Primitivity testing of finite nilpotent linear groups

AbstractWe describe a practical algorithm for primitivity testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For an imprimitive group, a system of imprimitivity can be constructed. An implementation of the algorithm in Magma is publicly available.

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Invariant subfields of rational function fields

Kodai Mathematical Seminar Reports ◽

10.2996/kmj/1138845501 ◽

1967 ◽

Vol 19 (4) ◽

pp. 451-453 ◽

Cited By ~ 1

Author(s):

Tsuneo Kanno

Keyword(s):

Rational Function ◽

Function Fields

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Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

Glasnik Matematicki ◽

10.3336/gm.56.1.06 ◽

2021 ◽

Vol 56 (1) ◽

pp. 79-94

Author(s):

Nikola Lelas ◽

Keyword(s):

Rational Function ◽

Finite Fields ◽

Function Fields ◽

Mean Value ◽

Multiplicative Functions ◽

Irreducible Polynomials ◽

Liouville Function ◽

The Mean

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

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New Binary Codes From Rational Function Fields

IEEE Transactions on Information Theory ◽

10.1109/tit.2014.2352306 ◽

2015 ◽

Vol 61 (1) ◽

pp. 60-65 ◽

Cited By ~ 1

Author(s):

Lingfei Jin ◽

Chaoping Xing

Keyword(s):

Rational Function ◽

Function Fields ◽

Binary Codes

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Algebraic systems of quadratic forms of number fields and function fields

manuscripta mathematica ◽

10.1007/bf01168301 ◽

1989 ◽

Vol 65 (2) ◽

pp. 225-243 ◽

Cited By ~ 5

Author(s):

Martin Kr�skemper

Keyword(s):

Quadratic Forms ◽

Function Fields ◽

Number Fields ◽

Algebraic Systems ◽

And Function

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Introduction

Berkeley Lectures on p-adic Geometry ◽

10.23943/princeton/9780691202082.003.0001 ◽

2020 ◽

pp. 1-6

Author(s):

Peter Scholze ◽

Jared Weinstein

Keyword(s):

Moduli Spaces ◽

Function Fields ◽

Number Fields ◽

Important Special Case ◽

Langlands Correspondence ◽

Key Innovation ◽

Perfectoid Spaces ◽

Definition Of ◽

Special Case

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.

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Simple Algebras Over Rational Function Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-078-1 ◽

1979 ◽

Vol 31 (4) ◽

pp. 831-835 ◽

Cited By ~ 1

Author(s):

T. Nyman ◽

G. Whaples

Keyword(s):

Rational Function ◽

Number Field ◽

Matrix Algebra ◽

Function Fields ◽

Simple Algebra ◽

Noether Theorem ◽

Full Matrix ◽

Full Matrix Algebra ◽

Rank One ◽

Simple Algebras

The well-known Hasse-Brauer-Noether theorem states that a simple algebra with center a number field k splits over k (i.e., is a full matrix algebra) if and only if it splits over the completion of k at every rank one valuation of k. It is natural to ask whether this principle can be extended to a broader class of fields. In particular, we prove here the following extension.

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Lifting n-Dimensional Galois Representations

Canadian Journal of Mathematics ◽

10.4153/cjm-2008-046-4 ◽

2008 ◽

Vol 60 (5) ◽

pp. 1028-1049 ◽

Cited By ~ 4

Author(s):

Spencer Hamblen

Keyword(s):

Characteristic Zero ◽

Galois Representations ◽

Galois Groups ◽

Geometric Shape ◽

General Linear ◽

Linear Groups ◽

Selmer Group ◽

General Linear Groups

AbstractWe investigate the problem of deforming n-dimensional mod p Galois representations to characteristic zero. The existence of 2-dimensional deformations has been proven under certain conditions by allowing ramification at additional primes in order to annihilate a dual Selmer group. We use the same general methods to prove the existence of n-dimensional deformations.We then examine under which conditions we may place restrictions on the shape of our deformations at p, with the goal of showing that under the correct conditions, the deformations may have locally geometric shape. We also use the existence of these deformations to prove the existence as Galois groups over ℚ of certain infinite subgroups of p-adic general linear groups.

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