Recent advances in isogeny-based cryptography have motivated the development of algorithms for higher-dimensional analogues of elliptic curves, called abelian varieties. Due to their rich geometric structure, a central challenge is to identify efficient and
workable representations of these objects. Among the available approaches, we focus on the framework of Hermitian modules. In this talk, we present an algorithm for computing a basis of such modules, and we show how it can be interpreted as a solver
for the principal ideal problem in the endomorphism ring of the associated abelian variety. We also highlight how this module-based representation enables improvements of state-of-the-art algorithms, such as KLPT2 .

Joint work with Aurel Page and Damien Robert.