For higher class field theory Witt and Kummer extensions are very important. In fact, Parshin’s construction of class field theory for higher local fields of prime characteristic [P] is based on an explicit (Artin–Schreier–Witt) pairing; see [F] for a generalization to the case of a perfect residue field. Kummer extensions in the mixed characteristic case can be described by using class field t...

We give oneand two-dimensional scalar multiplication algorithms for Jacobians of genus 2 curves that operate by projecting to Kummer surfaces, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper “signed” output back on the Jacobian. This extends the work of López and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional ...

In the early 1990s, Flynn gave an explicit description of the Jacobian of a genus 2 hyperelliptic curve to perform efficient arithmetic on these objects. In this paper, we give a generalization of Flynn’s work when the ground field has characteristic 2. More precisely, we give an explicit description of both the Jacobian and the Kummer surface. We also give (and explain how we found) explicit f...

We compute the Buhler-Reichstein essential dimension of a complex abelian variety using Kummer theory.

We test R. van Luijk’s method for computing the Picard group of a K3 surface. The examples considered are the resolutions of Kummer quartics in P. Using the theory of abelian varieties, in this case the Picard group may be computed directly. Our experiments show that the upper bounds provided by R. van Luijk’s method are sharp when sufficiently large primes are used. In fact, there are many pri...

In this paper, we revisit the recent small characteristic discrete logarithm algorithms. We show that a simplified description of the algorithm, together with some additional ideas, permits to obtain an improved complexity for the polynomial time precomputation that arises during the discrete logarithm computation. With our new improvements, this is reduced to O(q), where q is the cardinality o...

c W W L Chen, 1984, 2013. This chapter originates from material used by the author at Imperial College London between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and ...

We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice II1,25 of rank 26 and signature (1, 25). The generators are related to reflections with respect to some Leech roots. A similar observation was made fir...

Let p be a prime. In this paper, we present detailed p-adic analysis to factorials and double factorials and their congruences. We give good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli number B̂n n when n is divisible by p−1. Using these we then establish the universal Kummer congruences modulo powers of a prime p for the divided universal Bernoulli numbers...