On the automorphisms of the non-split Cartan modular curves of prime level

We study the automorphisms of the non-split Cartan modular curves $X_{ns}(p)$ of prime level $p$. We prove that if $p\geq 37$ all the automorphisms preserve the cusps. Furthermore, if $p\equiv 1\text{ mod }12$ and $p\neq 13$, the automorphism group is generated by the modular involution given by the normalizer of a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_p)$. We also prove that for every $p\geq 37$ such that $X_{ns}(p)$ has a CM rational point, the existence of an exceptional rational automorphism would give rise to an exceptional rational point on the modular curve $X_{ns}^+(p)$ associated to the normalizer of a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_p)$

Automorphisms of Cartan modular curves of prime and composite level

We study the automorphisms of modular curves associated to Cartan subgroups of $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ and certain subgroups of their normalizers. We prove that if $n$ is large enough, all the automorphisms are induced by the ramified covering of the complex upper half-plane. We get new results for non-split curves of prime level $p\ge 13$: the curve $X_{\text{ns}}^+(p)$ has no non-trivial automorphisms, whereas the curve $X_{\text{ns}}(p)$ has exactly one non-trivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of $X_0^*(n):=X_0(n)/W$, where $W$ is the group generated by the Atkin-Lehner involutions of $X_0(n)$ and $n$ is a large enough square.Comment: 31 pages, 2 tables. Some proofs rely on MAGMA scripts available at https://github.com/guidoshore/automorphisms_of_Cartan_modular_curve

Recent Progress in the Computational Many-Body Theory of Metal Surfaces

In this article we describe recent progress in the computational many-body theory of metal surfaces, and focus on current techniques beyond the local-density approximation of density-functional theory. We overview various applications to ground and excited states. We discuss the exchange-correlation hole, the surface energy, and the work function of jellium surfaces, as obtained within the random-phase approximation, a time-dependent density-functional approach, and quantum Monte Carlo methods. We also present a survey of recent quasiparticle calculations of unoccupied states at both jellium and real surfaces.Comment: 17 pages, 1 figure, to appear in Comput. Phys. Commu

A high throughput molecular force assay for protein-DNA interactions.

An accurate and genome-wide characterization of protein–DNA interactions such as transcription factor binding is of utmost importance for modern biology. Powerful screening methods emerged. But the vast majority of these techniques depend on special labels or markers against the ligand of interest and moreover most of them are not suitable for detecting low-affinity binders. In this article a molecular force assay is described based on measuring comparative unbinding forces of biomolecules for the detection of protein–DNA interactions. The measurement of binding or unbinding forces has several unique advantages in biological applications since the interaction between certain molecules and not the mere presence of one of them is detected. No label or marker against the protein is needed and only specifically bound ligands are detected. In addition the force-based assay permits the detection of ligands over a broad range of affinities in a crowded and opaque ambient environment. We demonstrate that the molecular force assay allows highly sensitive and fast detection of protein–DNA interactions. As a proof of principle, binding of the protein EcoRI to its DNA recognition sequence is measured and the corresponding dissociation constant in the sub-nanomolar range is determined. Furthermore, we introduce a new, simplified setup employing FRET pairs on the molecular level and standard epi-fluorescence for readout. Due to these advancements we can now demonstrate that a feature size of a few microns is sufficient for the measurement process. This will open a new paradigm in high-throughput screening with all the advantages of force-based ligand detection. Graphical abstract: A high throughput molecular force assay for protein–DNA interaction

Confocal microscopy of colloidal particles: towards reliable, optimum coordinates

Over the last decade, the light microscope has become increasingly useful as a quantitative tool for studying colloidal systems. The ability to obtain particle coordinates in bulk samples from micrographs is particularly appealing. In this paper we review and extend methods for optimal image formation of colloidal samples, which is vital for particle coordinates of the highest accuracy, and for extracting the most reliable coordinates from these images. We discuss in depth the accuracy of the coordinates, which is sensitive to the details of the colloidal system and the imaging system. Moreover, this accuracy can vary between particles, particularly in dense systems. We introduce a previously unreported error estimate and use it to develop an iterative method for finding particle coordinates. This individual-particle accuracy assessment also allows comparison between particle locations obtained from different experiments. Though aimed primarily at confocal microscopy studies of colloidal systems, the methods outlined here should transfer readily to many other feature extraction problems, especially where features may overlap one another.Comment: Accepted by Advances in Colloid and Interface Scienc

Modular Curves with many Points over Finite Fields

We compute the number of points over finite fields of some classes of modular curves, namely $X_0(N)$, $X_0^+(N)$, without using explicit equations. In this way we could improve many lower bounds for the maximum number of points of a curve over finite fields