On the Selmer groups of abelian varieties over function fields of characteristic p>0

In this paper, we study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We propose also a conjecture which might be considered as a counterpart of the principal conjecture in the case over a number field. \par This is a preprint which is distributed since 2005 which is still in the process of submision. Following a recent modification of some technical mistakes in the previous version of the paper as well as an amelioration of the presentation of the paper, we decide wider distribution via the archive.Comment: 21 page

Local density of states and superconducting gap in the iron chalcogenide superconductor Fe1+δ_{1+\delta}Se1−x_{1-x}Tex_{x} observed by scanning tunneling spectroscopy

We report on the first investigation of the quasiparticle local density of states and superconducting gap in the iron chalcogenide superconductor Fe$_{1+\delta}$Se$_{1-x}$Te$_{x}$ ($T_{\mathrm{c}} \sim 14$ K). The surface of a cleaved crystal revealed an atomic square lattice, superimposed on the inhomogeneous background, with a lattice constant of $\sim 3.8$ \AA without any reconstruction. Tunneling spectra measured at 4.2 K exhibit the superconducting gap, which completely disappears at 18 K, with a magnitude of $\sim 2.3$ meV, corresponding to $2\Delta / k_{\mathrm{B}}T_{\mathrm{c}}=3.8$.In stark contrast to the cuprate superconductors, the value of the observed superconducting gap is relatively homogeneous, following a sharp distribution with a small standard deviation of 0.23 meV. Conversely, the normal-state local density of states observed above $T_{\mathrm{c}}$ shows spatial variation over a wide energy range of more than 1 eV, probably due to the excess iron present in the crystal.Comment: 4 pages, 5 figure

Local Measurement of Microwave Response with Local Tunneling Spectra Using Near Field Microwave Microscopy

We have designed and built a near-field scanning microwave microscope, which has been used to measure the local microwave response and the local density-of-states (LDOS) in the area including the boundary between the gold deposited and the non-deposited region on highly-orientated pyrolytic graphite at a frequency of about 7.3 GHz. We have succeeded in measuring the spatial variation of both the LDOS and the surface resistance. It can be observed that the surface resistance in gold deposited region with the metallic tunneling spectra is smaller than that in the non-deposited region with the U-shaped tunneling spectra.Comment: 3 pages, 3 figures