On class number of quadratic extensions over function fields
Let \({\mathbb{F}} _p (T)\) be a rational function field over the field of \(p\) elements, \(p \geq 5\) and let \(k\) be a finite extension of \({\mathbb{F}} _p (T)\). Let \(n\) be a natural number and let \(\alpha : M _\infty \rightarrow \prod _{v \in M _\infty} Q _v\) be any map, where \(M _\infty\) denotes the set of infinite prime divisors of \(k\) and \(Q _v\) denotes the finite set of extensions over \(k _v\), the completion of \(k\) at \(v\), whose degrees over \(k _v\) are bounded by 2. Let \({\mathcal Q} _\alpha ^\ast (n)\) denote the set of quadratic extensions \(F\) over \(k\) such that for any place \(v \in M _\infty\), \(v\) is ramified, splits or is inert in \(F\) according to whether \(\alpha(v)\) is a ramified quadratic extension over \(k _v\), \(k _v\) itself or an unramified quadratic extension over \(k _v\), and \(D _{F/k} = q ^{2n}\). Here \(D _{F/k}\) denotes the norm in \(k\) of the discriminant of \(F/k\). Let \(S _0\), \(S _1\), \(S _2\) be mutually disjoint finite sets of finite places of \(k\). The main results of the paper are the following density theorems: \[ \lim _{n \rightarrow \infty} {{1} \over {| {\mathcal Q} _\alpha ^\ast (n)| }} \sum _{F \in {\mathcal Q} _\alpha ^\ast (n)} h _{F, 3} = 1 + {{1} \over 3 ^{r(\alpha)}} \tag{I} \] and \[ \liminf _{n \rightarrow \infty} {{| \left\{F \in {\mathcal Q} _\alpha ^\ast (n) \mid 3 \nmid h _{F/k}\right\}| } \over {| {\mathcal Q} _\alpha ^\ast (n)| }} \geq \left(1 - {{1}\over{2 \cdot 3 ^{r(\alpha)}}}\right) \left(\prod _{v \in S _0} {{1} \over {q _v + 1}} \right) \left(\prod _{v \in S _1 \cup S _2} {{q _v} \over {2 (q _v + 1)}} \right), \tag{II} \] where \(r(\alpha)\) denotes the number of places \(v \in M _\infty\) with \(\alpha(v) = k _v\) and \(h _{F,3}\) denotes the order of the maximal elementary abelian 3-group in the kernel of the norm map from the class group of \(F\) to that of \(k\). As consequences of the above results the author shows that there exist infinitely many quadratic extensions \(F\) over \(k\) with \(3\nmid h _{F/k}\) in which every place of \(S _i\) is ramified, is inert, or splits according to whether \(i = 0, 1\) or \(2\). In particular, if the class number \(h\) of \(k\) is prime to \(3\), then there exist infinitely many quadratic extensions over \(k\) with class number prime to \(3\) in which every place of \(S _i\) is ramified, is inert or splits according to whether \(i = 0, 1\) or \(2\). Also, it follows that there exist infinitely many quadratic extensions \(F\) over \(k\) with \(3 | h _{F/k}\) in which every place of \(S _i\) is ramified, is inert, or splits according to whether \(i = 0, 1\) or \(2\). The author notices that if one can prove that there exist infinitely many quadratic extensions \(F\) with prescribed ramification, decomposition and inertia over \(k\) whose class numbers are not divisible by a prime number \(p\), then this would imply that there exist infinitely many quadratic extensions \(F\) over \(k\) such that the Iwasawa invariants of the geometric \({\mathbb Z} _p\)-extensions over \(F\) vanish.
scientific article; zbMATH DE number 1215552