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Schertz style class invariants for quart...
Enge, Andreas...

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Schertz style class invariants for quartic CM fields by Enge, Andreas ( Author )
Lemonjar Open Access
Australian National University
10-08-2023
Values of Siegel modular functions for $\operatorname{Sp}_4 (\mathbb Z)$ in CM period matrices generate certain unramified abelian class fields of quartic CM fields, and they yield invariants of principally polarised abelian surfaces with a known endomorphism ring. Smaller alternative class invariants, values of modular functions for subgroups generating the same class fields, thus help to speed up constructions in explicit class field theory and public-key cryptography. Generalising results due to Schertz from elliptic curves to abelian surfaces and from classical modular functions to Siegel modular functions, we show that modular functions for the congruence subgroup $\Gamma^0 (N)$ yield class invariants under some splitting conditions on~$N$. We show how to obtain all Galois conjugates of a class invariant by evaluating the same modular function in CM period matrices derived from an $N$-system. Such a system consists of binary quadratic forms with coefficients in the real-quadratic subfield satisfying certain congruence conditions modulo~$N$. We examine conditions under which the minimal polynomial of a class invariant is real. Examples show that we may obtain class invariants that are much smaller than in previous constructions.
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Article
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30.00 KB
English
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MYR 0.01
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http://arxiv.org/abs/1610.04505
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