Quadratic forms, generalized Hamming weights of codes and curves with many points

We use the relations between quadrics, trace codes and algebraic curves to construct algebraic curves over finite fields with many points and to compute generalized Hamming weights of codes.Comment: 14 pages, Plain Te

Relations between some invariants of algebraic varieties in positive characteristic

We discuss relations between certain invariants of varieties in positive characteristic, like the a-number and the height of the Artin-Mazur formal group. We calculate the a-number for Fermat surfacesComment: 13 page

Effective divisors on projectivized Hodge bundles and modular Forms

We construct vector‐valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In particular, we construct basic modular forms for genus 2 and 3. We also discuss modular forms on the moduli of hyperelliptic curves. In that case, the relative canonical bundle is a pull back of a line bundle on a ℙ1‐bundle over the moduli of hyperelliptic curves and we extend that line bundle to a compactification so that its push down is (close to) the Hodge bundle and use this to construct modular forms. In the Appendix, we use our method to calculate divisor classes in the dual projectivized k‐Hodge bundle determined by Gheorghita-Tarasca and by Korotkin-Sauvaget-Zograf

Generating Picard modular forms by means of invariant theory

We use the description of the Picard modular surface for discriminant -3 as a moduli space of curves of genus 3 to generate all vector-valued Picard modular forms from bi-covariants for the action of GL2 on the space of pairs of binary forms of bi-degree (4, 1). The universal binary forms of degree 4 and 1 correspond to a meromorphic modular form of weight (4, -2)and a holomorphic Eisenstein series of weight (1, 1)

Concept of a laser-plasma based electron source for sub-10 fs electron diffraction

We propose a new concept of an electron source for ultrafast electron diffraction with sub-10~fs temporal resolution. Electrons are generated in a laser-plasma accelerator, able to deliver femtosecond electron bunches at 5 MeV energy with kHz repetition rate. The possibility of producing this electron source is demonstrated using Particle-In-Cell simulations. We then use particle tracking simulations to show that this electron beam can be transported and manipulated in a realistic beamline, in order to reach parameters suitable for electron diffraction. The beamline consists of realistic static magnetic optics and introduces no temporal jitter. We demonstrate numerically that electron bunches with 5~fs duration and containing 1.5~fC per bunch can be produced, with a transverse coherence length exceeding 2~nm, as required for electron diffraction

A bright ultracold atoms-based electron source

An important application of pulsed electron sources is Ultrafast Electron Diffraction [1]. In this technique, used e.g. in chemistry, biology and condensed matter physics, one can observe processes that take place at the microscopic level with sub-ps resolution. To reach the holy grail of UED, single-shot diffraction images of biologically relevant molecules, electron bunches of 1pC charge, 100fs length and 10nm coherence length are required. Conventional pulsed electron sources cannot fulfil these requirements, but according to the simulations reported in [2] and [3] a new type of source can.The new source combines the use of magneto-optical atom trapping with fast high voltage technology. We start by cooling and trapping rubidium atoms, followed by ionisation just above threshold, leading to an ultracold plasma. Another possibility is to excite the atoms into a high Rydberg level, from which they spontaneously evolve into an ultracold plasma. Applying a fast high voltage pulse, electron bunches can be extracted. In an initial study [2] it has been shown that this type of source can provide a very high brightness. Depending on the initial particle distribution, the reduced brightness can be in the order of 1x109 A/(rad2m2V), which is orders of magnitude higher than established technology such as an electron photogun can provide.Here we report the first experiments toward realisation of the source. Here, a simple accelerator structure consists of four bars surrounding a MOT, on which an 800V pulsed voltage with a rise time of 1ƒÝs is applied. An MCP together with a phosphor screen and a CCD camera are used as detection system. The bunch size obtained from the phosphor screen is fitted with a Gaussian distribution, from which the electron temperature is extracted. For small extracted charges, the electron temperature is found to have an upper limit of 500K, the measurement being limited by stray magnetic fields due to the low electron energy (10eV). We have also extracted a pulsed ion beam by reversing the sign of the accelerating voltage. Since ions are heavier, they obtain higher energy and are less influenced by the magnetic fields. The temperature in this case is found to b