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

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.