The rational solutions on an elliptic curve form a finitely generated abelian group, but the maximum number of generators needed is not known. Goldfeld conjectured that if one also fixes the j-invariant (i.e. the complex structure), then 50% of such curves should require 1 generator and 50% should have only the trivial solution. Smith has recently made substantial progress towards this conjecture in the special case of elliptic curves in Legendre form. I'll discuss recent work with Lemke Oliver, which bounds the average number of generators for general j-invariants.

Global Chang's Conjecture

Yair Hayut - (joint with Monroe Eskew)

For $\kappa < \lambda$ infinite cardinals let us consider the following generalization of the Lowenheim-Skolem theorem:
"For every algebra with countably many operations over $\lambda^+$ there is a sub-algebra with order type exactly $\kappa^+$".

We will discuss the consistency and inconsistency of some global versions of this statement and present some open questions. 

The goal of my talk will be to formulate local and global Langlands conjectures Key words: smooth representations of p-adic groups, supercuspidal representations, automorphic representations, cusp forms, Galois representations.