relation with different arithmetic-geometric problems: computation of Hilbert class polynomials, determination of abelian surfaces with complex multiplication... Examples of such computations have been already performed, but the results were not completely checked, due to the lack of such bounds. It remains yet to bound the exponents of the primes in the denominator; partial advance in this direction is announced in the paper.
Given a quartic primitive CM field \(K\) and a prime number \(p\), the authors relate an embedding problem (the existence of a certain embedding of \(K\) into the endomorphism ring of a product of two supersingular elliptic curves over \(\mathbb F_p\)), with the reduction at \(p\) of stable genus 2 curves with CM by the ring of integers of \(K\). A bound on the primes for which the embedding problem is solvable in terms of \(K\) provides the bound on the primes appearing in the denominators of the Igusa invariants for genus 2 curves with CM by \(K\). The methods to prove these results are of geometric nature.
Besides the natural application of the results to the construction of CM genus 2 curves, the authors also consider the computation of the class invariants \(u(\Phi,\mathfrak{a},\mathfrak{b})\) associated to ideals \(\mathfrak{a},\mathfrak{b}\) and CM-types \(\Phi\) of \(K\), giving rise to certain units of \(K\). The paper includes some numerical examples.

11G15 Complex multiplication and moduli of abelian varieties
11G16 Elliptic and modular units
11G18 Arithmetic aspects of modular and Shimura varieties
11R27 Units and factorization
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