Zur Realisierbarkeit endlicher Gruppen als Automorphismengruppen algebraischer Funktionenk�rper

Stichtenoth, Henning

doi:10.1007/bf01161706

Cited by 9 publications

(7 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This inequality plays an important role in [7,8]. In Proposition 3.5 we give an analogous inequality for K /k not necessarily separable generated.…”

Section: Bound For the Genus Of A Compositum Of Fieldsmentioning

confidence: 92%

“…The next result is analogous to [8,Lemma 2] and [7, Lemma 2] and we will use it to find suitable C-improvements of a given finite separable extension E/k(x). PROPOSITION 4.1.…”

Section: Separable Extensions With a Prime Divisor Of Degree One Ramimentioning

confidence: 99%

“…The main result of Stichtenoth in [8] is that if E/k(x) a finite separable extension of degree greater than one, then, for every function field K /k of genus greater than one, there exist infinitely many nonisomorphic separable extensions L/K such that The results of Stichtenoth and of D'Mello and Madan can be combined to obtain the analogous of Greenberg's theorem provided that the group G is solvable and the genus of K is larger than one.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Finite Groups as Galois Groups of Function Fields With Infinite Field of Constants

Álvarez-García

Villa‐Salvador

2010

J. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…This inequality plays an important role in [7,8]. In Proposition 3.5 we give an analogous inequality for K /k not necessarily separable generated.…”

Section: Bound For the Genus Of A Compositum Of Fieldsmentioning

confidence: 92%

“…The next result is analogous to [8,Lemma 2] and [7, Lemma 2] and we will use it to find suitable C-improvements of a given finite separable extension E/k(x). PROPOSITION 4.1.…”

Section: Separable Extensions With a Prime Divisor Of Degree One Ramimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Finite Groups as Galois Groups of Function Fields With Infinite Field of Constants

Álvarez-García

Villa‐Salvador

2010

J. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…For example, given a finite group G and an algebraically closed field k, there are infinitely many non-isomorphic function fields L in one variable over k with Aut(L/k) = G (Madden-Valentini [MV83]). In fact, given a non-trivial finite group G and a function field K in one variable over an arbitrary algebraically closed field k, there are infinitely many Galois field extensions L/K with Gal(L/K) = Aut(L/k) = G, as proved by Greenberg [Gre74] if k = C, Stichtenoth if K has genus at least 2 [Sti84], and Madan-Rosen in general [MR92].…”

Section: Introductionmentioning

confidence: 99%

On a variant of the Beckmann--Black problem

Legrand¹

2021

Preprint

View full text Add to dashboard Cite

Given a field k and a finite group G, the Beckmann-Black problem asks whether every Galois field extension F/k with group G is the specialization at some t 0 ∈ k of some Galois field extension E/k(T ) with group G and E ∩ k = k. We show that the answer is positive for arbitrary k and G, if one waives the requirement that E/k(T ) is normal. In fact, our result holds if Gal(F/k) is any given subgroup H of G and, in the special case H = G, we provide a similar conclusion even if F/k is not normal. We next derive that, given a division ring H and an automorphism σ of H of finite order, all finite groups occur as automorphism groups over the skew field of fractions H(T, σ) of the twisted polynomial ring H[T, σ].

show abstract

“…It has been known for a long time that every finite group occurs in this way, since, for any ground field K and any finite group G, there exists X such that Aut(X ) ∼ = G; see [14] for K = C and [28] for p 0; see also [27,38].…”

Section: Introductionmentioning

confidence: 99%

Automorphism groups of algebraic curves with p -rank zero

Giulietti

Korchmáros

2009

Journal of the London Mathematical Society

View full text Add to dashboard Cite

The Hurwitz bound on the size of the K-automorphism group Aut(X ) of an algebraic curve X of genus g 2 defined over a field K of zero characteristic is |Aut(X )| 84(g − 1). For a positive characteristic, algebraic curves can have many more automorphisms than expected from the Hurwitz bound. There even exist algebraic curves of arbitrary high genus g with more than 16g 4 automorphisms. It has been observed on many occasions that the most anomalous examples of algebraic curves with very large automorphism groups invariably have zero p-rank. In this paper, the K-automorphism group Aut(X ) of a zero 2-rank algebraic curve X defined over an algebraically closed field K of characteristic 2 is investigated. The main result is that, if the curve has genus g 2 and |Aut(X )| > 24g(g − 1), then Aut(X ) has a fixed point on X , apart from a few exceptions. In the exceptional cases, the possibilities for Aut(X ) and g are determined.

show abstract

Zur Realisierbarkeit endlicher Gruppen als Automorphismengruppen algebraischer Funktionenk�rper

Cited by 9 publications

References 4 publications

Finite Groups as Galois Groups of Function Fields With Infinite Field of Constants

Finite Groups as Galois Groups of Function Fields With Infinite Field of Constants

On a variant of the Beckmann--Black problem

Automorphism groups of algebraic curves with p -rank zero

Contact Info

Product

Resources

About