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The original example:

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”.

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The original example:

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  1. Regularity of Automorphic Distributions_______________________CRM Workshop May 3, 2004Stephen Miller (Rutgers University)Wilfried Schmid (Harvard University)

  2. 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.

  3. Graph of Riemann’s Function (Influential in the development of calculus) Note replication • (Sidenote) Mandelbrot: The “crisis” caused by this function launched fractals.

  4. 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

  5. Automorphic Distributions of holomorphic modular forms In general start with a q-expansion Restrict to x-axis The distribution F inherits automorphy from t :

  6. 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.

  7. Weight one antiderivative for Gamma_0(23) This is the image in the complex plane of the antiderivative’s values on the real line

  8. Cruder View

  9. Maass Form Antiderivative

  10. Zoom near origin

  11. Weight one antiderivative

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