A topological realization of the congruence subgroup Kernel A

A number of years ago, Kumar Murty pointed out to me that the computation of the fundamental group of a Hilbert modular surface ([7],IV,${\S}$6), and the computation of the congruence subgroup kernel of SL(2) ([6]) were surprisingly similar. We puzzled over this, in particular over the role of elementary matrices in both computations. We formulated a very general result on the fundamental group of a Satake compactification of a locally symmetric space. This lead to our joint paper [1] with Lizhen Ji and Les Saper on these fundamental groups. Although the results in it were intriguingly similar to the corresponding calculations of the congruence subgroup kernel of the underlying algebraic group in [5], we were not able to demonstrate a direct connection (cf. [1], ${\S}$7). The purpose of this note is to explain such a connection. A covering space is constructed from inverse limits of reductive Borel-Serre compactifications. The congruence subgroup kernel then appears as the group of deck transformations of this covering. The key to this is the computation of the fundamental group in [1]

Digital electric field induced switching of plasmonic nanorods using an electro-optic fluid fiber

We demonstrate the digital electric field induced switching of plasmonic nanorods between 1 and 0 orthogonal aligned states using an electro-optic fluid fiber component. We show by digitally switching the nanorods, that thermal rotational diffusion of the nanorods can be circumvented, demonstrating an approach to achieve submicrosecond switching times. We also show, from an initial unaligned state, that the nanorods can be aligned into the applied electric field direction in 110 nanoseconds. The high-speed digital switching of plasmonic nanorods integrated into an all-fiber optical component may provide novel opportunities for remote sensing and signaling applications

Values of quadratic forms at primitive integral points

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