This paper concerns the Galois theoretic behavior of the p-primary subgroup SelA(F)p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map sK/F′ SelA(F′)p →  SelA(K)pG(K/F′) where F′ is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B. Mazurand M. Harris.

The tripeptide glutathione is proposed to be protective against a number of chronic diseases including cardiovascular disease and cancer. However, there have been few studies of plasma glutathione levels in humans and in those studies the numbers of participants have been very small. In an exploratory analysis the determinants of plasma total glutathione (GSHt) were investigated in a group of 100 volunteers aged 18–61 years in Atlanta, Georgia, USA during June and July 1989. Data on demographic and health-related factors were collected by interview and plasma GSHt was measured using a recently modified laboratory method. The mean concentration of plasma GSHt for all 100 participants was 761 μg/l, with a standard deviation of 451 μg/l, a range of 86–2889 μg/1 and a median of 649 μg/l. Men had significantly higher levels of plasma GSHt than women (924 v. 692 μg/l; P = 0·006). Seventh-day Adventists participating in the present study had higher plasma GSHt levels than other subgroups defined by race and/or religion. Among Seventh-day Adventists consumption of a vegetarian diet was associated with increased plasma GSHt concentration (P = 0·002). Plasma GSHt levels also appeared to vary by race, but relationships with race could not be clearly disassociated from relationships with religion. Among white participants plasma GSHt concentration decreased with age in women but increased with age in men (P = 0·05). Few other factors were associated with plasma GSHt concentration, although use of oral contraceptives (P = 0·10) was somewhat associated with decreased plasma GSHt levels. These findings suggest that plasma GSHt levels may vary with several demographic and health-related attributes and support the need for further research on this potentially important disease-preventive compound.

In this paper we will discuss p-adic Artin L-functions. The existence of these functions is a simple consequence of a theorem of Deligne and Ribet [4]. One can formulate a “p-adic Artin conjecture” for these functions. Our primary purpose here is to relate this conjecture to the “main conjecture” discussed by Coates in [3]. We will describe the precise formulations of these conjectures that we will use later. Our main result will be that in fact the main conjecture implies the p-adic Artin conjecture.

Let p be a prime. If one adjoins to Q all pn-th roots of unity for n = 1,2,3, …, then the resulting field will contain a unique subfield Q∞ such that Q∞ is a Galois extension of Q with Gal (Q∞/Q) Zp, the additive group of p-adic integers. We will denote Gal (Q∞/Q) by Γ. In a previous paper [6], we discussed a conjecture relating p-adic L-functions to certain arithmetically defined representation spaces for Γ. Now by using some results of Iwasawa, one can reformulate that conjecture in terms of certain other representation spaces for Γ. This new conjecture, which we believe may be more susceptible to generalization, will be stated below.

Let p be a prime. If one adjoins to Q all pn-th roots of unity for n = 1, 2, 3, …, then the resulting field will contain a unique subfield Q∞ such that Q∞ is a Galois extension of Q with Gal the additive group of p-adic integers. We will denote Gal(Q∞/Q) by Γ.