Isogeny graphs with maximal real multiplication
Throughout, let \(K\) be a primitive CM-field of degree \(4\) with maximal totally real subfield \(K_0\), and let \(\mathcal{O}_K\) (resp. \(\mathcal{O}_{K_0}\)) be the ring of integers of \(K\) (resp. \(K_0\)). Assume further that \(\mathcal{O}_{K_0}\) is principal and \(K_0\) has class number one. Also, let \(l\) be a prime number and \(\mathfrak{l}\) a prime ideal of \(\mathcal{O}_{K_0}\) of norm \(l\), that is generated by a totally positive algebraic integer. Finally, let \(\mathcal{O}\) be an order in \(K,\) and let \(A\) be a principally polarizable abelian surface over a perfect field \(F\) with complex multiplication by \(\mathcal{O}.\) Then one of the main results of the paper under review describes the structure of the \(\mathfrak{l}\)-isogeny graph of principally polarized abelian surfaces (involving \(A\)) for the case when \(F=\mathbb{C}, K \ne \mathbb{Q}(\zeta_5),\) and \(\mathcal{O}\) has locally maximal real multiplication at \(\mathfrak{l}\) (see Proposition 21, from which we deduce that the structure of a connected component of the \(\mathfrak{l}\)-isogeny graph is precisely that of an \(l\)-isogeny graph for elliptic curves, called volcano (see e.g., [\textit{M. Fouquet} and \textit{F. Morain}, Lect. Notes Comput. Sci. 2369, 276--291 (2002; Zbl 1058.11041)]). The description is two-fold and depends on whether \(\mathfrak{l}\) is prime to the conductor \(\mathfrak{f}\) of \(\mathcal{O}\) or not, and if \(\mathfrak{l}\) is prime to \(\mathfrak{f}\), then there are three types of graphs that can occur, depending on the ramification behavior of \(\mathfrak{l}\) in \(\mathcal{O}_K.\) (More precisely, if \(\mathfrak{l}\) splits in \(\mathcal{O}_K\) (ramifies, is inert, respectively), then up to isomorphism, there are exactly two (one, zero, respectively) horizontal \(\mathfrak{l}\)-isogenies starting from \(A\), and all others are descending.) For another interesting result of the paper under review, we also consider the case of the base field being a finite field. Namely, let \(A\) be an ordinary principally polarized abelian surface over a finite field \(\mathbb{F}_q\) (where \(q\) is a prime power) such that \(\mathbb{Z}[\pi, \overline{\pi}] \subset \text{End}(A) \subset \mathcal{O}_K\), where \(\pi\) is a \(q\)-Weil number corresponding to the Frobenius endomorphism of \(A,\) and such that \(\text{End}(A)\) has locally maximal real multiplication at \(\mathfrak{l}.\) Then the second main result of the paper exhibits the structure of the \(\mathfrak{l}\)-isogeny graph in this case in the sense that there is a graph isomorphism between an \(\mathfrak{l}\)-isogeny graph for ordinary principally polarized abelian surfaces over finite fields whose endomorphism ring shares the properties of \(\text{End}(A)\) (\(A\) as above) and a graph whose vertices are isomorphism classes of principally polarized abelian surfaces over a number field (with certain further properties). (For more precise statements, see Corollary 27.) Furthermore, from the two aforementioned results, the authors could explain the structure of connected components of the non-oriented \(\mathfrak{l}\)-isogeny graph over a finite field \(\mathbb{F}_q\) (\(q=p^a\) for some prime \(p\) and an integer \(a \geq 1\)) with the property that \(p\) does not divide the conductor of the order \(\mathbb{Z}[\pi, \overline{\pi}].\) Finally, at the end of the paper, the authors provide a depth-first algorithm for computing the endomorphism ring of the Jacobian variety of a curve of genus \(2\) over a finite field under suitable assumptions, and give complexity analysis, together with practical experiments for the Jacobian variety of a hyperelliptic curve over the prime field \(\mathbb{F}_{p}\) with \(p=53050573\).
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