7,795 research outputs found
Modular polynomials for genus 2
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
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
It has been conjectured that for sufficiently large, there are no
quadratic polynomials in with rational periodic points of period
. Morton proved there were none with , by showing that the genus~
algebraic curve that classifies periodic points of period~4 is birational to
, whose rational points had been previously computed. We prove there
are none with . Here the relevant curve has genus~, but it has a
genus~ quotient, whose rational points we compute by performing
a~-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 Gal-stable -cycles, and show that
there exist Gal-stable -cycles for infinitely many .
Furthermore, we answer a question of Morton by showing that the genus~
curve and its quotient are not modular. Finally, we mention some partial
results for
Computing the 2-adic Canonical Lift of Genus 2 Curves
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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