Cusp forms of level one and weight zero

A theme in number theory is the non-existence of objects which are "too unramified".  For instance, by Minkowski there are no everywhere unramified extensions of Q, and by Fontaine and Abrashkin there are no abelian varieties over Q with everywhere good reduction.  Such results may be viewed (possibly conditionally) through the lens of the Stark-Odlyzko positivity method in the theory of L-functions.

 

After reviewing these things, I will turn to the question of this talk: for n>1 do there exist cuspidal automorphic forms for GL_n which are everywhere unramified and have lowest regular weight (cohomological weight 0)?  For n=2 these are more familiarly holomorphic cuspforms of level 1 and weight 2.  This question may be rephrased in terms of the existence of cuspidal cohomology of GL_n(Z) or (at least conjecturally) in terms of the existence of certain motives or Galois representations.  In 1997, Stephen Miller used the positivity method to show that they do not exist for n<27.  In the other direction, in joint work with Frank Calegari and Toby Gee, we prove that they do exist for some n, including n=79,105, and 106.