Mathematics – Number Theory
Scientific paper
2010-08-24
Mathematics
Number Theory
64 pages
Scientific paper
A path space realization of $PSL(2, \lvZ)$ is given, by means of which an action of this group on sections of the universal prounipotent bundle with connection on $\P^1\setminus\{0,1,\infty\}$ is produced. Under this action, the distinguished section given by the polylogarithm generating series $Li(z)$ is mapped to a family of formal power series each multiplied by $Li(z)$ itself. The resulting expressions are prototypical of statements of the quasi-modularity defined in the paper. In this case, the automorphy factors corresponding to the involutive and free generators of $PSL(2, \lvZ)$ are respectively given by a transform of the Drinfel'd associator and an $R$-matrix. Since we prove also that the associated mapping of $PSL(2, \lvZ)$ into formal power series is injective, a power series realization of $PSL(2, \lvZ)$ emerges which constitutes both an embedding of this group into the prounipotent completion of the fundamental group of $\P^1\setminus\{0,1,\infty\},$ and also into Drinfel'd's formal power series model of the quasi-triangular quasi-Hopf algebras. For certain qtqH algebras (satisfying an hypothesis on the $R$-matrix), a mapping of representations of such algebras into the representations of $PSL(2,\lvZ)$ arises. The quasi-modularity provides an echo of usual modularity in an important example: Riemann's contour integral proof of the functional equation of the zeta function $\zeta(s)$ is shown to belong to a family of proofs following from the quasi-modularity of $Li(z)$, reminiscent of the way in which the modularity of the theta function yields his Fourier analysis proof.
Joyner Sheldon T.
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