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.

Risk and utility functionals are fundamental building blocks in economics and finance. In this paper we investigate under which conditions a risk or utility functional is sensitive to the accumulation of losses in the sense that any sufficiently large multiple of a position that exposes an agent to future losses has positive risk or negative utility. We call this property sensitivity to large losses and provide necessary and sufficient conditions thereof that are easy to check for a very large class of risk and utility functionals. In particular, our results do not rely on convexity and can therefore also be applied to most examples discussed in the recent literature, including (non-convex) star-shaped risk measures or S-shaped utility functions encountered in prospect theory. As expected, Value at Risk generally fails to be sensitive to large losses. More surprisingly, this is also true of Expected Shortfall. By contrast, expected utility functionals as well as (optimized) certainty equivalents are proved to be sensitive to large losses for many standard choices of concave and nonconcave utility functions, including S-shaped utility functions. We also show that Value at Risk and Expected Shortfall become sensitive to large losses if they are either properly adjusted or if the property is suitably localized.