Field of moduli and field of definition

2012 ◽  
pp. 113-119
Keyword(s):  
Field Of Moduli ◽  
Field Of Definition

scholarly journals A Class of Pseudo-Real Riemann Surfaces with Diagonal Automorphism Group

2020 ◽  
Vol 27 (02) ◽  
pp. 247-262
Author(s):  
Eslam Badr
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Riemann Surfaces ◽  
Real Plane ◽  
Plane Model ◽  
Field Of Moduli ◽  
Field Of Definition ◽  
Smooth Plane ◽  
Real Riemann Surfaces ◽  
Even Order

A Riemann surface [Formula: see text] having field of moduli ℝ, but not a field of definition, is called pseudo-real. This means that [Formula: see text] has anticonformal automorphisms, but none of them is an involution. A Riemann surface is said to be plane if it can be described by a smooth plane model of some degree d ≥ 4 in [Formula: see text]. We characterize pseudo-real-plane Riemann surfaces [Formula: see text], whose conformal automorphism group Aut+([Formula: see text]) is PGL3(ℂ)-conjugate to a finite non-trivial group that leaves invariant infinitely many points of [Formula: see text]. In particular, we show that such pseudo-real-plane Riemann surfaces exist only if Aut+([Formula: see text]) is cyclic of even order n dividing the degree d. Explicit families of pseudo-real-plane Riemann surfaces are given for any degree d = 2pm with m > 1 odd, p prime and n = d/p.

scholarly journals Hyperelliptic Curves with Extra Involutions

2005 ◽  
Vol 8 ◽  
pp. 102-115 ◽  
Author(s):  
J. Gutierrez ◽  
T. Shaska
Keyword(s):  
Automorphism Group ◽  
Moduli Space ◽  
Hyperelliptic Curves ◽  
Rational Model ◽  
Field Of Moduli ◽  
Field Of Definition

AbstractThe purpose of this paper is to study hyperelliptic curves with extra involutions. The locusLgof such genus-ghyperelliptic curves is ag-dimensional subvariety of the moduli space of hyperelliptic curvesHg. The authors present a birational parameterization ofLgvia dihedral invariants, and show how these invariants can be used to determine the field of moduli of points p ∈ Lg. They conjecture that for p ∈Hgwith |Aut(p)| > 2, the field of moduli is a field of definition, and they prove this conjecture for any point p ∈Lgsuch that the Klein 4-group is embedded in the reduced automorphism group ofp. Further, forg= 3, they show that for every moduli point p ∈H3such that |Aut(p)| > 4, the field of moduli is a field of definition. A rational model of the curve over its field of moduli is provided.

scholarly journals SOME SPECIAL FAMILIES OF HYPERELLIPTIC CURVES

2004 ◽  
Vol 03 (01) ◽  
pp. 75-89 ◽  
Author(s):  
TANUSH SHASKA
Keyword(s):  
Automorphism Group ◽  
Function Field ◽  
Hyperelliptic Curves ◽  
Necessary Condition ◽  
Field Of Moduli ◽  
Field Of Definition ◽  
G 12

Let [Formula: see text] denote the locus of hyperelliptic curves of genus g whose automorphism group contains a subgroup isomorphic to G. We study spaces [Formula: see text] for G≅ℤn, ℤ2⊕ℤn, ℤ2⊕A4, or SL2(3). We show that for G≅ℤn, ℤ2⊕ℤn, the space [Formula: see text] is a rational variety and find generators of its function field. For G≅ℤ2⊕A4, SL2(3) we find a necessary condition in terms of the coefficients, whether or not the curve belongs to [Formula: see text]. Further, we describe algebraically the loci of such curves for g≤12 and show that for all curves in these loci, the field of moduli is a field of definition.

scholarly journals Fields of Definition for Representations of Associative Algebras

2018 ◽  
Vol 62 (1) ◽  
pp. 291-304
Author(s):  
Dave Benson ◽  
Zinovy Reichstein
Keyword(s):  
Finite Extension ◽  
Representation Type ◽  
Essential Dimension ◽  
Extension Field ◽  
Associative Algebras ◽  
Separable Extension ◽  
Finite Representation ◽  
Field Of Definition ◽  
Finite Dimensional ◽  
A Finite Group

AbstractWe examine situations, where representations of a finite-dimensionalF-algebraAdefined over a separable extension fieldK/F, have a unique minimal field of definition. Here the base fieldFis assumed to be a field of dimension ≼1. In particular,Fcould be a finite field ork(t) ork((t)), wherekis algebraically closed. We show that a unique minimal field of definition exists if (a)K/Fis an algebraic extension or (b)Ais of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension ofF. This is not the case ifAis of infinite representation type orFfails to be of dimension ≼1. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of Karpenko, Pevtsova and the second author.

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