Covering collections and a challenge problem of Serre

We answer a challenge of Serre by showing that every rational point on the projective curve X$^4$ + Y$^4$ = 17 Z$^4$ is of the form ($\pm$1, $\pm$2, 1) or ($\pm$2, $\pm$1, 1). Our approach builds on recent ideas from both Nils Bruin and the authors on the application of covering collections and Chabauty arguments to curves of high rank. This is the only value of c$\le$81 for which the Fermat quartic X$^4$ + Y$^4$ = c Z$^4$ cannot be solved trivially, either by local considerations or maps to elliptic curves of rank 0, and it seems likely that our approach should give a method of attack for other nontrivial values of c

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form $Y^2 = f_3 X^6 + f_2 X^4 + f_1 X^2 + f_0$, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field Q(a). If each of these elliptic curves has rank less than the degree of Q(a) : Q, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over Q(a) on the elliptic curves, for which x is in Q. This in turn allows us to find all Q-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over Q), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over Q

Lattice-Constrained Parametrizations of Form Factors for Semileptonic and Rare Radiative B Decays

We describe the form factors for semileptonic B to rho l nu and radiative B to K* gamma decays with just two parameters and the two form factors for semileptonic B to pi l nu decays with three parameters. The parametrizations are constrained by lattice results and are consistent with heavy quark symmetry, kinematic constraints and light cone sum rule scaling relations.Comment: 3 pages, latex, 2 eps files, uses epsf.sty and espcrc2.sty, poster presented at Lattice 97, Edinburgh, 22-26 July 199

A study of trace contaminant identification by microwave double resonance spectroscopy

Trace contaminant identification using microwave double resonance spectroscop

Highly Ionized High-Velocity Clouds toward PKS 2155-304 and Markarian 509

To gain insight into four highly ionized high-velocity clouds (HVCs) discovered by Sembach et al. (1999), we have analyzed data from the Hubble Space Telescope (HST) and Far Ultraviolet Spectroscopic Explorer (FUSE) for the PKS 2155-304 and Mrk 509 sight lines. We measure strong absorption in OVI and column densities of multiple ionization stages of silicon (SiII/III/IV) and carbon (CII/III/IV). We interpret this ionization pattern as a multiphase medium that contains both collisionally ionized and photoionized gas. Toward PKS 2155-304, for HVCs at -140 and -270 km/s, respectively, we measure logN(OVI)=13.80+/-0.03 and log N(OVI)=13.56+/-0.06; from Lyman series absorption, we find log N(HI)=16.37^(+0.22)_(-0.14) and 15.23^(+0.38)_(-0.22). The presence of high-velocity OVI spread over a broad (100 km/s) profile, together with large amounts of low-ionization species, is difficult to reconcile with the low densities, n=5x10^(-6) cm^(-3), in the collisional/photoionization models of Nicastro et al. (2002), although the HVCs show a similar relation in N(SiIV)/N(CIV) versus N(CII)/N(CIV) as high-z intergalactic clouds. Our results suggest that the high-velocity OVI in these absorbers do not necessarily trace the WHIM, but instead may trace HVCs with low total hydrogen column density. We propose that the broad high-velocity OVI absorption arises from shock ionization, at bowshock interfaces produced from infalling clumps of gas with velocity shear. The similar ratios of high ions for HVC Complex C and these highly ionized HVCs suggest a common production mechanism in the Galactic halo.Comment: 38 pages, including 10 figures. ApJ, 10 April, 2004. Replaced with accepted versio

A New Method for Obtaining Binary Pulsar Distances and its Implications for Tests of General Relativity

We demonstrate how measuring orbital period derivatives can lead to more accurate distance estimates and transverse velocities for some nearby binary pulsars. In many cases this method will estimate distances more accurately than is possible by annual parallax, as the relative error decreases as t^-5/2. Unfortunately, distance uncertainties limit the degree to which nearby relativistic binary pulsars can be used for testing the general relativistic prediction of orbital period decay to a few percent. Nevertheless, the measured orbital period derivative of PSR B1534+12 agrees within the observational uncertainties with that predicted by general relativity if the proper-motion contribution is accounted for.Comment: 4 pages, latex, uuencoded compressed postscript + source, no figures, uses aaspptwo.sty and dec.sty, accepted for publication in ApJL, omitted reference now include

Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular jacobians of genus 2 curves

This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute

Towards a lattice determination of the B∗BπB^\ast B \pi coupling

The coupling $g_{B^\ast B \pi}$ is related to the form factor at zero momentum of the axial current between $B^\ast$- and $B$-states. This form factor is evaluated on the lattice using static heavy quarks and light quark propagators determined by a stochastic inversion of the fermionic bilinear. The \gBBP coupling is related to the coupling $g$ between heavy mesons and low-momentum pions in the effective heavy meson chiral lagrangian. The coupling of the effective theory can therefore be computed by numerical simulations. We find the value $g = 0.42(4)(8)$. Besides its theoretical interest, the phenomenological implications of such a determination are discussed.Comment: 20 pages, 6 figure

A Kuiper Belt Source for Solar Flare Track-Rich Interplanetary Dust Particles

The presence of solar flare particle tracks in mineral grains within interplanetary dust particles (IDPs) has long been accepted as proof of their extraterrestrial origin [e.g. 1]. The 10-20 micrometers diameter IDPs released by dust producing objects in the solar system (mainly comets and asteroids) spiral in towards the Sun under the influence of Poynting-Robertson (PR) drag forces [2] and accumulate solar flare energetic particle tracks during their journey. The number of IDPs with well-constrained track density measurements is small, owing to the difficulty in the measurements and the lack of appropriatelysized crystals in which to image them. In order to use track densities as a chronometer of space exposure, the track production rate must be known. All previous work relied on track production rates determined by chemical etching techniques [e.g. 3], but tracks in IDPs are measured using TEM imaging. Here we report measurements of track densities in IDPs from both the anhydrous and hydrated IDP groups. Using the track production calibration determined from TEM observations of anorthite and olivine in lunar rock 64455 [4] we estimate space exposure times for these IDPs to constrain their parent body sources