We'll use formal properties of "correspondences with a core" to give "conceptual" (i.e. non-computational) proofs of statements like the following.

1. Any two supersingular elliptic curves over \bar{F_p} are related by an l-primary isogeny for any l\neq p.

2. A Hecke correspondence of compactified modular curves is always ramified at at least one cusp.

3. There is no canonical lift of supersingular points on a (projective) Shimura curve. (In particular, this provides yet another conceptual reason why there is not a canonical lift of supersingular elliptic curves.)

To do this, we'll introduce the concepts of "invariant line bundles" and of "invariant sections" on a correspondence without a core. Then (1), (2), and (3) will be implied by the following:

Theorem 1: Let X<-Z->Y be a correspondence of curves without a core over a field k. There is at most one etale clump.

Theorem 2. Let X<-Z->Y be an etale correspondence of curves without a core over a field of characteristic 0. Then there are no clumps. We'll end with several open questions.

### Algebra Seminar

Monday, October 9, 2017 - 3:15pm

#### Raju Krishnamoorthy

Free University Berlin