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Computing functions on Jacobians and their quotients. (English) Zbl 1333.14038

This article considers the problem of evaluation of the quotient of a Jacobian variety \(J\) of a curve \(C\) by a maximal isotropic subgroup \(V\) in its \(\ell\)-torsion for \(\ell\) an odd prime integer different from the characteristic of \(K\). Firstly, the problem is considered generally. It has been considered how to quickly design and compute standard functions on \(J/V\). It has been shown that if the dimension \(g\) of \(J\) equals two, the quotient is the Jacobian of another curve \(D\). The complexity of evaluating standard functions on Jacobians including Weil functions and algebraic theta functions has been bounded. A formula for the divisor of certain functions on J defined using determinants has been given. An expression for eta functions as combinations of these determinants has been deduced. The resulting algorithm for evaluating Eta function has been detailed. One has deduced a bound for the complexity of computing a basis of sections for the bundle associated with a multiple of the natural polarization of \(J\). The algebraic definition of canonical theta functions has been recalled and the evaluating canonical theta functions has been given. One has been bounded the complexity evaluating functions on the quotient of \(J\) by a maximal isotropic subgroup \(V\) in \(J [\ell]\) when \(\ell\) is and odd prime different from the charscteristic of \(\mathbf K\). Assuming that the characteristic p of \(\mathbf K\) is odd one has been bounded the complexity of computing an isogeny \(J_{C} \to J_{D}\) between two Jacobians of dimension two, the expected form of such isogeny has been given. A specific algorithm for genus 2 curves have been presented. A complete example with \(\mathbf K\) being the field with 1009 elements has been given.

MSC:

14K02 Isogeny
14Q05 Computational aspects of algebraic curves
14Q20 Effectivity, complexity and computational aspects of algebraic geometry
14K25 Theta functions and abelian varieties
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