Animating quadratic and bilinear maps on abelian varieties

If $A/k$ is an abelian variety, there are no non trivial maps (linear, bilinear, quadratic) from $A$ (or $A \times B$) to $G_m$. However, seeing these objects as fppf sheafs of anima (i.e., $\infty$-groupoids) rather than fppf sheafs of sets, the space/anima of linear maps, bilinear and quadratic maps is highly non trivial. Using the Dold-Kan correspondance, we can interpret their $\pi_1$ as, respectively:

- linear maps $A \to BG_m$, i.e. as elements of the dual abelian variety $\widehat{A}=Hom(A, BG_m)$

- biextensions of $A \times B$ by $G_m$

- cubical structures on $G_m$-torsors on $A$

This talk will be divided in three part.

In the first elementary part, we will sketch the many analogies between bilinear and quadratic maps on one hand, and polarisations and line bundles on an abelian variety on the other hand.

In the second part, we will give a sketch of the animation procedure and why it explains the above analogies.

Finally, in the third part, we will give algorithmic applications. In particular, cubical arithmetic serves as a swiss-knife toolbox for abelian varieties, since it can be used to recover the biextension arithmetic and theta group arithmetic, and allows to compute pairings, isogenies, radical isogenies, isogeny preimages, change of level... If time permits, we'll also give an example on how it sheds new lights on the DLP, notably via the monodromy leak attack.