1,406 research outputs found

    Linearizing torsion classes in the Picard group of algebraic curves over finite fields

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    We address the problem of computing in the group of â„“k\ell^k-torsion rational points of the jacobian variety of algebraic curves over finite fields, with a view toward computing modular representations.Comment: To appear in Journal of Algebr

    Computing Hilbert class polynomials with the Chinese Remainder Theorem

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    We present a space-efficient algorithm to compute the Hilbert class polynomial H_D(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D|^(1/2+o(1))log P) space and has an expected running time of O(|D|^(1+o(1)). We describe practical optimizations that allow us to handle larger discriminants than other methods, with |D| as large as 10^13 and h(D) up to 10^6. We apply these results to construct pairing-friendly elliptic curves of prime order, using the CM method.Comment: 37 pages, corrected a typo that misstated the heuristic complexit
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