7,793 research outputs found

    Modular polynomials for genus 2

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    Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them

    Topological string amplitudes for the local half K3 surface

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    We study topological string amplitudes for the local half K3 surface. We develop a method of computing higher-genus amplitudes along the lines of the direct integration formalism, making full use of the Seiberg-Witten curve expressed in terms of modular forms and E_8-invariant Jacobi forms. The Seiberg-Witten curve was constructed previously for the low-energy effective theory of the non-critical E-string theory in R^4 x T^2. We clarify how the amplitudes are written as polynomials in a finite number of generators expressed in terms of the Seiberg-Witten curve. We determine the coefficients of the polynomials by solving the holomorphic anomaly equation and the gap condition, and construct the amplitudes explicitly up to genus three. The results encompass topological string amplitudes for all local del Pezzo surfaces.Comment: 35 pages, v2: several clarifications made, an equation and references added, v3: published versio

    Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve

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    It has been conjectured that for NN sufficiently large, there are no quadratic polynomials in Q[z]\bold Q[z] with rational periodic points of period NN. Morton proved there were none with N=4N=4, by showing that the genus~22 algebraic curve that classifies periodic points of period~4 is birational to X1(16)X_1(16), whose rational points had been previously computed. We prove there are none with N=5N=5. Here the relevant curve has genus~1414, but it has a genus~22 quotient, whose rational points we compute by performing a~22-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible GalQ_{\bold Q}-stable 55-cycles, and show that there exist GalQ_{\bold Q}-stable NN-cycles for infinitely many NN. Furthermore, we answer a question of Morton by showing that the genus~1414 curve and its quotient are not modular. Finally, we mention some partial results for N=6N=6

    Computing the 2-adic Canonical Lift of Genus 2 Curves

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    International audienceLet k be a field of even characteristic and M2(k) the moduli space of the genus 2 curves defined over k. We first compute modular polynomials in function of invariants with good reduction modulo two. We then use these modular polynomials to compute the canonical lift of genus 2 curves in even characteristic. The lifted Frobenius is characterized by the reduction behaviors of the Weierstrass points over k. This allows us to compute the cardinality of the Jacobian variety. We give a detailed description with the necessary optimizations for an efficient implementation
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