Towards the Sato–Tate groups of trinomial hyperelliptic curves

On the variation of Tate–Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=Fq(t) having conductor (p)·∞, where (p) is a prime ideal in Fq [t]. Let d ∈ Fq [t] be an irreducible polynomial of odd degree, and let K=F( √ d). Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗FK, 1). As a consequence, we obtain a formula for the order of the Tate–Shafarevich group I(E/K) when L(E⊗FK, 1) =...

The Arithmetic of Hyperelliptic Curves

We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve.

Arithmetic of Hyperelliptic Curves

Families of Hyperelliptic Curves

Throughout this work we deal with a natural number g ≥ 2 and with an algebraically closed field k whose characteristic differs from 2. A hyperelliptic curve of genus g over k is a smooth curve of genus g, that is a double cover of the projective line P. The Riemann-Hurwitz formula implies that this covering should be ramified at 2g + 2 points. Because of this explicit description, hyperelliptic...

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...