For an algebraic number field and a prime , define the number to be the maximal number such that there exists a Galois extension of whose Galois group is a free pro--group of rank . The Leopoldt conjecture implies , ( denotes the number of complex places of ). Some examples of and with have been known so far. In this note, the invariant is studied, and among other things some examples with are given.
We study the behaviour of the absolute Weil height of algebraic numbers in certain infinite extensions of . In particular, we obtain a Northcott type property for infinite abelian extensions of finite exponent and also a Bogomolov type property for certain fields which are a -adic analog of totally real fields. Moreover, we obtain a non-archimedean analog of a uniform distribution theorem of Bilu in the archimedean case.
We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of Macsyma, with there being a unique rational nondegenerate solution. For the second case...