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201. CJM 1997 (vol 49 pp. 722)

Elder, G. Griffith; Madan, Manohar L.
 Galois module structure of the integers in wildly ramified $C_p\times C_p$ extensions Let $L/K$ be a finite Galois extension of local fields which are finite extensions of $\bQ_p$, the field of $p$-adic numbers. Let $\Gal (L/K)=G$, and $\euO_L$ and $\bZ_p$ be the rings of integers in $L$ and $\bQ_p$, respectively. And let $\euP_L$ denote the maximal ideal of $\euO_L$. We determine, explicitly in terms of specific indecomposable $\bZ_p[G]$-modules, the $\bZ_p[G]$-module structure of $\euO_L$ and $\euP_L$, for $L$, a composite of two arithmetically disjoint, ramified cyclic extensions of $K$, one of which is only weakly ramified in the sense of Erez \cite{erez}. Keywords:Galois module structure---integral representation.Categories:11S15, 20C32

202. CJM 1997 (vol 49 pp. 749)

Howe, Lawrence
 Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups Following a method outlined by Greenberg, root number computations give a conjectural lower bound for the ranks of certain Mordell-Weil groups of elliptic curves. More specifically, for $\PQ_{n}$ a \pgl{{\bf Z}/p^{n}{\bf Z}}-extension of ${\bf Q}$ and $E$ an elliptic curve over {\bf Q}, define the motive $E \otimes \rho$, where $\rho$ is any irreducible representation of $\Gal (\PQ_{n}/{\bf Q})$. Under some restrictions, the root number in the conjectural functional equation for the $L$-function of $E \otimes \rho$ is easily computible, and a `$-1$' implies, by the Birch and Swinnerton-Dyer conjecture, that $\rho$ is found in $E(\PQ_{n}) \otimes {\bf C}$. Summing the dimensions of such $\rho$ gives a conjectural lower bound of $$p^{2n} - p^{2n - 1} - p - 1$$ for the rank of $E(\PQ_{n})$. Categories:11G05, 14G10

203. CJM 1997 (vol 49 pp. 468)

Burris, Stanley; Sárközy, András
 Fine spectra and limit laws I. First-order laws Using Feferman-Vaught techniques we show a certain property of the fine spectrum of an admissible class of structures leads to a first-order law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order law. We present three conditions for verifying that the above property actually holds. The first condition is that the count function of an admissible class has regular variation with a certain uniformity of convergence. This applies to a wide range of admissible classes, including those satisfying Knopfmacher's Axiom A, and those satisfying Bateman and Diamond's condition. The second condition is similar to the first condition, but designed to handle the discrete case, {\it i.e.}, when the sizes of the structures in an admissible class $K$ are all powers of a single integer. It applies when either the class of indecomposables or the whole class satisfies Knopfmacher's Axiom A$^\#$. The third condition is also for the discrete case, when there is a uniform bound on the number of $K$-indecomposables of any given size. Keywords:First order limit laws, generalized number theoryCategories:O3C13, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81

204. CJM 1997 (vol 49 pp. 499)

Fitzgerald, Robert W.
 Gorenstein Witt rings II The abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classifiy the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type. Categories:11E81, 13H10

205. CJM 1997 (vol 49 pp. 283)

McCall, Thomas M.; Parry, Charles J.; Ranalli, Ramona R.
 The $2$-rank of the class group of imaginary bicyclic biquadratic fields A formula is obtained for the rank of the $2$-Sylow subgroup of the ideal class group of imaginary bicyclic biquadratic fields. This formula involves the number of primes that ramify in the field, the ranks of the $2$-Sylow subgroups of the ideal class groups of the quadratic subfields and the rank of a $Z_2$-matrix determined by Legendre symbols involving pairs of ramified primes. As applications, all subfields with both $2$-class and class group $Z_2 \times Z_2$ are determined. The final results assume the completeness of D.~A.~Buell's list of imaginary fields with small class numbers. Categories:11R16, 11R29, 11R20

206. CJM 1997 (vol 49 pp. 405)

Vulakh, L. Ya.
 On Hurwitz constants for Fuchsian groups Explicit bounds for the Hurwitz constants for general cofinite Fuchsian groups have been found. It is shown that the bounds obtained are exact for the Hecke groups and triangular groups with signature $(0:2,p,q)$. Categories:11J04, 20H10
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