Dirichlet series and hyperelliptic curves
نویسندگان
چکیده
For a fixed hyperelliptic curve C given by the equation y 1⁄4 f ðxÞ with f A Z1⁄2x 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 my 2 1⁄4 f ðxÞ. More precisely, we study the Dirichlet series Df ðsÞ 1⁄4 P 0 m00aCmðQÞjmj s where the summation is over all non-zero squarefree integers. We show that Df ðsÞ converges for <ðsÞ > 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 Z1⁄2x . 2000 Mathematics Subject Classification: 11G30; 11M41.
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