## Algebraic Geometry / Number Theory Seminar

We'll use formal properties of "correspondences without a core" to give "conceptual" (i.e. non-computational) proofs of statements like the following.

1. Any two supersingular elliptic curves over $\overline{\textbf{F}_p}$ are related by an $\ell$-primary isogeny for any $\ell \neq p$.

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 $X \leftarrow Z \to Y$ be a correspondence of curves without a core over a field k. There is at most one étale clump.

Theorem 2. Let $X \leftarrow Z \to Y$ be an étale correspondence of curves without a core over a field of characteristic 0. Then there are no clumps.

We'll end with several open questions.