Counterexamples to the $\ell$-modular secrecy function conjecture

The secrecy function conjecture of Belfiore and Solé for unimodular lattices, which arose from wireless communications using lattice coset coding on gaussian wiretap channels, claims that for an $n$-dimensional unimodular lattice $\Gamma$, the quotient $\vartheta_\Gamma/\vartheta_{\mathbb Z^n}$ has, on the positive imaginary axis, a unique global minimum at the symmetry point $i$. Similarly, for integers $\ell\geqslant2$, a natural extension of this conjecture would be to claim that for an $\ell$-modular lattice, the quotient $\vartheta_\Gamma/\vartheta_{(\ell^{1/4}\mathbb Z)^n}$ has, on the positive imaginary axis, a unique global minimum at the symmetry point $i/\sqrt\ell$. Alas, Ernvall-Hytönen and Sethuraman showed that this is not the case as the behaviour of this quotient for the 4-modular lattice $\mathbb Z\oplus\sqrt2\,\mathbb Z\oplus2\mathbb Z$ is the opposite to the conjectured behaviour: there is a unique global maximum at the point $y=1/2$.
In this talk, we discuss the secrecy function conjecture. In particular, we describe an infinite family of $\ell$-modular counterexamples with $\ell\geqslant2$, constructed from direct products of dilates of $\mathbb Z$, for which we show that the behaviour is again opposite to what the extended secrecy function conjecture requires. One of the key ideas is using convexity properties of classical $\vartheta$-functions. The results described are joint work with A.-M. Ernvall-Hytönen.