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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)].

MSC:

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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[1] Abhyankar, S.B., Galois theory on the line in nonzero characteristic, Bull. amer. math. soc., 27, 68-133, (1992) · Zbl 0760.12002
[2] Artin, E., Quadratische Körper im gebiete der höheren kongruenzen, I, II, Math. Z., 19, 153-246, (1924) · JFM 50.0631.03
[3] Barrucand, P.; Cohn, H., Primes of type x2+32y2, class number and residuacity, J. reine angew. math., 238, 67-70, (1969) · Zbl 0207.36202
[4] Cornelissen, G., Zeros of Eisenstein series, quadratic class numbers and supersingularity for rational function fields, Math. ann., 314, 175-196, (1999) · Zbl 0979.11029
[5] Lejeune-Dirichlet, P., Collected works, (1969), Chelsea New York
[6] van der Geer, G.; van der Vlugt, M., Kloosterman sums and the p-torsion of certain Jacobians, Math. ann., 290, 549-563, (1991) · Zbl 0731.14014
[7] Gekeler, E.-U., Über Drinfeld’sche modulkurven vom hecke – typ, Compositio math., 57, 219-236, (1986) · Zbl 0599.14032
[8] Gekeler, E.-U., On finite Drinfeld modules, J. algebra, 141, 187-203, (1991) · Zbl 0731.11034
[9] Geyer, W.D.; Jarden, M., Bounded realization of l-groups over global fields. the method of scholz and reichardt, Nagoya math. J., 150, 13-62, (1998) · Zbl 0906.12002
[10] Hasse, H., Über die klassenzahl des Körpers P(\(−p\)) mit einer primzahl p≡(1mod23), Aequationes math., 3, 231-234, (1969) · Zbl 0167.32302
[11] Hayes, D.R., Explicit class field theory in global function fields, (), 173-217
[12] Kaplan, P., Divisibilité par 8 du nombre des classes des corps quadratiques dont le 2-groupe des classes est cyclique, et réciprocité biquadratique, J. math. soc. Japan, 25, 596-608, (1973) · Zbl 0276.12006
[13] Neumann, P.M., Some primitive permutation groups, Proc. London math. soc., 50, 265-281, (1985) · Zbl 0555.20003
[14] Rédei, L., Ein neues zahlentheoretisches symbol mit anwendungen auf die theorie der quadratischen zahlkörper, J. reine angew. math., 180, 1-43, (1939) · JFM 65.0106.03
[15] Serre, J.-P., Corps locaux, Actualités scientifiques et industrielles, 1296, (1968), Hermann Paris
[16] Shimura, G., Introduction to the arithmetic theory of automorphic functions, Kanô memorial lectures, no. 1. publications of the mathematical society of Japan, no. 11, iwanami shoten, Tokyo, (1971), Princeton University Press Princeton
[17] Stevenhagen, P., Divisibility by 2-powers of certain quadratic class numbers, J. number theory, 43, 1-19, (1993) · Zbl 0767.11054
[18] Weil, A., Basic number theory, Classics in mathematics, (1995), Springer-Verlag Berlin
[19] Wielandt, H., Finite permutation groups, Werke, I, (1994), de Gruyter Berlin/New York, p. 119-198
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