Kneser's method of neighbors

In a landmark paper published in 1957, Kneser introduced a method for enumerating classes in the genus of a definite, integral quadratic form. This method has been deeply influential, on account of its theoretical importance as well as its practicality. In this survey, we exhibit Kneser's method of neighbors and indicate some of its applications in number theory.Comment: 18 page

A canonical form for positive definite matrices

We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software. The algorithm runs in a number of arithmetic operations that is exponential in the dimension n, but it is practical and more efficient than canonical forms based on Minkowski reduction

Theta series of ternary quadratic lattice cosets

In this paper, we consider the decomposition of theta series for lattice cosets of ternary lattices. We show that the natural decomposition into an Eisenstein series, a unary theta function, and a cuspidal form which is orthogonal to unary theta functions correspond to the theta series for the genus, the deficiency of the theta series for the spinor genus from that of the genus, and the deficiency of the theta series for the class from that of the spinor genus, respectively. These three pieces are hence invariants of the genus, spinor genus, and class, respectively, extending known results for lattices and verifying a conjecture of the first author and Haensch. We furthermore extend the definition of $p$-neighbors to include lattice cosets and construct an algorithm to compute respresentatives for the classes in the genus or spinor genus via the $p$-neighborhoods

Algorithmic enumeration of ideal classes for quaternion orders

We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of two-sided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2.Comment: 39 pages, includes 2 tables; corrections made to Table 8.

Lattice methods for algebraic modular forms on classical groups

We use Kneser's neighbor method and isometry testing for lattices due to Plesken and Souveigner to compute systems of Hecke eigenvalues associated to definite forms of classical reductive algebraic groups

Lattice methods for algebraic modular forms on classical groups

Abstract We use Kneser's neighbor method and isometry testing for lattices due to Plesken and Souveigner to compute systems of Hecke eigenvalues associated to definite forms of classical reductive algebraic groups