Reduction of Polarizations on Products of Elliptic Curves

Let $A = E_1 \times E_2$ be a product of elliptic curves. Polarizations on $A$ can be represented by positive definite Hermitian matrices $H =\begin{pmatrix}a ~~ h\\ \widehat{h}~~ b \end{pmatrix}$, with $a,b \in \mathbb{Z}$ and $h \in \operatorname{Hom}(E_1,E_2)$. Equivalence of polarizations corresponds to the natural congruence action of $\operatorname{Aut}(A)$ on these matrices.

We develop algorithms for reducing such matrix representatives within their equivalence classes. The algorithms transform a given polarization into an equivalent representative with controlled coefficients. By analyzing the reduction process, we derive explicit bounds on $a$, $b$, and $\deg(h)$ for the resulting representatives.

This talk is based on ongoing joint work with Harun Kir, Eli Orvis, and Benjamin Wesolowski.