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Computing isogenies between Jacobians of curves of genus 2 and 3. (English) Zbl 1433.14043
Let \(\mathcal{C}\) be a projective, smooth, absolutely integral curve of genus \(g \in \{2,3\}\) over a finite field \(\mathbf{F}_{\!q}\). Denote by \(\ell\) an odd prime different from the characteristic of \(\mathbf{F}_{\!q}\). Let \(\mathcal{V}\) be a maximal isotropic subgroup of the \(\ell\)-torsion in the Jacobian \(J_{\mathcal{C}}\) of \(\mathcal{C}\) with respect to the Weil pairing. The author describes algorithms for computing the equation of a curve \(\mathcal{D}\) such that the quotient \(J_{\mathcal{C}}/\mathcal{V}\) is equal to the Jacobian \(J_{\mathcal{D}}\) of \(\mathcal{D}\). Rational fractions expliciting the quotient isogeny \(f : J_{\mathcal{C}} \rightarrow J_{\mathcal{D}}\) are also given.
Actually this work is based on a paper by J.-M. Couveignes and T. Ezome [LMS J. Comput. Math. 18, 555–577 (2015; Zbl 1333.14038)] which relates how to efficiently evaluate eta and theta functions on Jacobian varieties with dimension \(\ge 2\), and compute \((\ell, \ell)\)-isogenies between Jacobians of genus 2 curves. Milio starts by recalling Couveignes-Ezome algorithm for computing isogenies between \(2\)-dimensional Jacobians. It happens that this algorithm requires the equation of the Kummer surface associated to \(\mathcal{D}\), however Milio shows that there is actually no need for this equation to output the results in dimension 2. Also he optimized the computation of the equation of the curve \(\mathcal{D}\), and the computation of the rational functions describing \(f\). Furthermore he extended the Couveignes-Ezome algorithm to the case of hyperelliptic curves of genus 3. But the generalization faces obstacles concerning genus 3 non-hyperelliptic curves: there are some uncertainties to recover the equation of the curve \(\mathcal{D}\) from data about its Kummer surface. To avoid this, the author uses theta based formulas and the reconstruction of plane quartics from bitangents.
MSC:
14K02 Isogeny
14K25 Theta functions and abelian varieties
14Q05 Computational aspects of algebraic curves
14Q20 Effectivity, complexity and computational aspects of algebraic geometry
Software:
AVIsogenies; Magma
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