Title:

Algorithmics of lattices in euclidean spaces with application to computations with siegel modular forms

In computational number theory I developed an algorithm to compute simultaneous
Hecke eigenforms. This problem transforms to a computational problem in the theory
of lattices in Euclidean spaces, which, as I discovered, is connected to classical group such
as the special orthogonal group over a finite field. By extending the Kneser neighboring
process, I introduced a family of commuting, selfadjoint linear operators that act on the
vector spaces of the isometry classes of lattices. To get their representation explicitly,
instead of using a trivial, slow technique of enumerating all neighbor lattices of a given
lattice, which consists in bruteforce search in a finite set of candidates for a solution of
the problem and in rather a simple test for any such candidate whether it is a solution,
I designed an efficient algorithm that is also easy to implement.
In this work I focused towards advancing theory that is also verified computationally
by implementation of the algorithms in Magma. Thereby I computed the action of algebra
of the Hecke operators on the Hilbert spaces of modular forms spanned by the Siegel
theta series and produced simultaneous eigenfunctions explicitly as linear combinations
of Siegel theta series attached to positive definite integral quadratic forms of nontrivial
level. Not only could I then numerically confirm the results that were theoretically
derived by Walling for eigenvalues of the average Siegel theta series, but also explain
them via the aforementioned connection between the problem to construct neighbor
lattices and the theory of buildings of groups. Moreover, in order to identify nonzero
cusp forms among eigenfunctions, I extended the PoorYuen method for determining
support sets for the Fourier coefficients. Last but not least, my numerical observations
suggest a strategy for constructing new cases of Hecke eigenforms that are cusp forms
and describing their eigenvalues by a formula.
