On the dimension of an Abelian group

We introduce a measure of dimensionality of an Abelian group. Our definition of dimension is based on studying perpendicularity relations in an Abelian group. For G ≅ ℤn, dimension and rank coincide but in general they are different. For example, we show that dimension is sensitive to the overall dimensional structure of a finite or finitely generated Abelian group, whereas rank ignores the torsion subgroup completely.

A Theorem About Maximal Cohen–Macaulay Modules

It is shown that for any local strongly $F$-regular ring there exists natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen–Macaulay module, then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite.

Elliptic surfaces over ℙ1 and large class groups of number fields

Given a non-isotrivial elliptic curve over [Formula: see text] with large Mordell–Weil rank, we explain how one can build, for suitable small primes [Formula: see text], infinitely many fields of degree [Formula: see text] whose ideal class group has a large [Formula: see text]-torsion subgroup. As an example, we show the existence of infinitely many cubic fields whose ideal class group contains a subgroup isomorphic to [Formula: see text].

Generators and number fields for torsion points of a special elliptic curve

Let E be an elliptic curve with Weierstrass form y2=x3−px, where p is a prime number and let E[m] be its m-torsion subgroup. Let p1=(x1,y1) and p2=(x2,y2) be a basis for E[m], then we prove that ℚ(E[m])=ℚ(x1,x2,ξm,y1) in general. We also find all the generators and degrees of the extensions ℚ(E[m])/ℚ for m=3 and m=4.

Low-complexity computations for nilpotent subgroup problems

We solve the following algorithmic problems using [Formula: see text] circuits, or in logspace and quasilinear time, uniformly in the class of nilpotent groups with bounded nilpotency class and rank: subgroup conjugacy, computing the normalizer and isolator of a subgroup, coset intersection, and computing the torsion subgroup. Additionally, if any input words are provided in compressed form as straight-line programs or in Mal’cev coordinates, the algorithms run in quartic time.