An archimedian analog of Iwasawa theory

We will show a conjecture which reduces Mazur-Tate-Teitelbaum conjecture to the known cases. In order to explain its background we will develop an archimedian analog of Iwasawa theory. Moreover consequences of the conjecture which are related to Birch and Swinnerton-Dyer conjecture will be discussed.Comment: 18 page

Tentative Appraisal of Compatibility of Small-Scale CMB Anisotropy Detections in the Context of COBE-DMR-Normalized Open and Flat Λ\Lambda CDM Cosmogonies

Goodness-of-fit statistics are used to quantitatively establish the compatibility of CMB anisotropy predictions in a wide range of DMR-normalized, open and spatially-flat $\Lambda$, CDM cosmogonies with the set of all presently available small-scale CMB anisotropy detection data. Conclusions regarding model viability depend sensitively on the prescription used to account for the 1$\sigma$ uncertainty in the assumed value of the DMR normalization, except for low-density, $\Omega_0 \sim 0.3$ -- 0.4, open models which are compatible with the data for all prescriptions used. While large baryon-density (\Omega_B \gap 0.0175 h^{-2}), old (t_0 \gap 15 -- 16 Gyr), low-density ($\Omega_0 \sim 0.2$ -- 0.4), flat-$\Lambda$ models might be incompatible, no model is incompatible with the data for all prescriptions. In fact, some open models seem to fit the data better than should be expected, and this might be an indication that some error bars are mildly overconservative.Comment: 15 page PostScript file, including 6 included figures. Also available via anonymous ftp from ftp://astro.caltech.edu/users/kmg/chi.p

The Ihara zeta functions of a Ramanujan graph (Geometry and Analysis of Discrete Groups and Hyperbolic Spaces)

"Geometry and Analysis of Discrete Groups and Hyperbolic Spaces". June 22～26, 2015. edited by Michihiko Fujii, Nariya Kawazumi and Ken'ichi Ohshika. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We will discuss the relationship between Ihara's zeta functions of Ramanujan graphs and Hasse-Weil's congruent congruent zeta functions of modular curves. The residue of the Hasse-Weil's congruent zeta functions at t = 1 will be described by the number of supersingular points and the complexity of the associated graphs