Representation of cyclic groups in positive characteristic and Weierstrass semigroups

Karanikolopoulos, Sotiris; Kontogeorgis, Aristides

doi:10.1016/j.jnt.2012.05.039

Cited by 12 publications

(23 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As Lemma , Lemma , and the results of Tamagawa are valid over a perfect field, and the standard form of Lemma is enough to prove [, Theorem 1], we find that [, Theorem 7] (by the same arguments) holds over any perfect field of characteristic

p > 0

containing the requisite roots of unity. Precisely: Proposition Suppose that an extension L is a global standard function field (Definition ) such that

L / K

is cyclic.…”

Section: Standard Formmentioning

confidence: 75%

“…Remark The invariants in the previous theorem are the same as those in [, Theorem 7] (see Remark ), as

d_{scriptP} = 1

whenever k algebraically closed.…”

Section: Standard Formmentioning

confidence: 86%

“…We therefore prove the existence of a weaker standard form than the global standard form: Precisely, we are able to find a generator for each

L_{i} / L_{i - 1}

such that, for this choice of generator, all of the ramified places of the tower are simultaneously in local standard form. This is the only ingredient necessary for us to extend the proof of [, Lemma 8] from an algebraically closed constant field k to any perfect constant field which contains the requisite roots of unity. Lemma Let K be any function field of characteristic

p > 0

with perfect field of constants k , and let

L / K

be an Artin–Schreier extension.…”

Section: Standard Formmentioning

confidence: 99%

“…To each value of μ, we associate the p ‐adic expansion

\begin{matrix} true \\ right μ_{p} = μ_{A_{s}} + p μ_{A_{s - 1}} + \dots + μ_{A_{1}} p^{s - 1}, \end{matrix}

so that μ is associated with a unique

(μ_{p}, μ_{K u})

. We define the

k [G]

‐indecomposable module

Δ_{μ}

(see also [, Proposition 2]) to be the

μ_{p}

‐dimensional k ‐vector space with basis

{v_{1}, \dots, v_{μ_{p}}}

and Galois action given by

σ false(v_{i} false) = ξ^{μ_{K u}} v_{i} + v_{i + 1}

for all

1 \leq i \leq μ_{p} - 1

and

σ true(v_{μ_{p}} true) = ξ^{μ_{K u}} v_{μ_{p}}

. We obtain the decomposition in indecomposable

k [G]

‐modules

\begin{matrix} true \\ right {normalΩ}_{L} ≅ ⨁_{μ \in Γ} {normalΔ}_{μ}^{d_{μ}}, \end{matrix}

where

d_{μ}

denotes the number of times that the module

Δ_{μ}

appears in this decomposition of

Ω_{L}

.…”

Section: Standard Formmentioning

confidence: 99%

“…Proof This follows easily by [, Theorem 7], replacing the use of [, Theorem 1] with Lemma if k is not algebraically closed.

□

…”

Section: Standard Formmentioning

confidence: 99%

See 4 more Smart Citations

Holomorphic differentials of certain solvable covers of the projective line over a perfect field

Marques

Ward

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

p > 0

containing the requisite roots of unity. Precisely: Proposition Suppose that an extension L is a global standard function field (Definition ) such that

L / K

is cyclic.…”

Section: Standard Formmentioning

confidence: 75%

“…Remark The invariants in the previous theorem are the same as those in [, Theorem 7] (see Remark ), as

d_{scriptP} = 1

whenever k algebraically closed.…”

Section: Standard Formmentioning

confidence: 86%

“…We therefore prove the existence of a weaker standard form than the global standard form: Precisely, we are able to find a generator for each

L_{i} / L_{i - 1}

p > 0

with perfect field of constants k , and let

L / K

be an Artin–Schreier extension.…”

Section: Standard Formmentioning

confidence: 99%

“…To each value of μ, we associate the p ‐adic expansion

\begin{matrix} true \\ right μ_{p} = μ_{A_{s}} + p μ_{A_{s - 1}} + \dots + μ_{A_{1}} p^{s - 1}, \end{matrix}

so that μ is associated with a unique

(μ_{p}, μ_{K u})

. We define the

k [G]

‐indecomposable module

Δ_{μ}

(see also [, Proposition 2]) to be the

μ_{p}

‐dimensional k ‐vector space with basis

{v_{1}, \dots, v_{μ_{p}}}

and Galois action given by

σ false(v_{i} false) = ξ^{μ_{K u}} v_{i} + v_{i + 1}

for all

1 \leq i \leq μ_{p} - 1

and

σ true(v_{μ_{p}} true) = ξ^{μ_{K u}} v_{μ_{p}}

. We obtain the decomposition in indecomposable

k [G]

‐modules

\begin{matrix} true \\ right {normalΩ}_{L} ≅ ⨁_{μ \in Γ} {normalΔ}_{μ}^{d_{μ}}, \end{matrix}

where

d_{μ}

denotes the number of times that the module

Δ_{μ}

appears in this decomposition of

Ω_{L}

.…”

Section: Standard Formmentioning

confidence: 99%

“…Proof This follows easily by [, Theorem 7], replacing the use of [, Theorem 1] with Lemma if k is not algebraically closed.

□

…”

Section: Standard Formmentioning

confidence: 99%

See 3 more Smart Citations

Holomorphic differentials of certain solvable covers of the projective line over a perfect field

Marques

Ward

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

The canonical representation of the Drinfeld curve

Laurent,

Köck

2024

Mathematische Nachrichten

View full text Add to dashboard Cite

If is a smooth projective curve over an algebraically closed field and is a group of automorphisms of , the canonical representation of is given by the induced ‐linear action of on the vector space of holomorphic differentials on . Computing it is still an open problem in general when the cover is wildly ramified. In this paper, we fix a prime power , we consider the Drinfeld curve, that is, the curve given by the equation over together with its standard action by , and decompose as a direct sum of indecomposable representations of , thus solving the aforementioned problem in this case.

show abstract

Automorphisms of Curves

Antoniadis

Kontogeorgis

2017

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

Representation of cyclic groups in positive characteristic and Weierstrass semigroups

Cited by 12 publications

References 17 publications

Holomorphic differentials of certain solvable covers of the projective line over a perfect field

Holomorphic differentials of certain solvable covers of the projective line over a perfect field

The canonical representation of the Drinfeld curve

Automorphisms of Curves

Contact Info

Product

Resources

About