Automorphic forms and differentiability properties

Abstract

Abstract. We consider Fourier series given by a type of fractional integral of automorphic forms, and we study their local and global properties, especially differentiability and fractal dimension of the graph of their real and imaginary parts. In this way we can construct fractal objects and continuous non-differentiable functions associated with elliptic curves and theta functions.

The Hölder exponent of some Fourier series

“…The conditions are met by certain series arising from modular forms. These and other related series has been treated by several authors in connection with fractal and multifractal analysis [Cha04], [HT91], [Jaf96], [MS04], [Ota10], [Pet13], [RC14]. We deal with them in Section 3 and we deduce that the spectra of singularities of fractional integrals of cusp forms have a discrete image, in this sense they are pure fractals.…”

“…The proof of this result requires especial considerations when α − r/2 is a positive integer. In this case we need the following lemma that, in some sense, completes [Cha04] (cf. Corollary 2.1.1) including an extremal case.…”

Multifractal behavior of polynomial Fourier series

“…More recently several authors have shown interest on the global properties of R and allied functions (this interest was initially linked to wavelet methods [JM96], [HT91]). For instance, it is known that the (box counting) dimension of the graph of R is 5/4, in particular it is a fractal [Cha04], [CC99].…”

“…Basically R α is a fractional integral of the automorphic theta function θ(z) = ∞ n=−∞ e(n 2 z) and the local behavior of R α at a given x is determined by the convergents in the continued fraction of x and by the Fourier expansion of θ at the cusps (every rational number is equivalent to a cusp). This approach is developed in [Cha04] and [MS04] in a broader context.…”

The Highly Oscillatory Behavior of Automorphic Distributions for SL(2)

“…The absolute value of φ(x) also oscillates rapidly, as is illustrated by figure 2. Near the origin |φ(x)| evidently displays fractal behavior -see figure 3.…”