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In this paper, we study simple cubic fields in the function field setting, and also generalize the notion of a set of exceptional units to cubic function fields, namely the notion of $k$-exceptional units. We give a simple proof that the Galois simple cubic function fields are the immediate analog of Shanks simplest cubic number fields. In addition to computing the invariants, including a formula for the regulator, we compute the class numbers of the Galois simple cubic function fields over $mathbb{F}_{5}$ and $mathbb{F}_{7}$ using truncated Euler products. Finally, as an additional application, we determine all Galois simple cubic function fields with class number one, subject to a mild restriction.
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We present computational results on the divisor class number and the regulator of a cubic function field over a large base field. The underlying method is based on approximations of the Euler product representation of the zeta function of such a field. We give details on the implementation for purely cubic function fields of signatures $(3,1)$ and $(1, 1; 1, 2)$, operating in the ideal class group and infrastructure of the function field, respectively. Our implementation provides numerical evidence of the computational effectiveness of this algorithm. With the exception of special cases, such as purely cubic function fields defined by superelliptic curves, the examples provided are the largest divisor class numbers and regulators ever computed for a cubic function field over a large prime field. The ideas underlying the optimization of the class number algorithm can in turn be used to analyze the distribution of the zeros of the function fields zeta function. We provide a variety of data on a certain distribution of the divisor class number that verify heuristics by Katz and Sarnak on the distribution of the zeroes of the zeta function.
We describe and give computational results of a procedure to compute the divisor class number and regulator of most purely cubic function fields of unit rank 2. Our implementation is an improvement to Pollards Kangaroo method in infrastructures, using distribution results of class numbers as well as information on the congruence class of the divisor class number, and an adaptation that efficiently navigates these torus-shaped infrastructures. Moreover, this is the first time that an efficient square-root algorithm has been applied to the infrastructure of a global field of unit rank 2. With the exception of certain function fields defined by Picard curves, our examples are the largest known divisor class numbers and regulators ever computed for a function field of genus 3.
We present a method for tabulating all cubic function fields over $mathbb{F}_q(t)$ whose discriminant $D$ has either odd degree or even degree and the leading coefficient of $-3D$ is a non-square in $mathbb{F}_{q}^*$, up to a given bound $B$ on the degree of $D$. Our method is based on a generalization of Belabas method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires $O(B^4 q^B)$ field operations as $B rightarrow infty$. The algorithm, examples and numerical data for $q=5,7,11,13$ are included.
For a cubic algebraic extension $K$ of $mathbb{Q}$, the behavior of the ideal counting function is considered in this paper. Let $a_{K}(n)$ be the number of integral ideals of the field $K$ with norm $n$. An asymptotic formula is given for the sum $$ sumlimits_{n_{1}^2+n_{2}^2leq x}a_{K}(n_{1}^2+n_{2}^2). $$
We consider the Galois group $G_2(K)$ of the maximal unramified $2$-extension of $K$ where $K/mathbb{Q}$ is cyclic of degree $3$. We also consider the group $G^+_2(K)$ where ramification is allowed at infinity. In the spirit of the Cohen-Lenstra heuristics, we identify certain types of pro-$2$ group as the natural spaces where $G_2(K)$ and $G^+_2(K)$ live when the $2$-class group of $K$ is $2$-generated. While we do not have a theoretical scheme for assigning probabilities, we present data and make some observations and conjectures about the distribution of such groups.
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