Reduction mod l of Theta Series of Level l n Nils -

It is proved that the theta series of an even lattice whose level is a power of a prime l is congruent modulo l 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. 1 Statement of Results Let l be a prime. We assume throughout that l ≥ 5. It is well-known that every modular form of level l is congruent modulo l to a modular form of level one [Serre, Théorème 5.4]. This fact applies in particular to theta series associated to quadratic forms whose level is a power of l. The purpose of this note is to prove a slightly more precise statement and to discuss various consequences. Though the main result is actually a statement about modular forms, the proof presented here works only for theta series. The virtue of this method of proof, however, is that it admits generalizations to more general types of theta series. We shall pursue this elsewhere. In this article we shall prove the following theorem. Main Theorem. Let L = (L, b) be an even integral lattice whose level is a power of l, and let e(L) be the sum of the elementary divisors of L. Then there exists a modular form f of level 1 and weight e(L)/2 and with integral Fourier coefficients such that