Orbit parametrizations of theta characteristics on hypersurfaces over arbitrary fields

It is well-known that theta characteristics on smooth plane curves over a field of characteristic different from two are in bijection with certain smooth complete intersections of three quadrics. We generalize this bijection to possibly singular hypersurfaces of any dimension over arbitrary fields including those of characteristic two. It is accomplished in terms of linear orbits of tuples of symmetric matrices instead of smooth complete intersections of quadrics. As an application of our methods, we give a description of the projective automorphism groups of complete intersections of quadrics generalizing Beauville's results.Comment: 35 page

A positive proportion of cubic curves over Q admit linear determinantal representations

Can a smooth plane cubic be defined by the determinant of a square matrix with entries in linear forms in three variables? If we can, we say that it admits a linear determinantal representation. In this paper, we investigate linear determinantal representations of smooth plane cubics over various fields, and prove that any smooth plane cubic over a large field (or an ample field) admits a linear determinantal representation. Since local fields are large, any smooth plane cubic over a local field always admits a linear determinantal representation. As an application, we prove that a positive proportion of smooth plane cubics over Q, ordered by height, admit linear determinantal representations. We also prove that, if the conjecture of Bhargava-Kane-Lenstra-Poonen-Rains on the distribution of Selmer groups is true, a positive proportion of smooth plane cubics over Q fail the local-global principle for the existence of linear determinantal representations.Comment: 19 page

Super Kamiokande results: atmospheric and solar neutrinos

Atmospheric neutrino and solar neutrino data from the first phase of Super-Kamiokande (SK-I) are presented. The observed data are used to study atmospheric and solar neutrino oscillations. Zenith angle distributions from various atmospheric neutrino data samples are used to estimate the neutrino oscillation parameter region. In addition, a new result of the $L/E$ measurement is presented. A dip in the $L/E$ distribution was observed in the data, as predicted from the sinusoidal flavor transition probability of neutrino oscillation. The energy spectrum and the time variation such as day/night and seasonal differences of solar neutrino flux are measured in Super-Kamiokande. The neutrino oscillation parameters are strongly constrained from those measurements.Comment: 8 pages, 7 figures, Proceedings for the XXXIXth Rencontres de Moriond on Electroweak Interactions (2004

An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields

We give a brief report on our computations of linear determinantal representations of smooth plane cubics over finite fields. After recalling a classical interpretation of linear determinantal representations as rational points on the affine part of Jacobian varieties, we give an algorithm to obtain all linear determinantal representations up to equivalence. We also report our recent study on computations of linear determinantal representations of twisted Fermat cubics defined over the field of rational numbers. This paper is a summary of the author's talk at the JSIAM JANT workshop on algorithmic number theory in March, 2016. Details will appear elsewhere

The Hasse principle for flexes on cubics and the local-global divisibility problem

A point on a smooth cubic is called a flex (or an inflection point) if the tangent line at that point intersects with the cubic with multiplicity 3. We prove the flexes on a smooth cubic over a global field satisfy the Hasse principle. Moreover, we study the Hasse principle allowing exceptional sets of positive density, following a suggestion by J.-P. Serre. We prove a smooth cubic over a global field has a flex over the base field if and only if it has a flex locally at a set of places of Dirichlet density strictly larger than 8/9.Comment: 21 pages. The main theorem is generalized and the proof is revised according to J.-P. Serre's suggestion. This version also contains general results on the Hasse principle for Galois cohomology of finite group schemes allowing exceptional sets of positive densit

On the symmetric determinantal representations of the Fermat curves of prime degree

We prove that the defining equations of the Fermat curves of prime degree cannot be written as the determinant of symmetric matrices with entries in linear forms in three variables with rational coefficients. In the proof, we use a relation between symmetric matrices with entries in linear forms and non-effective theta characteristics on smooth plane curves. We also use some results of Gross-Rohrlich on the rational torsion points on the Jacobian varieties of the Fermat curves of prime degree.Comment: 11 pages; to appear in International Journal of Number Theor

Sheaf theoretic classifications of pairs of square matrices over arbitrary fields

We give classifications of linear orbits of pairs of square matrices with non-vanishing discriminant polynomials over a field in terms of certain coherent sheaves with additional data on closed subschemes of the projective line. Our results are valid in a uniform manner over arbitrary fields including those of characteristic two. This work is based on the previous work of the first author on theta characteristics on hypersurfaces. As an application, we give parametrizations of orbits of pairs of symmetric matrices under special linear groups with fixed discriminant polynomials generalizing some results of Wood and Bhargava-Gross-Wang.Comment: 22 page

Establishing neutrino mass hierarchy and CP violation by two identical detectors with different baselines using the J-PARC neutrino beam

We discuss how and to what extent one can determine the neutrino mass hierarchy, normal or inverted, and at the same time uncover CP violation in the lepton sector by using two identical detectors with different baselines in neutrino oscillation experiments using low energy superbeam from the J-PARC facility.Comment: 2 pages, 2 figures, talk given at NuFact05, 21-26 June, 2005, Frascati, Ital

Sub-Kelvin refrigeration with dry-coolers on a rotating system

We developed a cryogenic system on a rotating table that achieves sub-Kelvin conditions. The cryogenic system consists of a helium sorption cooler and a pulse tube cooler in a cryostat mounted on a rotating table. Two rotary-joint connectors for electricity and helium gas circulation enable the coolers to be operated and maintained with ease. We performed cool-down tests under a condition of continuous rotation at 20 rpm. We obtained a temperature of 0.23 K with a holding time of more than 24 hours, thus complying with catalog specifications. We monitored the system's performance for four weeks; two weeks with and without rotation. A few-percent difference in conditions was observed between these two states. Most applications can tolerate such a slight difference. The technology developed is useful for various scientific applications requiring sub-Kelvin conditions on rotating platforms.Comment: 3pages, 4 figures, to appear in Review of Scientific Instrument