Reduction mod $\ell$ of Theta Series of Level $\ell^n$

TL;DR

This paper proves that theta series of even lattices with level as a prime power are congruent modulo that prime to level 1 elliptic modular forms, using lattice properties rather than modular form theory.

Contribution

It introduces a lattice-based approach to congruences of theta series, enabling generalizations to Siegel and Hilbert modular forms.

Findings

Theta series of lattices with level $\ell^n$ are congruent modulo $\ell$ to level 1 modular forms.

The proof relies on arithmetic and algebraic properties of lattices, not modular form techniques.

Methods can be extended to more general theta series in advanced modular form theories.

Abstract

It is proved that the theta series of an even lattice whose level is a power of a prime $\ell$ is congruent modulo $\ell$ to an elliptic modular form of level~1. The proof uses arithmetic and algebraic properties of lattices rather than methods from the theory of modular forms. The methods presented here may therefore be especially pleasing to those working in the theory of quadratic forms, and they admit generalizations to more general types of theta series as they occur e.g. in the theory of Siegel or Hilbert modular forms.