The 2-primary class group of certain hyperelliptic curves. (English) Zbl 1010.11032

Consider the field \(\mathbb F_q(\mathbb T, \sqrt{e{\mathfrak p}})\) for \({\mathfrak p}\) a monic prime polynomial in \(\mathbb F_q(T)\) of even degree \(k\) and \(e\) nonsquare in \(\mathbb F_q\). Let \(h({\mathfrak p})\) denote the class number. Then \(h({\mathfrak p})\) is divisible by 2 or 4 (resp.) according as \(k\), but no divisibility condition for \(k\) assures divisibility by 8. This is like \(h(p)\) the class number of \(\mathbb Q(\sqrt {-p})\) (\(p\) prime), which is divisible by \(m=2^t\) when \(p\equiv 1\bmod 2m\) for \(m=2\) or 4 and when \(p\) splits in \(\mathbb Q(\sqrt i, \sqrt{(1+i)})\) for \(m=8\) (no divisibility condition for \(p)\). This result of P. Barrucand and H. Cohn [J. Reine Angew. Math. 238, 67–70 (1969; Zbl 0207.36202)] led to the concept of the “governing field” [see P. Stevenhagen, J. Number Theory 43, 1–19 (1993; Zbl 0767.11054)] which the author extends as part of a much larger program involving Jacobians, quaternion algebras, Drinfeld modules, etc., [see his earlier work in Math. Ann. 314, 175–196 (1999; Zbl 0979.11029)].


11G20 Curves over finite and local fields
11R29 Class numbers, class groups, discriminants
11R37 Class field theory
11G09 Drinfel’d modules; higher-dimensional motives, etc.
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