Global Field Of Characteristic Research Articles

Parity conjectures for elliptic curves over global fields of positive characteristic

AbstractWe prove thep-parity conjecture for elliptic curves over global fields of characteristicp>3. We also present partial results on theℓ-parity conjecture for primesℓ≠p.

On the admissibility of finite groups over global fields of finite characteristic

Let k be a global field of characteristic p. A finite group G is called k-admissible if there exists a division algebra finite dimensional and central over k which is a crossed product for G. Let G be a finite group with normal Sylow p-subgroup P. If the factor group G P is k-admissible, then G is k-admissible. A necessary condition is given for a group to be k-admissible: if a finite group G is k-admissible, then every Sylow l-subgroup of G for l ≠ p is metacyclic with some additional restriction. Then it is proved that a metacyclic group G generated by x and y is k-admissible if some relation between x and y is satisfied.

Galois groups of global fields of finite characteristic

Let G be a finite group having a normal p-Sylow subgroup P such that G P is the direct product of two cyclic groups of orders e and f, both prime to p. Let k be a global field of characteristic p, containing the eth roots of unity, and S a finite set of primes of k. Then there exists a Galois extension L of k with Galois group G( L k ) isomorphic to G, such that the local Galois group G( L v k v ) ≅ G for each v ϵ S. This result, together with work of Fein and Schacher, yield the following. Let k be a global field of finite characteristic p, G a finite group. Then a necessary and sufficient condition that every division ring with center k and index equal to the order of G be a crossed product for G, is that k and G satisfy the hypothesis of the above theorem.