Counting classes and characters of groups of prime exponent

2006 ◽  
Vol 156 (1) ◽  
pp. 205-220 ◽  
Author(s):  
Noboru Ito ◽  
Avinoam Mann
Keyword(s):  
Counting Classes ◽  
Prime Exponent

Stability of nilpotent groups of class 2 and prime exponent

1981 ◽  
Vol 46 (4) ◽  
pp. 781-788 ◽  
Author(s):  
Alan H. Mekler
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Stability Class ◽  
Similarity Type ◽  
Nilpotent Class ◽  
Prime Exponent

AbstractLet p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M.Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes.Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class.

Introduction to Counting Classes

2011 ◽  
pp. 247-260
Author(s):  
Steven Homer ◽  
Alan L. Selman
Keyword(s):  
Counting Classes

scholarly journals Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

2016 ◽  
Vol 68 (2) ◽  
pp. 361-394
Author(s):  
Francesc Fité ◽  
Josep González ◽  
Joan-Carles Lario
Keyword(s):  
Abelian Variety ◽  
Complex Multiplication ◽  
Cyclotomic Field ◽  
Explicit Method ◽  
Fermat Curve ◽  
Tate Conjecture ◽  
Tate Group ◽  
Fermat Curves ◽  
Prime Exponent ◽  
Simple Abelian Variety

AbstractLet denote the Fermat curve over ℚ of prime exponent ℓ. The Jacobian Jac() of splits over ℚ as the product of Jacobians Jac(k), 1 ≤ k ≤ ℓ −2, where k are curves obtained as quotients of by certain subgroups of automorphisms of . It is well known that Jac(k) is the power of an absolutely simple abelian variety Bk with complex multiplication. We call degenerate those pairs (ℓ, k) for which Bk has degenerate CM type. For a non-degenerate pair (ℓ, k), we compute the Sato–Tate group of Jac(Ck), prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether (ℓ, k) is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the ℓ-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

Central simple algebras of prime exponent and divided power operations

2013 ◽  
Vol 11 (1) ◽  
pp. 113-123 ◽  
Author(s):  
A.S. Sivatski
Keyword(s):  
Division Algebra ◽  
Roots Of Unity ◽  
Central Simple Algebras ◽  
Divided Power ◽  
Open Questions ◽  
Primary Roots ◽  
Power Operations ◽  
Simple Algebras ◽  
Central Simple ◽  
Prime Exponent

AbstractLet p be a prime and F a field of characteristic different from p. Suppose all p-primary roots of unity are contained in F. Let α ∈ pBr(F) which has a cyclic splitting field. We prove that γi(α) = 0 for all i ≥ 2, where γi : pBr(F) → K2i(F)/pK2i(F) are the divided power operations of degree p. We also show that if char F ≠ 2, √−1 ∈ F*. D ∈2 Br(F), indD = 8 and a ∈ F* such that ind DF(√a) = 4, then γ3(D) = {a,s}γ2(D) for some s ∈ F*. Consequently, we prove that if D, considered as a division algebra, has a subfield of degree 4 of certain type, then γ3(D) = 0. At the end of the paper we pose a few open questions.

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