In a category enriched in a closed symmetric monoidal category, the power

object construction, if it is representable, gives a contravariant monoidal

action. We first survey the construction, due to Serre, of the power object

by (projective) Hermitian modules on abelian varieties. The resulting

action, when applied to a primitively oriented elliptic curve, gives a

contravariant equivalence of category (Jordan, Keeton, Poonen, Rains,

Shepherd-Barron and Tate).

We then give several applications of this module action:

1) We first explain how it allows to describe purely algebraically the

ideal class group action on an elliptic curve or the Shimura class group

action on a CM abelian variety over a finite field, without lifting to

characteristic 0.

2) We then extend the usual algorithms for the ideal action to the case of

modules, and use it to explore isogeny graphs of powers of an elliptic

curve in dimension up to 4. This allows us to find new examples of curves

with many points. (This is a joint work with Kirschmer, Narbonne and

Ritzenthaler)

3) Finally, we give new applications for isogeny based cryptography. We

explain how, via the Weil restriction, the supersingular isogeny path

problem can be recast as a rank 2 module action inversion problem. We also

propose ⊗-MIKE a novel NIKE (non interactive isogeny key exchange) that only

needs to send j-invariants of supersingular curves, and compute a dimension

4 abelian variety as the shared secret.