John Baez and James Dolan, 2022-07-11

Classifying complex abelian surfaces in order of genericity, and the ranks of their Néron-Severi groups:

https://en.wikipedia.org/wiki/Abelian_surface
https://en.wikipedia.org/wiki/N%C3%A9ron%E2%80%93Severi_group

The generic case gives a Néron-Severi group of rank 1, the cartesian product of two distinct elliptic curves gives rank 2, the cartesian square of a generic elliptic curve gives rank 3 and the cartesian square of an elliptic curve with complex multiplication gives rank 4.

To understand this, we should look at the endomorphism ring of the abelian surface and tensor it with the reals, giving the "endomorphism algebra". Upon picking an polarization this algebra gets a Rosati involution:

https://en.wikipedia.org/wiki/Rosati_involution

which makes the algebra into a star-algebra, allowing us to split endomorphisms into a self-adjoint and skew-adjoint part.

The moduli stack of elliptic curves:

https://en.wikipedia.org/wiki/Moduli_stack_of_elliptic_curves

and the associated Coxeter group. How can we generalize this to higher-dimensional principally polarized abelian varieties? How do modular curves generalize to the higher-dimensional case, giving Siegel modular varieties?

https://en.wikipedia.org/wiki/Siegel_modular_variety

Видео John Baez and James Dolan, 2022-07-11 канала John Baez