Transcript Regularity of Automorphic Distributions _______________________ CRM Workshop

Regularity of
Automorphic Distributions
_______________________
CRM Workshop
May 3, 2004
Stephen Miller (Rutgers University)
Wilfried Schmid (Harvard University)
The original example:
Riemann/Weierstrass “Non-differentiable Function”
• Historical claim: f(x) is non-differentiable at all real x !
• Hardy(1916): proven for almost all x
• Gerver (1970) disproven!
f’(x) = - p
for x = 2p/q,
p and q
Graph of Riemann’s Function
(Influential in the development of calculus)
Note replication
(Sidenote) Mandelbrot: The “crisis” caused
by this function launched fractals.
What does this have to do with
automorphic forms?
f’(x) is essentially the q-function
restricted to the real axis
which exists as an automorphic distribution.
This automorphy explains the replication.
In fact Gerver’s points x=2p/q are the orbit of x=1/2
under G0(4)
and q is cuspidal in exactly this of the three cusps
(0,1/2,∞) of G0(4)\H
Automorphic Distributions of
holomorphic modular forms
In general start with a q-expansion
Restrict to x-axis
The distribution F
inherits automorphy from t :
Regularity of Automorphic
Distributions
For holomorphic cusp forms of weight one (and
Maass forms…), t is the first anti-derivative
of a continuous function
whose Hölder properties can be
nearly-exactly characterized.
This is the image in the complex plane of the antiderivative’s values on the real line
Cruder View
Maass Form Antiderivative
Zoom near origin
Weight one antiderivative