# Mathematics Calendar

Wednesday, May 30, 2018 - 13:00

HU, Insitut für Mathematik

Rudower Chaussee 25, 12489 Berlin-Adlershof, 1.114, 1. Stock

Forschungsseminar "Algebraische Geometrie"

Prof. Dr. Gavril Farkas, Prof. Dr. Bruno Klingler

We'll use formal properties of "correspondences without a core" to sketch "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.

submitted by Kristina Schulze (schulze@math.hu-berlin.de)