Standard sequence subgroups in finite fields

Nonstandard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes

Let q = pr be a prime power, and let f(x) = xm − σm−1x m−1 − · · · − σ1x− σ0 be an irreducible polynomial over the finite field GF(q) of size q. A zero ξ of f is called nonstandard (of degree m) over GF(q) if the recurrence relation um = σm−1um−1 + · · · + σ1u1 + σ0u0 with characteristic polynomial f can generate the powers of ξ in a nontrivial way, that is, with u0 = 1 and f(u1) 6= 0. In 2003,...

Elliptic curves over finite fields with fixed subgroups

We prove that for any given group Zm⊕Zn, where m divides n, and any rational elliptic curve, for a positive density of the rational primes p ∈ P, Zm ⊕ Zn is isomorphic to a subgroup of E(Fp). Our methods are effective and we demonstrate how to construct elliptic curves such that for a large density of the primes p, the given group is isomorphic to a subgroup of E(Fp). We show that for some grou...

Rep#2a: Finite subgroups of multiplicative groups of fields

This theorem generalizes the (well-known) fact that the multiplicative group of a finite field is cyclic. Most proofs of this fact can actually be used to prove Theorem 1 in all its generality, so there is not much need to provide another proof here. But yet, let us sketch a proof of Theorem 1 that requires only basic number theory. The downside is that it is very ugly. First, an easy number-th...

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ` is a prime different from p, and for some difference closed field (K, σ) the `-primary torsion of G(K)...

CONGRUENCE SUBGROUPS AND TWISTED COHOMOLOGY OF SLn(F [t℄) II: FINITE FIELDS AND NUMBER FIELDS