Twists of X(7) and primitive solutions to x^2+y^3=z^7

We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve X. To restrict the set of relevant twists, we exploit the isomorphism between X and the modular curve X(7), and use modularity of elliptic curves and level lowering. This leaves 10 genus-3 curves, whose rational points are found by a combination of methods.Comment: 47 page

Class Groups and Selmer Groups

AbstractIt is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is to their intersection. In this paper we compute the two groups and their intersection explicitly in the local case and put together the local information to get sharp upper bounds in the global case. The techniques in this paper can be used for arbitrary abelian varieties, isogenies and number fields assuming a frequently occurring condition. Several examples are worked out for the Jacobians of elliptic and hyperelliptic curves

Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve

It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\bold Q[z]$ with rational periodic points of period $N$. Morton proved there were none with $N=4$, by showing that the genus~$2$ algebraic curve that classifies periodic points of period~4 is birational to $X_1(16)$, whose rational points had been previously computed. We prove there are none with $N=5$. Here the relevant curve has genus~$14$, but it has a genus~$2$ quotient, whose rational points we compute by performing a~$2$-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal$_{\bold Q}$-stable $5$-cycles, and show that there exist Gal$_{\bold Q}$-stable $N$-cycles for infinitely many $N$. Furthermore, we answer a question of Morton by showing that the genus~$14$ curve and its quotient are not modular. Finally, we mention some partial results for $N=6$

Large emissions from floodplain trees close the Amazon methane budget

Wetlands are the largest global source of atmospheric methane (CH4), a potent greenhouse gas. However, methane emission inventories from the Amazon floodplain, the largest natural geographic source of CH4 in the tropics, consistently underestimate the atmospheric burden of CH4 determined via remote sensing and inversion modelling, pointing to a major gap in our understanding of the contribution of these ecosystems to CH4 emissions. Here we report CH4 fluxes from the stems of 2,357 individual Amazonian floodplain trees from 13 locations across the central Amazon basin. We find that escape of soil gas through wetland trees is the dominant source of regional CH4 emissions. Methane fluxes from Amazon tree stems were up to 200 times larger than emissions reported for temperate wet forests6 and tropical peat swamp forests, representing the largest non-ebullitive wetland fluxes observed. Emissions from trees had an average stable carbon isotope value (δ13C) of −66.2 ± 6.4 per mil, consistent with a soil biogenic origin. We estimate that floodplain trees emit 15.1 ± 1.8 to 21.2 ± 2.5 teragrams of CH4 a year, in addition to the 20.5 ± 5.3 teragrams a year emitted regionally from other sources. Furthermore, we provide a ‘top-down’ regional estimate of CH4 emissions of 42.7 ± 5.6 teragrams of CH4 a year for the Amazon basin, based on regular vertical lower-troposphere CH4 profiles covering the period 2010–2013. We find close agreement between our ‘top-down’ and combined ‘bottom-up’ estimates, indicating that large CH4 emissions from trees adapted to permanent or seasonal inundation can account for the emission source that is required to close the Amazon CH4 budget. Our findings demonstrate the importance of tree stem surfaces in mediating approximately half of all wetland CH4 emissions in the Amazon floodplain, a region that represents up to one-third of the global wetland CH4 source when trees are combined with other emission sources

The SAMI Galaxy Survey : spatially resolving the main sequence of star formation

We present the ∼800 star formation rate maps for the Sydney-AAO Multi-object Integral field spectrograph (SAMI) Galaxy Survey based on H α emission maps, corrected for dust attenuation via the Balmer decrement, that are included in the SAMI Public Data Release 1. We mask out spaxels contaminated by non-stellar emission using the [O iii]/H β, [N ii]/H α, [S ii]/H α, and [O i]/H α line ratios. Using these maps, we examine the global and resolved star-forming main sequences of SAMI galaxies as a function of morphology, environmental density, and stellar mass. Galaxies further below the star-forming main sequence are more likely to have flatter star formation profiles. Early-type galaxies split into two populations with similar stellar masses and central stellar mass surface densities. The main-sequence population has centrally concentrated star formation similar to late-type galaxies, while galaxies >3σ below the main sequence show significantly reduced star formation most strikingly in the nuclear regions. The split populations support a two-step quenching mechanism, wherein halo mass first cuts off the gas supply and remaining gas continues to form stars until the local stellar mass surface density can stabilize the reduced remaining fuel against further star formation. Across all morphologies, galaxies in denser environments show a decreased specific star formation rate from the outside in, supporting an environmental cause for quenching, such as ram-pressure stripping or galaxy interactions.Publisher PDFPeer reviewe

Communications Biophysics

Contains reports on nine research projects split into four sections.National Institutes of Health (Grant 5 P01 NS13126)National Institutes of Health (Grant 5 K04 NS00113)National Institutes of Health (Training Grant 5 T32 NS07047)National Institutes of Health (Grant 5 ROl NS11153-03)National Institutes of Health (Fellowship 1 T32 NS07099-01)National Science Foundation (Grant BNS77-16861)National Institutes of Health (Grant 5 ROl NS10916)National Institutes of Health (Grant 5 ROl NS12846)National Science Foundation (Grant BNS77-21751)National Institutes of Health (Grant 1 RO1 NS14092)Health Sciences FundNational Institutes of Health (Grant 2 R01 NS11680)National Institutes of Health (Grant 2 RO1 NS11080)National Institutes of Health (Training Grant 5 T32 GM07301