Absch�tzung der Automorphismenanzahl von Funktionenk�rpern bei Primzahlcharakteristik

Roquette, Peter

doi:10.1007/bf01109838

Cited by 58 publications

(37 citation statements)

References 7 publications

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“…In particular, p > 3. As PSL(2, p) is simple for p > 3, from Roquette's theorem [25], see also [13, p. 545], we have that |G| ≤ 84(g − 1) = 42(p − 3).…”

Section: Then Every Conjugatementioning

confidence: 99%

Algebraic curves with a large non-tame automorphism group fixing no point

Giulietti

Korchmáros

2010

Trans. Amer. Math. Soc.

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Abstract. Let K be an algebraically closed field of characteristic p > 0, and let X be a curve over K of genus g ≥ 2. Assume that the automorphism group Aut(X ) of X over K fixes no point of X . The following result is proven. If there is a point P on X whose stabilizer in Aut(X ) contains a p-subgroup of order greater than gp/(p − 1), then X is birationally equivalent over K to one of the irreducible plane curves (II), (III), (IV), (V) listed in the Introduction.

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“…In particular, p > 3. As PSL(2, p) is simple for p > 3, from Roquette's theorem [25], see also [13, p. 545], we have that |G| ≤ 84(g − 1) = 42(p − 3).…”

Section: Then Every Conjugatementioning

confidence: 99%

Algebraic curves with a large non-tame automorphism group fixing no point

Giulietti

Korchmáros

2010

Trans. Amer. Math. Soc.

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“…The gap sequence at the unique place Q of F above P x=0 is 1, 2, 3, 4, 6,7,8,9,11,12,13,14,16,18. The pole numbers up to first pole number not divisible by 5 are 0, 5, 10, 15, 17, i.e., m = 17, and (17Q) = 5, as one can compute using the Magma program [1].…”

Section: Remarkmentioning

confidence: 99%

“…We see that if g < p − 1 then g p−1 = 0 and the dimension is at least 2-dimensional. Roquette in [11] proved that if a curve has a wild ramification point then p ≤ g + 1 with only one exception, the hyperelliptic curve…”

Section: Two-dimensional Representationsmentioning

confidence: 99%

The ramification sequence for a fixed point of an automorphism of a curve and the Weierstrass gap sequence

Kontogeorgis

2007

Math. Z.

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For nonsingular projective curves defined over algebraically closed fields of positive characteristic the dependence of the ramification filtration of decomposition groups of the automorphism group with Weierstrass semigroups attached at wild ramification points is studied. A faithful representation of the p-part of the decomposition group at each wild ramified point to a Riemann-Roch space is defined.

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“…Schmid [10] was able to prove the finiteness of the group G of automorphisms. Concerning the order of G, Roquette [8] showed that the Hurwitz bound is valid if p > g + 1, with one exception. Henn [-4] has recently proved that [GI _-< 3(2g) ~/2, if F does not belong to one of four exceptional classes.…”

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confidence: 98%

Automorphism groups of algebraic function fields

Valentini

Madan

1981

Math Z

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The conformal mappings of a compact Riemann surface R onto itself form a finite group, if the genus g of the surface is greater than 1. This is a classical result commonly attributed to Poincar6 and Klein. It was first suggested by H.A. Schwarz. There are several proofs, including those by Weierstrass [13], M. Noether [7] and Hurwitz [5]. In fact, Hurwitz proved that the order of this group cannot exceed 84(g-1) and that every finite group is a group of conformal mappings for a suitable compact Riemann surface. Considering the well-known correspondence between compact Riemann surfaces and fields of algebraic functions of one variable over the field of complex numbers, the above results can also be stated as follows: The group of automorphisms of a field of algebraic functions in one variable over the field of complex numbers is a finite group, provided its genus g is greater than one. Its order does not exceed 84(g -1). Every finite group is a group of automorphisms of a suitable function field.To extend these results to function fields F/K of characteristic p, it was natural to look for the analogue of Weierstrass points on a compact Riemann surface. This was done by F.K. Schmidt [11]. The theory of Weierstrass points in characteristic p is more complicated. Nevertheless, H.L. Schmid [10] was able to prove the finiteness of the group G of automorphisms. Concerning the order of G, Roquette [8] showed that the Hurwitz bound is valid if p > g + 1, with one exception. Henn [-4] has recently proved that [GI _-< 3(2g) ~/2, if F does not belong to one of four exceptional classes. The question concerning the realizability of a given group as a full group of automorphisms of a suitable function field is an open problem. Our principal result in this paper is an affirmative solution of this problem in the case of abelian groups.An abelian group is a direct product of cyclic groups of prime power order. Corresponding to the direct factors, we construct cyclic extensions of a rational function field K(x). The ramification is chosen to give arithmetically disjoint fields. This insures that the Galois group of the composite over K(x) is the direct product of the Galois groups of the factors. The ramification is further specified to produce Weierstrass points with suitable gap sequences. (See [1] or [11] for the definitions and basic properties.) This guarantees that any automorphism of the composite fixes K(x) elementwise.

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Absch�tzung der Automorphismenanzahl von Funktionenk�rpern bei Primzahlcharakteristik

Cited by 58 publications

References 7 publications

Algebraic curves with a large non-tame automorphism group fixing no point

Algebraic curves with a large non-tame automorphism group fixing no point

The ramification sequence for a fixed point of an automorphism of a curve and the Weierstrass gap sequence

Automorphism groups of algebraic function fields

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