Class groups of global function fields with certain splitting behaviors of the infinite prime

Abstract:

For certain two cases of splitting behaviors of the prime at infinity with unit rank $r$, given positive integers $m, n$, we construct infinitely many global function fields $K$ such that the ideal class group of $K$ of degree $m$ over $\mathbb {F}(T)$ has $n$-rank at least $m-r-1$ and the prime at infinity splits in $K$ as given, where $\mathbb {F}$ denotes a finite field and $T$ a transcendental element over $\mathbb {F}$. In detail, for positive integers $m$, $n$ and $r$ with $0 \le r \le m-1$ and a given signature $(e_i, \mathfrak {f}_i)$, $1 \le i \le r+1$, such that $\sum _{i=1}^{r+1}{e_i\mathfrak {f}_i} =m$, in the following two cases where $e_i$ is arbitrary and $\mathfrak {f}_i =1$ for each $i$, or $e_i =1$ and $\mathfrak {f}_i$’s are the same for each $i$, we construct infinitely many global function fields $K$ of degree $m$ over $\mathbb {F}(T)$ such that the ideal class group of $K$ contains a subgroup isomorphic to $(\mathbb {Z}/n\mathbb {Z})^{m-r-1}$ and the prime at infinity ${\wp _\infty }$ splits into $r+1$ primes $\mathfrak {P}_1, \mathfrak {P}_2, \cdots , \mathfrak {P}_{r+1}$ in $K$ with $e(\mathfrak {P}_i/{\wp _\infty }) = e_i$ and $\mathfrak {f}(\mathfrak {P}_i/{\wp _\infty }) = \mathfrak {f}_i$ for $1 \le i \le r+1$ (so, $K$ is of unit rank $r$).