Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various Gaussian integers. For any Gaussian integer $u+vi$, let $h_P(u+vi)$ be the number of lattice point in $(u+vi) \cdot P$. I would like to say that $h_P(u+vi)$ is in some sense "quasi-polynomial modulo $N$". To emphasize: The reason that this does not follow from the ordinary Ehrhart polynomial theorem is that I am multiplying by a Gaussian integer, not just an ordinary integer.

The case $N=1$ can be studied using Pick's theorem. The area of $P$ transforms simply under multiplication by Gaussian integers, the number of lattice points on the boundary is only a little messier. In particular, it is easy to show that there is some non-zero Gaussian integer $D$ such that, as long as $GCD(D, u+vi)=1$ and $GCD(u,v)=1$, then $h_P(u+vi) = (u^2+v^2) \mathrm{Area}(P) + \mbox{constant}$.

A result that would satisfy me is that there is some non-zero non-zero Gaussian integer $D$ such that, as long as $GCD(D, u+vi)=1$ and $GCD(u,v)=1$, then $$h_p(u,v) = (u^2+v^2) \mathrm{Area}(P) + a(u,v) u + b(u,v) v + c(u,v)$$ where $a$, $b$ and $c$ are periodic modulo $N$ in each input.

Motivation: This is enough to give an elementary deduction that the quartic residue symbol $\left[ \frac{z}{u+vi} \right]$ is periodic as a function of $u+vi$.

Motivation behind the motivation: There are a series of papers: Habicht, Kubota, Hill which promise to prove $m$-th power reciprocity laws roughly by counting lattice points. I can't follow any of them, so I thought I would true to reverse engineer what might be true in order to make the $\mathbb{Z}[i]$ proof come out.

For this reason, I would also be interested in statements for other number fields, but I don't have a precise question worked out in that setting.

  • $\begingroup$ I was under impression that the quasiperiodicity of Ehrhart polynomial holds for any lattice $\Lambda$ in $\mathbb{R}^d$ and a polytope with vertices in $\Lambda$. How does your setting differ? $\endgroup$ – Dima Pasechnik May 12 '14 at 8:15
  • 3
    $\begingroup$ @DimaPasechnik Because I'm multiplying by Gaussian integers, not ordinary integers. Edited to clarify. $\endgroup$ – David E Speyer May 12 '14 at 13:19
  • $\begingroup$ naively, at least if $|a+ib|\in\mathbb{Z}$, this should follow from the classical case; indeed, by multiplying by $a+ib$ you apply to your $\mathbb{C}$-plane the composition of a rotation and the scalar matrix $C=|a+ib|I$, and so the integer points would behave in the same way as by scaling with $C$ alone. $\endgroup$ – Dima Pasechnik May 13 '14 at 14:43
  • $\begingroup$ I do not see this. A rotation can change the lattice in a quite messy way. What am I missing? $\endgroup$ – David E Speyer May 14 '14 at 7:04
  • $\begingroup$ Yeah, right, I didn't think straight. Sorry for noise. $\endgroup$ – Dima Pasechnik May 15 '14 at 9:24

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