From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond

The discovery of connections between the distribution of energy levels of heavy nuclei and spacings between prime numbers has been one of the most surprising and fruitful observations in the twentieth century. The connection between the two areas was first observed through Montgomery's work on the pair correlation of zeros of the Riemann zeta function. As its generalizations and consequences have motivated much of the following work, and to this day remains one of the most important outstanding conjectures in the field, it occupies a central role in our discussion below. We describe some of the many techniques and results from the past sixty years, especially the important roles played by numerical and experimental investigations, that led to the discovery of the connections and progress towards understanding the behaviors. In our survey of these two areas, we describe the common mathematics that explains the remarkable universality. We conclude with some thoughts on what might lie ahead in the pair correlation of zeros of the zeta function, and other similar quantities.Comment: Version 1.1, 50 pages, 6 figures. To appear in "Open Problems in Mathematics", Editors John Nash and Michael Th. Rassias. arXiv admin note: text overlap with arXiv:0909.491

Local systems and Suzuki groups

We study geometric monodromy groups G_{\geo,\sF_q} of the local sheaves \sF_q on the affine line over \F_2 of rank $D=\sqrt{q}(q-1)$, $q=2^{2n+1}$, constructed in \cite{Ka-ERS}. The main result of the paper shows that G_{\geo,\sF_q} is either the Suzuki simple group \tw2 B_2(q), or the special linear group \SL_D. We also show that \sF_8 has geometric monodromy group \tw2B_2(8), and arithmetic monodromy group \Aut(\tw2 B_2(8)) over \F_2, thus establishing \cite[Conjecture 2.2]{Ka-ERS} in full in the case $q=8$