Adjoint modular Galois representations and their Selmer groups

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(φ) of a two-dimensional modular Galois representation φ. We start with the p-adic Galois representation φ0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(φ0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(φ0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let φ denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting φ0, and the characteristic power series of the Selmer group Sel(ad(φ)) is given by a p-adic L-function interpolating L(1, ad(φk)) for weight k + 2 specialization φk of φ. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(φ) ⊗ ν−1), where ν is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.

Zeta functions and Eisenstein series on classical groups

We construct an Euler product from the Hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a CM field and the eigenform is holomorphic. We also prove analytic continuation of an Eisenstein series on another unitary group, containing the group just mentioned defined with such an eigenform. As an application of our methods, we prove an explicit class number formula for a totally definite hermitian form over a CM field.

Congruences between modular forms: Raising the level and dropping Euler factors

We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida’s result accounts for congruences in terms of the value of an L-function, and Ribet’s result is related to the behavior of the period that appears there. Wiles’ theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at “nonminimal level” is obtained from one at “minimal level” by dropping Euler factors from the L-function.

Anticholinergic medication use and cognitive impairment in the older population:the Medical Research Council cognitive function and ageing study

OBJECTIVES: To determine whether the use of medications with possible and definite anticholinergic activity increases the risk of cognitive impairment and mortality in older people and whether risk is cumulative. DESIGN: A 2-year longitudinal study of participants enrolled in the Medical Research Council Cognitive Function and Ageing Study between 1991 and 1993. SETTING: Community-dwelling and institutionalized participants. PARTICIPANTS: Thirteen thousand four participants aged 65 and older. MEASUREMENTS: Baseline use of possible or definite anticholinergics determined according to the Anticholinergic Cognitive Burden Scale and cognition determined using the Mini-Mental State Examination (MMSE). The main outcome measure was decline in the MMSE score at 2 years. RESULTS: At baseline, 47% of the population used a medication with possible anticholinergic properties, and 4% used a drug with definite anticholinergic properties. After adjusting for age, sex, educational level, social class, number of nonanticholinergic medications, number of comorbid health conditions, and cognitive performance at baseline, use of medication with definite anticholinergic effects was associated with a 0.33-point greater decline in MMSE score (95% confidence interval (CI)=0.03–0.64, P=.03) than not taking anticholinergics, whereas the use of possible anticholinergics at baseline was not associated with further decline (0.02, 95% CI=-0.14–0.11, P=.79). Two-year mortality was greater for those taking definite (OR=1.68; 95% CI=1.30–2.16; P<.001) and possible (OR=1.56; 95% CI=1.36–1.79; P<.001) anticholinergics. CONCLUSION: The use of medications with anticholinergic activity increases the cumulative risk of cognitive impairment and mortality.

(Table 1) Physical oceanography during Lance cruise LA99/2 in the marginal ice zone of the Barents Sea

Zooplankton was studied on eight stations in the marginal ice zone (MIZ) of the Barents Sea, in May 1999, along two transects across the ice edge. On each station, physical background measurements and zooplankton samples were taken every 6 h over a 24 h period at five discrete depth intervals. Cluster analysis revealed separation of open water stations from all ice stations as well as high similarity level among replicates belonging to particular station. Based on five replicates per station, analysis of variance (ANOVA) confirmed significant differences (P < 0.05) in abundances of the main mesozooplankton taxa among stations. Relations between the zooplankton community and environmental parameters were established using redundancy analysis (CANOCO). In total, 55% of mesozooplankton variability within studied area was explained by eight variables with significant conditional effects: depth stratum, fluorescence, temperature, salinity, bottom depth, latitude, bloom situation, and ice concentration. GLM models supported supposition about clear and negative relationship between concentration of Oithona similis, and overall mesozooplankton diversity The analyses showed a dynamic relationship between mesozooplankton distribution and hydrological conditions on short-term scale. Furthermore, our study demonstrated that variability in the physical environment of dynamic MIZ of the Barents Sea has measurable effect on the Arctic pelagic ecosystem.

Quarternary tephra layers in sediment cores from the Bellingshausen Sea and Scotia Sea

The Southern Ocean is a region of the world's ocean which is fundamental to the generation of cold deep ocean water which drives the global therrno-haline circulation. Previous investigations of deep-sea sediments south of the Polar Front have been significantly constrained by the lack of a suitable correlation and dating technique. In this study, deep-sea sediment cores from the Bellingshausen, Scotia and Weddell seas have been investigated for the presence of tephra layers. The major oxide and trace element composition of glass shards have been used to correlate tephra isochrons over distances in excess of 600 km. The source volcanoes for individual tephra layers have been identified. Atmospheric transport distances greater than 1500 km for >32 pm shards are reported. One megascopic tephra is identified and correlated across 7 sediment drifts on the continental rise in the Bellingshausen Sea. Its occurrence in a sedimentary unit that has been biostratigraphically dated to delta 18O substage 5e identifies it as a key regional marker horizon for that stage. An unusual bimodal megascopic ash layer erupted from Deception Island, South Shetland Islands, has been correlated between 6 sediment cores which form a 600 km NE-SW transect from the central Scotia Sea to Jane Basin. This megascopic ash layer has been 14C dated at c. 10,670 years BP. It represents the last significant input of tephra into the Scotia Sea or Jane Basin from that volcano and forms an important early Holocene marker horizon for the region. Five disseminated tephras can be correlated to varying extents across the central Scotia Sea cores. Together with the megascopic tephra they form a tephrostratigraphic framework that will greatly aid palaeoclimatic, palaeoenvironrnental and palaeoceanographic investigations in the region.