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Hilbert modular polynomials. (English) Zbl 1442.11171
This paper proposes a generalization of the classical $$l$$-modular polynomial for elliptic curves, the set of Hilbert modular polynomials, for principally polarized abelian varieties of dimension $$g$$ with maximal real multiplication for a totally real field $$K_0$$. It also provides an algorithm to compute them.
The $$l$$-modular polynomial for elliptic curves parametrises $$l$$-isogenies. Analogously the set of Hilbert modular polynomials are related with isogenies of cyclic kernel, the $$\mu$$-isogenies (Definition 2.2), with $$\mu\in K_0$$ a totally positive prime.
Section 1 gives an summary of the proposal and states the main result (Theorem 1.5, proved in Section 6), concerning the existence of Hilbert modular polynomials. Section 2 summarizes the notions of maximal real multiplication and Hilbert modular forms.
Section 3 studies the existence of RM isomorphism invariants (Proposition 3.1). Section 4 gives an if and only if condition for the existence of a $$\mu$$-isogeny (Proposition 4.5). Section 5 considers the computation of RM isomorphism invariants in the case $$g=2$$.
Section 6 gives an algorithm to compute a set of Hilbert polynomials (Algorithm 6.3) and provides the proof of Theorem 1.5. In the case of surfaces ($$g=2$$) Section 7 presents improvements of the algorithm (Algorithm 7.8) and details of an implementation in MAGMA for $$K_0= Q(\sqrt 5)$$ and $$K_0= Q(\sqrt 2)$$.
Finally Section 8 shows three possible applications: point counting for curves of genus two with maximal real multiplication, see [S. Ballentine et al., Assoc. Women Math. Ser. 9, 63–94 (2017; Zbl 1414.11076)], walking on isogeny graphs for genus 2 curves, see [A. Dudeanu et al., “Cyclic isogenies for abelian varieties with real multiplication”, Preprint, arXiv:1710.05147] and finally computing Hilbert class polynomials to genus 2, generalizing a method of A. V. Sutherland [Math. Comput. 80, No. 273, 501–538 (2011; Zbl 1231.11144)].
##### MSC:
 11Y16 Number-theoretic algorithms; complexity 11G10 Abelian varieties of dimension $$> 1$$ 11G15 Complex multiplication and moduli of abelian varieties 14K02 Isogeny
Echidna; Magma
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