# Automorphism groups

Using Professor Shimada’s work on automorphism groups of K3 surfaces as a starting point, we produced a generalized and fully automated implementation of the Borcherds’ method. This implementation uses process-based parallelism to compute a generating set of $\aut(X)$, a set containing representatives of all the orbits of smooth rational curves on $X$ and much more.

The only input data required consists of a Gram matrix of $S=\text{NS}(X)$, of an ample class, and vectors defining a primitive embedding of $S$ into a suitable ambient even hyperbolic lattice, as explained in the first point below.

## Parallelism & The Borcherds’ method

Coming soon :

NB: The Poolized Borcherds’ method enforces parallelism at the level of the internal procedures of the Borcherds’ method. When we write “towards a parallelized Borcherds’ method”, we aim for also enforcing parallelism at the level of the Borcherds’ method itself: That is, parallelizing the exploration & processing of the chamber structure, while remaining able to make use of parallelism within the method, i.e. at the level of its core components, like in our Poolized implementation of the Borcherds’ method which takes advantage of process-based parallelism. We fulfill this goal by enforcing our PFB / APFB strategy.