Dirichlet series and hyperelliptic curves

Dirichlet series and hyperelliptic curves

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...

Computing L-Series of Hyperelliptic Curves

We discuss the computation of coefficients of the L-series associated to a hyperelliptic curve over Q of genus at most 3, using point counting, generic group algorithms, and p-adic methods.

Hyperelliptic Curves and Cryptography

In 1989, Koblitz proposed using the jacobian of a hyperelliptic curve defined over a finite field to implement discrete logarithm cryptographic protocols. This paper provides an overview of algorithms for performing the group law (which are necessary for the efficient implementation of the protocols), and algorithms for solving the hyperelliptic curve discrete logarithm problem (whose intractab...

Computing L-series of geometrically hyperelliptic curves of genus three

Let C/Q be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of Q, but may not have a hyperelliptic model of the usual form over Q. We describe an algorithm that computes the local zeta functions of C at all odd primes of good reduction up to a prescribed bound N . The algorithm relies on an adaptation of the ‘accumulating...

Arithmetic of Hyperelliptic Curves