The distribution of the zeros of the Goss zeta-function for $A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)$

Abstract

Let $F$ be a global function field over a finite constant field and $\infty$ a place of $F$. The ring $A$ of functions regular away from $\infty$ in $F$ is a Dedekind domain. For such $A$ Goss defined a $\zeta$-function which is a continuous function from $\mathbb{Z }_p$ to the ring of entire power series with coefficients in the completion $F_\infty$ of $F$ at $\infty$. He asks what one can say about the distribution of the zeros of the entire function at any parameter of $\mathbb{Z }_p$. In the simplest case $A$ is the polynomial ring in one variable over a finite field. Here the question was settled completely by J. Sheats, after previous work by J. Diaz-Vargas, B. Poonen and D. Wan: for any parameter in $\mathbb{Z }_p$ the zeros of the power series have pairwise different valuations and they lie in $F_\infty$. In the present article we completely determine the distribution of zeros for the simplest case different from polynomial rings, namely $A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)$—this $A$ has class number $1$, it is the affine coordinate ring of a supersingular elliptic curve and the place $\infty$ is $\mathbb{F }\,\!{}_2$-rational. The answer is slightly different from the above case of polynomial rings. For arbitrary $A$ such that $\infty$ is a rational place of $F$, we describe a pattern in the distribution of zeros which we observed in some computational experiments. Finally, we present some precise conjectures on the fields of rationality of these zeroes for one particular hyperelliptic $A$ of genus $2$.