What are shifted lattices?

Question

As a part of a project, I have been mentioned the work of deriving the functional equation of some $L$-functions associated with shifted lattices. Functional equations are not hard to obtain, but what it means to associate it with shifted lattices. Before that, what exactly are shifted lattices. Is it connected to what we study in Modular Forms?

Answer 1

Take $A\in GL_k(\Bbb{R})$ such that $A^\top A$ has integer entries then $$f_A(x) = \sum_{m\in\Bbb{Z}^n} e^{-\pi \|Am\|^2 x}$$ is $2i$-periodic and the Poisson summation formula gives $$f_A(x)= \det(A)^{-1} x^{-k/2} f_{A^{-\top}}(1/x)$$ Thus it is $\in M_{k/2}(\Gamma_1(n)))$.

What they are asking is to investigate, for $b\in \Bbb{Q}^k$, the modularity of $$f_{A,b}(x) = \sum_{m\in\Bbb{Z}^k} e^{-\pi \|A(m+b)\|^2 x}$$ Using the dual functions $$f_{A,b,c} (x)= \sum_{m\in\Bbb{Z}^k} e^{-\pi \|A(m+b)\|^2 x}e^{2i\pi c^\top x}$$ you'll find it is in $M_{k/2}(\Gamma_1(N))$ for some $N$.

Those things are useful because they lead to the Hecke L-functions of imaginary quadratic number fields, the first cases of the Galois-automorphic correspondence as well as the modularity of CM elliptic curves.