Dirichlet series and hyperelliptic curves
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and
Abstract
For a fixed hyperelliptic curve C given by the equation y2 = f(x) with f ∈ ℤ[x] having distinct roots and degree at least 5, we study the variation of rational points on the quadratic twists Cm whose equation is given by my2 = f(x). More precisely, we study the Dirichlet series 
where the summation is over all non-zero squarefree integers. We show that 
converges for ℜ(s) > 1. We extend its range of convergence assuming the ABC conjecture. This leads us to study related Dirichlet series attached to binary forms. We are then led to investigate the variation of rational points on twists of superelliptic curves. We apply this study to certain classical problems of analytic number theory such as the number of powerfree values of a fixed polynomial in ℤ[x].
© Walter de Gruyter
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- Symmetric spaces with convex metrics
- Gradient estimates for parabolic problems with unbounded coefficients in non convex unbounded domains
- Global wellposedness and scattering for 3D energy critical Schrödinger equation with repulsive potential and radial data
- Dirichlet series and hyperelliptic curves
- Embeddability of quadratic extensions in cyclic extensions
- Weakly half-factorial sets in finite abelian groups
- Completeness of cotorsion pairs
- Erratum to: Continuous control and the algebraic L-theory assembly map
Articles in the same Issue
- Symmetric spaces with convex metrics
- Gradient estimates for parabolic problems with unbounded coefficients in non convex unbounded domains
- Global wellposedness and scattering for 3D energy critical Schrödinger equation with repulsive potential and radial data
- Dirichlet series and hyperelliptic curves
- Embeddability of quadratic extensions in cyclic extensions
- Weakly half-factorial sets in finite abelian groups
- Completeness of cotorsion pairs
- Erratum to: Continuous control and the algebraic L-theory assembly map
