Zeta functions of a class of elliptic curves over a rational function field of characteristic two

We show how to calculate the zeta functions and the orders \III\ of Tate-Sh afarevich groups of the elliptic curves with equation Y-2 + XY = X-3 + alph a X-2 + const . T-k over the rational function field F-q(T), where q is a p ower of 2, In the range q = 2, k less than or equal to 37, alpha epsilon F- 2[T-1] odd of degree less than or equal to 19, the largest values obtained for \III\ are 47(2) (one case), 39(2) (one case) and 27(2) (three cases). We observe and discuss a remarkable pattern for the distributions of signs in the functional equation and of fudge factors at places of bad reduction. These imply strong restrictions on the precise form of the Langlands corre spondence for GL(2) over local or global fields of characteristic two.