36 citations · 90 across the 10 of their papers we have counts for
12 papers
Realizing orders as group rings
H. W. Lenstra, A. Silverberg, D. M. H. van Gent
An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use…
Arakelov class groups of random number fields
Alex Bartel, Henri Johnston, Hendrik W. Lenstra
The main purpose of the paper is to formulate a probabilistic model for Arakelov class groups in families of number fields, offering a correction to the Cohen--Lenstra--Martinet he…
On class groups of random number fields
Alex Bartel, Hendrik W. Lenstra
The main aim of the present paper is to disprove the Cohen--Lenstra--Martinet heuristics in two different ways and to offer possible corrections. We also recast the heuristics in t…
Testing isomorphism of lattices over CM-orders
Hendrik W. Lenstra, Alice Silverberg
A CM-order is a reduced order equipped with an involution that mimics complex conjugation. The Witt-Picard group of such an order is a certain group of ideal classes that is closel…
Universal gradings of orders
H. W. Lenstra, A. Silverberg
For commutative rings, we introduce the notion of a {\em universal grading}, which can be viewed as the "largest possible grading". While not every commutative ring (or order) has…
The unit theorem for finite-dimensional algebras
Hendrik Lenstra
The "unit theorem" to which the present mini-course is devoted is a theorem from algebra that has a combinatorial flavour, and that originated in fact from algebraic combinatorics.…
Commensurability of automorphism groups
Alex Bartel, Hendrik W. Lenstra
We develop a theory of commensurability of groups, of rings, and of modules. It allows us, in certain cases, to compare sizes of automorphism groups of modules, even when those are…
Algorithms for commutative algebras over the rational numbers
H. W. Lenstra, A. Silverberg
The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present determin…
Roots of unity in orders
H. W. Lenstra, A. Silverberg
We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the or…
Determining cyclicity of finite modules
H. W. Lenstra, A. Silverberg
We present a deterministic polynomial-time algorithm that determines whether a finite module over a finite commutative ring is cyclic, and if it is, outputs a generator.
Lattices with Symmetry
H. W. Lenstra, A. Silverberg
For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a pr…
Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves
Manjul Bhargava, Daniel M. Kane, Hendrik W. Lenstra +2
Using maximal isotropic submodules in a quadratic module over Z_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short…