Roughly speaking, class field theory for a number field K describes the abelianization of its absolute Galois group in terms of the idele class group of K. Geometric class field theory is what we get when K is instead the function field of a smooth projective geometrically connected curve X over a finite field. In this talk, I give a precise statement of geometric class field theory in the unramified case and describe how one can prove it by showing the Picard stack of X is the “free dualizable commutative group stack on X”. A key part is to show that the usual “divisor class group exact sequence“ can be done in families to give the adelic uniformization of the Picard stack by the moduli space of Cartier divisors on X. 

The Hecke algebra is an algebraic gadget for studying the smooth complex representations of locally profinite groups. We demonstrate the spherical Hecke algebra of GL(n,F) is commutative and present a combinatorial proof of the Satake isomorphism. We apply this to the classification of spherical representations of GL(2,F).