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This workshop by Stephen Miller and Wilfried Schmid delves into the regularity of automorphic distributions, specifically in relation to holomorphic modular forms and their connections to functions like the Riemann/Weierstrass non-differentiable function. The talk highlights historical claims around non-differentiability and showcases the significance of such functions in calculus and fractals. Key topics include the q-function, automorphy explanations for replication, and the characteristics of antiderivatives for various cusp forms. A fascinating exploration of mathematical structures and behavior awaits.
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Regularity of Automorphic Distributions_______________________CRM Workshop May 3, 2004Stephen 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 odd.
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.
Weight one antiderivative for Gamma_0(23) This is the image in the complex plane of the antiderivative’s values on the real line
