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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
Full Text:
##### References:
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