853 research outputs found

    Solving Sparse Integer Linear Systems

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    We propose a new algorithm to solve sparse linear systems of equations over the integers. This algorithm is based on a pp-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox-based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art

    Algorithms for Mumford curves

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    Mumford showed that Schottky subgroups of PGL(2,K)PGL(2,K) give rise to certain curves, now called Mumford curves, over a non-Archimedean field K. Such curves are foundational to subjects dealing with non-Archimedean varieties, including Berkovich theory and tropical geometry. We develop and implement numerical algorithms for Mumford curves over the field of p-adic numbers. A crucial and difficult step is finding a good set of generators for a Schottky group, a problem solved in this paper. This result allows us to design and implement algorithms for tasks such as: approximating the period matrices of the Jacobians of Mumford curves; computing the Berkovich skeleta of their analytifications; and approximating points in canonical embeddings. We also discuss specific methods and future work for hyperelliptic Mumford curves.Comment: 32 pages, 4 figure

    Counting points on curves using a map to P^1

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    We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends Kedlaya's algorithm to a very general class of curves using a map to the projective line. We develop all the necessary bounds, analyse the complexity of the algorithm and provide some examples computed with our implementation

    A p-adic quasi-quadratic point counting algorithm

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    In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality qq with time complexity O(n2+o(1))O(n^{2+o(1)}) and space complexity O(n2)O(n^2), where n=log(q)n=\log(q). In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of theta constants. The theta null values are computed with respect to a semi-canonical theta structure of level 2νp2^\nu p where ν>0\nu >0 is an integer and p=\mathrm{char}(\F_q)>2. The results of this paper suggest a global positive answer to the question whether there exists a quasi-quadratic time algorithm for the computation of the number of rational points on a generic ordinary abelian variety defined over a finite field.Comment: 32 page

    Explicit computations of Hida families via overconvergent modular symbols

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    In [Pollack-Stevens 2011], efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of pp-adic LL-functions and have further been applied to compute rational points on elliptic curves (e.g. [Darmon-Pollack 2006, Trifkovi\'c 2006]). In this paper, we generalize these algorithms to the case of families of overconvergent modular symbols. As a consequence, we can compute pp-adic families of Hecke-eigenvalues, two-variable pp-adic LL-functions, LL-invariants, as well as the shape and structure of ordinary Hida-Hecke algebras.Comment: 51 pages. To appear in Research in Number Theory. This version has added some comments and clarifications, a new example, and further explanations of the previous example
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