Representations by quadratic forms and the
Eichler Commutation Relation

@article{Walling2019RepresentationsBQ,
title={Representations by quadratic forms and the
Eichler Commutation Relation},
author={Lynne Walling},
journal={Automorphic Forms and Related Topics},
year={2019}
}

Let Q be a positive definite quadratic form on a lattice L = Zx1 ⊕ · · · ⊕ Zxm, with associated symmetric bilinear form B so that B(x, x) = Q(x). For convenience, we assume that L is even integral, meaning that Q(L) ⊆ 2Z. Question: Given an n-dimensional quadratic form T , on how many sublattices Λ of L does Q restrict to T? To help us work toward an answer to this question, we define theta series, as follows. Theta series: For n = 1, and τ in the complex upper half-plane, we set

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