Rational divisors in rational divisor classes

We discuss the situation where a curve C, defined over a number field K, has a known K-rational divisor class of degree 1, and consider whether this class contains an actual K-rational divisor. When C has points everywhere locally, the local to global principle of the Brauer group gives the existence of such a divisor. In this situation, we give an alternative, more down to earth, approach, which indicates how to compute this divisor in certain situations. We also discuss examples where C does not have points everywhere locally, and where no such K-rational divisor is contained in the K-rational divisor class

The arithmetic of hyperelliptic curves

We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve

Identities for hyperelliptic P-functions of genus one, two and three in covariant form

We give a covariant treatment of the quadratic differential identities satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of genera 1, 2 and 3

Serre's "formule de masse" in prime degree

For a local field F with finite residue field of characteristic p, we describe completely the structure of the filtered F_p[G]-module K^*/K^*p in characteristic 0 and $K^+/\wp(K^+) in characteristic p, where K=F(\root{p-1}\of F^*) and G=\Gal(K|F). As an application, we give an elementary proof of Serre's mass formula in degree p. We also determine the compositum C of all degree p separable extensions with solvable galoisian closure over an arbitrary base field, and show that C is K(\root p\of K^*) or K(\wp^{-1}(K)) respectively, in the case of the local field F. Our method allows us to compute the contribution of each character G\to\F_p^* to the degree p mass formula, and, for any given group \Gamma, the contribution of those degree p separable extensions of F whose galoisian closure has group \Gamma.Comment: 36 pages; most of the new material has been moved to the new Section

The proportion of failures of the Hasse norm principle

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field

Testing Hardy nonlocality proof with genuine energy-time entanglement

We show two experimental realizations of Hardy ladder test of quantum nonlocality using energy-time correlated photons, following the scheme proposed by A. Cabello \emph{et al.} [Phys. Rev. Lett. \textbf{102}, 040401 (2009)]. Unlike, previous energy-time Bell experiments, these tests require precise tailored nonmaximally entangled states. One of them is equivalent to the two-setting two-outcome Bell test requiring a minimum detection efficiency. The reported experiments are still affected by the locality and detection loopholes, but are free of the post-selection loophole of previous energy-time and time-bin Bell tests.Comment: 5 pages, revtex4, 6 figure

Cross-plane electronic and thermal transport properties of p-type La0.67Sr0.33MnO3/LaMnO3 perovskite oxide metal/semiconductor superlattices

Lanthanum strontium manganate (La0.67Sr0.33MnO3, i.e., LSMO)/lanthanum manganate (LaMnO3, i.e., LMO) perovskite oxide metal/semiconductor superlattices were investigated as a potential p-type thermoelectric material. Growth was performed using pulsed laser deposition to achieve epitaxial LSMO (metal)/LMO (p-type semiconductor) superlattices on (100)-strontium titanate (STO) substrates. The magnitude of the in-plane Seebeck coefficient of LSMO thin films (/K) is consistent with metallic behavior, while LMO thin films were p-type with a room temperature Seebeck coefficient of 140 mu V/K. Thermal conductivity measurements via the photo-acoustic (PA) technique showed that LSMO/LMO superlattices exhibit a room temperature cross-plane thermal conductivity (0.89 W/m.K) that is significantly lower than the thermal conductivity of individual thin films of either LSMO (1.60 W/m.K) or LMO (1.29 W/m.K). The lower thermal conductivity of LSMO/LMO superlattices may help overcome one of the major limitations of oxides as thermoelectrics. In addition to a low cross-plane thermal conductivity, a high ZT requires a high power factor (S-2 sigma). Cross-plane electrical transport measurements were carried out on cylindrical pillars etched in LSMO/LMO superlattices via inductively coupled plasma reactive ion etching. Cross-plane electrical resistivity data for LSMO/LMO superlattices showed a magnetic phase transition temperature (T-P) or metal-semiconductor transition at similar to 330 K, which is similar to 80K higher than the T-P observed for in-plane resistivity of LSMO, LMO, or LSMO/LMO thin films. The room temperature cross-plane resistivity (rho(c)) was found to be greater than the in-plane resistivity by about three orders of magnitude. The magnitude and temperature dependence of the cross-plane conductivity of LSMO/LMO superlattices suggests the presence of a barrier with the effective barrier height of similar to 300 meV. Although the magnitude of the cross-plane power factor is too low for thermoelectric applications by a factor of approximately 10(-4)-in part because the growth conditions chosen for this study yielded relatively high resistivity films-the temperature dependence of the resistivity and the potential for tuning the power factor by engineering strain, oxygen stoichiometry, and electronic band structure suggest that these epitaxial metal/semiconductor superlattices are deserving of further investigation. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4754514

Rational approximation and arithmetic progressions

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed

Cyclotomic integers, fusion categories, and subfactors

Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the Frobenius-Perron dimension of an object in a fusion category. The smallest number on this list is realized in a new fusion category which is constructed in the appendix written by V. Ostrik, while the others are all realized by known examples. The second application proves that in any family of graphs obtained by adding a 2-valent tree to a fixed graph, either only finitely many graphs are principal graphs of subfactors or the family consists of the A_n or D_n Dynkin diagrams. This result is effective, and we apply it to several families arising in the classification of subfactors of index less then 5.Comment: 47 pages, with an appendix by Victor Ostri