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 $|\Russian{X}|$ of Tate-Shafarevich groups of the elliptic curves with equation $Y^2+XY=X^3+\alpha X^2+\mbox{const}\cdot T^{-k}$ over the rational function field $\mathbf{F}_q(T)$, where $q$ is a power of 2. In the range $q=2$, $k \leq 37$, $\alpha \in \mathbf{F}_2\lbrack T^{-1}\rbrack$ odd of degree $\leq 19$, the largest values obtained for $|\Russian{X}|$ 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 correspondence for GL$(2)$ over local or global fields of characteristic two.