Spanning the isogeny class of a power of an elliptic curve. (English) Zbl 1486.14046
Let \(g, m \geq 1\) be integers, \(p\) be a prime, \(q = p^m\) and \(\mathcal{W}\) (resp. \(\mathcal{W}_1\)) be a set of the \(\mathbb{F}_q\)-isomorphism classes of (resp. principally polarized) abelian varieties isogenous to a given \(g\)-dimensional abelian variety \(A_0\) over \(\mathbb{F}_q\). In general, it is difficult to give an explicit description of \(\mathcal{W}\) and \(\mathcal{W}_1\) and only partial results have been obtained so far. In the article under review the authors present an algorithm to obtain, under some technical assumptions, such a description for \(A_0 = E^g\), where \(E\) is an ordinary elliptic curve defined over \(\mathbb{F}_q\). Using the functorial relations developed in [B. W. Jordan et al., Compos. Math. 154, No. 5, 934–959 (2018; Zbl 1400.14116)] they show that elements in \(\mathcal{W}_1\) are in the correspondence with the unimodular positive definite hermitian \(R\)-lattices \((L,h)\) of rank \(g\), where \(R = \mathbb{Z}[\pi]\) with \(\pi\) being the Frobenius endomorphism of \(E\). Then, to obtain a geometric description, the authors give an algorithm to compute the theta null point of \((A, \mathcal{L}) \in \mathcal{W}_1\) from its lattice description. More precisely, it is possible to compute the theta null point for \(E^g\) with the product polarization \(\mathcal{L}_0\), construct an isogeny \(f: E^g \to A\) such that \(f^*\mathcal{L} = \mathcal{L}_0^l\) for some \(l\) and transport the theta null point from \((E^g, \mathcal{L}_0)\) to \((A, \mathcal{L})\) using the isogeny formula from [R. Cosset and D. Robert, Math. Comput. 84, No. 294, 1953–1975 (2015; Zbl 1315.11103)]. In addition, the authors show how to compute Siegel modular forms of even weight given as polynomials in the theta constants. This is done by choosing a certain affine lift of theta null point on \((A, \mathcal{L})\). As applications of the algorithms, the authors present an algebraic computation of Serre’s obstruction for any \((A, \mathcal{L})\) in the isogeny class of \(E^3\) and the Igusa modular form in dimension 4, which gives an alternative way to prove that a certain isogeny class does not contain Jacobians. A few interesting applications of the algorithms are presented for curves of genera 2 and 3 with many rational points over certain finite fields.
Reviewer: Anatoli Shatsila (Kraków)
MSC:
| 14H42 | Theta functions and curves; Schottky problem |
| 14G15 | Finite ground fields in algebraic geometry |
| 14H45 | Special algebraic curves and curves of low genus |
| 16H20 | Lattices over orders |
Keywords:
Hermitian lattice; order in quadratic field; isogeny class; polarization; curves with many points over finite fields; Siegel modular form; theta constant; theta null point; algorithm; Igusa modular form; Serre’s obstruction; Schottky locusReferences:
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