Theta series of a positive definite lattice

Let $L$ be a lattice and $Q$ a positive definite quadratic form on its real span such that $Q(L)\subseteq\mathbb Z$ . Its theta series, with the norm normalization used here, is

\Theta_{L,Q}(z)=\sum_{x\in L}q^{Q(x)} =\sum_{n\ge0}r_{L,Q}(n)q^n, \qquad q=e^{2\pi iz},\quad\operatorname{Im}z>0,

where $r_{L,Q}(n)=\#\{x\in L:Q(x)=n\}$ . Positive definiteness makes these counts finite and the series convergent.

Quaternion orders

For an order in a definite rational quaternion algebra, take $Q=\operatorname{nrd}$ . Its theta coefficients count elements of each reduced norm. Related lattices give the coefficients of Brandt matrices.

Normalization

If a lattice is presented with an even integral bilinear form $b$ , a common convention is $Q(x)=b(x,x)/2$ . Replacing $Q$ by $2Q$ replaces $z$ by $2z$ ; it can change the modular level. Consequently one must specify both the lattice and its quadratic form when comparing theta series, including those associated with $D_4$ and Hurwitz quaternions.

References

John Voight, Quaternion Algebras, §40.4, equation (40.4.1), Lemma 40.4.2, and the level convention in Definition 40.4.3.