Explicit endomorphism of the Jacobian of a hyperelliptic function field of genus 2 using base field operations

We present an efficient endomorphism for the Jacobian of a curve C of genus 2 for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulæ for divisor doubling and addition of divisors with disjoint support in JF(C) using only base field operations. Explicit formulæ is presented for this third case and a different approach for divisor doubling.

l-PARTS OF DIVISOR CLASS GROUPS OF CYCLIC FUNCTION FIELDS OF DEGREE l

Let l be a prime number and K be a cyclic extension of degree l of the rational function field 𝔽q(T) over a finite field of characteristic ≠ = l. Using class field theory we investigate the l-part of Pic 0(K), the group of divisor classes of degree 0 of K, considered as a Galois module. In particular we give deterministic algorithms that allow the computation of the so-called (σ - 1)-rank and the (σ - 1)2-rank of Pic 0(K), where σ denotes a generator of the Galois group of K/𝔽q(T). In the case l = 2 this yields the exact structure of the 2-torsion and the 4-torsion of Pic 0(K) for a hyperelliptic function field K (and hence of the 𝔽q-rational points on the Jacobian of the corresponding hyperelliptic curve over 𝔽q). In addition we develop similar results for l-parts of S-class groups, where S is a finite set of places of K. In many cases we are able to prove that our algorithms run in polynomial time.

Non-periodic continued fractions in hyperelliptic function fields

Dedicated to George Szekeres on his 90th birthdayWe discuss the exponential growth in the height of the coefficients of the partial quotients of the continued fraction expansion of the square root of a generic polynomial.