• ModularForms
• ThetaFunctions

A lattice is a free Z-module equipped with an integral positive-definite quadratic form.

We want to create a database of lattices useful for the purposes of creating tables of modular forms, e.g. classical, Siegel, Hilbert modular forms. Given level and weight, construct theta series of lattices that are modular forms of the given weight.

What we can do (in principle): Find the lattices, calculate some coefficients of the theta-series (Siegel, Hilbert ...) This might be improved, using automorphism groups (??) or decomposable lattices of small index, resp. both.

Here we first concentrate on finding the lattices.

What exists

• MAGMA

Tables

• All genera (Genus-Symbols) of given dimension and determinant (or elementary divisors)
• All lattices in the given genus.

Desired Algorithms

• Genus symbol for number fields (esp. for p\mid 2)

• Explicit construction of some orthogonally deomposable lattice in a genus

• Local => Global for quadratic spaces and lattices

• Neighbor method for (pos def & Hermitian) lattice

• Hecke eigenvalues from neighbor method
• Local orbit identification for fast neighbor generation

Necessary subalgorithms

• Calculation of theta series: Elkies(?) algorithm using small index, shortest vector and orthogonal complement
• Calculation of genus
• Isometry of lattices (easy test for not), automorphisms of lattices (C programs?)
  • Enumeration of small elements
  • Bernd Souvignier's existing program
  • Invariants for quickly showing if two lattices are not isometric (32-dimensional lattices example?)

• Mass formula: p-inclusion, local conditions (Shimura paper in Jon's thesis?)

• Given local data, compute a global representative?
  • Orthogonal decomposition of a lattice
  • Given an eigenform, can you associate to it a linear combination of lattices? (Cusp form or Eisenstein series).

Questions

• When does a genus given by local data contain an orthogonally decomposable lattice ?

Table data

• LLL-reduced basis
• Dimension
• Elementary Divisors
• Determinant
• Level
• Genus symbol (set of primes...)
• Class number if known
• Dimension of the linear span of the theta series of lattices in the genus
• First few coefficients of theta series (to determine it in the space of modular forms)

Look up by level, first few coefficients Low dimensions (binary quadratic forms, from quaternion algebras, ...)

For the future

Lattices which are not necessarily positive definite over Z: e.g. Hermitian/indefinite, or integral, or over number rings. Maybe Epstein zeta functions of lattices.

Algorithms

• [NeighborMethod The p-Neighbor method]

Definitions

An integral lattice L of rank n over \mathbb{Z} is a free abelian group with a non-degenerate symmetric bilinear form B: L \times L \rightarrow \mathbb{Z}. The associated quadratic form is q: L \rightarrow \mathbb{Z} is given by q(x) = B(x,x).

A lattice or quadratic module L over R is a free R-module equipped with a non-degenerate symmetric bilinear form L \times L \rightarrow R. Two lattices L and M over R are equivalent over R if there is an isomorphism of modules L \rightarrow M which is compatible with the forms.

The theta function \vartheta_L of an integral lattice L is

.