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201. CJM 1999 (vol 51 pp. 771)

Flicker, Yuval Z.
 Stable Bi-Period Summation Formula and Transfer Factors This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group $G(E)$, with periods by a subgroup $G(F)$, where $E/F$ is a quadratic extension of number fields. The split case, where $E = F \oplus F$, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups $H$ which occur in the case of standard conjugacy. The spectral side of the bi-period summation formula involves periods, namely integrals over the group of $F$-adele points of $G$, of cusp forms on the group of $E$-adele points on the group $G$. Our stabilization suggests that such cusp forms---with non vanishing periods---and the resulting bi-period distributions associated to periodic'' automorphic forms, are related to analogous bi-period distributions associated to periodic'' automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$. This offers a sharpening of the theory of liftings, where periods play a key role. The stabilization depends on the fundamental lemma'', which conjectures that the unit elements of the Hecke algebras on $G$ and $H$ have matching orbital integrals. Even in stating this conjecture, one needs to introduce a transfer factor''. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case. Finally, the fundamental lemma is verified for $\SL(2)$. Categories:11F72, 11F70, 14G27, 14L35

202. CJM 1999 (vol 51 pp. 835)

Kim, Henry H.
 Langlands-Shahidi Method and Poles of Automorphic $L$-Functions: Application to Exterior Square $L$-Functions In this paper we use Langlands-Shahidi method and the result of Langlands which says that non self-conjugate maximal parabolic subgroups do not contribute to the residual spectrum, to prove the holomorphy of several \emph{completed} automorphic $L$-functions on the whole complex plane which appear in constant terms of the Eisenstein series. They include the exterior square $L$-functions of $\GL_n$, $n$ odd, the Rankin-Selberg $L$-functions of $\GL_n\times \GL_m$, $n\ne m$, and $L$-functions $L(s,\sigma,r)$, where $\sigma$ is a generic cuspidal representation of $\SO_{10}$ and $r$ is the half-spin representation of $\GSpin(10, \mathbb{C})$. The main part is proving the holomorphy and non-vanishing of the local normalized intertwining operators by reducing them to natural conjectures in harmonic analysis, such as standard module conjecture. Categories:11F, 22E

203. CJM 1999 (vol 51 pp. 225)

Betke, U.; Böröczky, K.
 Asymptotic Formulae for the Lattice Point Enumerator Let $M$ be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large $\lambda$ the number of lattice points in $\lambda M$ is given by $G(\lambda M) =V(\lambda M) + O(\lambda^{d-1-\varepsilon (d)})$ for some positive $\varepsilon(d)$. Here we give for general convex bodies the weaker estimate $\left| G(\lambda M) -V(\lambda M) \right | \le \frac{1}{2} S_{\Z^d}(M) \lambda^{d-1}+o(\lambda^{d-1})$ where $S_{\Z^d}(M)$ denotes the lattice surface area of $M$. The term $S_{\Z^d}(M)$ is optimal for all convex bodies and $o(\lambda^{d-1})$ cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of $M$. Further we deal with families $\{P_\lambda\}$ of convex bodies where the only condition is that the inradius tends to infinity. Here we have $\left| G(P_\lambda)-V(P_\lambda) \right| \le dV(P_\lambda,K;1)+o \bigl( S(P_\lambda) \bigr)$ where the convex body $K$ satisfies some simple condition, $V(P_\lambda,K;1)$ is some mixed volume and $S(P_\lambda)$ is the surface area of $P_\lambda$. Categories:11P21, 52C07

204. CJM 1999 (vol 51 pp. 266)

Deitmar, Anton; Hoffman, Werner
 Spectral Estimates for Towers of Noncompact Quotients We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Ga$-automorphic Laplacian below a given bound when $\Ga$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Ga$ in the family is assumed to contain a principal congruence subgroup whose index in $\Ga$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Ga$ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice~$\Ga$. Categories:11F72, 58G25, 22E40

205. CJM 1999 (vol 51 pp. 10)

Chacron, M.; Tignol, J.-P.; Wadsworth, A. R.
 Tractable Fields A field $F$ is said to be tractable when a condition described below on the simultaneous representation of quaternion algebras holds over $F$. It is shown that a global field $F$ is tractable i{f}f $F$ has at most one dyadic place. Several other examples of tractable and nontractable fields are given. Categories:12E15, 11R52

206. CJM 1999 (vol 51 pp. 164)

Tan, Victor
 Poles of Siegel Eisenstein Series on $U(n,n)$ Let $U(n,n)$ be the rank $n$ quasi-split unitary group over a number field. We show that the normalized Siegel Eisenstein series of $U(n,n)$ has at most simple poles at the integers or half integers in certain strip of the complex plane. Categories:11F70, 11F27, 22E50

207. CJM 1999 (vol 51 pp. 176)

van der Poorten, Alfred; Williams, Kenneth S.
 Values of the Dedekind Eta Function at Quadratic Irrationalities Let $d$ be the discriminant of an imaginary quadratic field. Let $a$, $b$, $c$ be integers such that $$b^2 - 4ac = d, \quad a > 0, \quad \gcd (a,b,c) = 1.$$ The value of $\bigl|\eta \bigl( (b + \sqrt{d})/2a \bigr) \bigr|$ is determined explicitly, where $\eta(z)$ is Dedekind's eta function $$\eta (z) = e^{\pi iz/12} \prod^\ty_{m=1} (1 - e^{2\pi imz}) \qquad \bigl( \im(z) > 0 \bigr). %\eqno({\rm im}(z)>0).$$ Keywords:Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class groupCategories:11F20, 11E45

208. CJM 1999 (vol 51 pp. 130)

Savin, Gordan; Gan, Wee Teck
 The Dual Pair $G_2 \times \PU_3 (D)$ ($p$-Adic Case) We study the correspondence of representations arising by restricting the minimal representation of the linear group of type $E_7$ and relative rank $4$. The main tool is computations of the Jacquet modules of the minimal representation with respect to maximal parabolic subgroups of $G_2$ and $\PU_3(D)$. Categories:22E35, 22E50, 11F70

209. CJM 1998 (vol 50 pp. 1253)

López-Bautista, Pedro Ricardo; Villa-Salvador, Gabriel Daniel
 Integral representation of $p$-class groups in ${\Bbb Z}_p$-extensions and the Jacobian variety For an arbitrary finite Galois $p$-extension $L/K$ of $\zp$-cyclotomic number fields of $\CM$-type with Galois group $G = \Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $\mu_L^-$ are zero, we obtain unconditionally and explicitly the Galois module structure of $\clases$, the minus part of the $p$-subgroup of the class group of $L$. For an arbitrary finite Galois $p$-extension $L/K$ of algebraic function fields of one variable over an algebraically closed field $k$ of characteristic $p$ as its exact field of constants with Galois group $G = \Gal(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated to $L/k$. Keywords:${\Bbb Z}_p$-extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structureCategories:11R33, 11R23, 11R58, 14H40

210. CJM 1998 (vol 50 pp. 1323)

Morales, Jorge
 L'invariant de Hasse-Witt de la forme de Killing Nous montrons que l'invariant de Hasse-Witt de la forme de Killing d'une alg{\e}bre de Lie semi-simple $L$ s'exprime {\a} l'aide de l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de $L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines positives), et des invariants associ{\'e}s au groupe des sym{\'e}tries du diagramme de Dynkin de $L$. Categories:11E04, 11E72, 17B10, 17B20, 11E88, 15A66

211. CJM 1998 (vol 50 pp. 1007)

Elder, G. Griffith
 Galois module structure of ambiguous ideals in biquadratic extensions Let $N/K$ be a biquadratic extension of algebraic number fields, and $G=\Gal (N/K)$. Under a weak restriction on the ramification filtration associated with each prime of $K$ above $2$, we explicitly describe the $\bZ[G]$-module structure of each ambiguous ideal of $N$. We find under this restriction that in the representation of each ambiguous ideal as a $\bZ[G]$-module, the exponent (or multiplicity) of each indecomposable module is determined by the invariants of ramification, alone. For a given group, $G$, define ${\cal S}_G$ to be the set of indecomposable $\bZ[G]$-modules, ${\cal M}$, such that there is an extension, $N/K$, for which $G\cong\Gal (N/K)$, and ${\cal M}$ is a $\bZ[G]$-module summand of an ambiguous ideal of $N$. Can ${\cal S}_G$ ever be infinite? In this paper we answer this question of Chinburg in the affirmative. Keywords:Galois module structure, wild ramificationCategories:11R33, 11S15, 20C32

212. CJM 1998 (vol 50 pp. 1105)

Roberts, Brooks
 Tempered representations and the theta correspondence Let $V$ be an even dimensional nondegenerate symmetric bilinear space over a nonarchimedean local field $F$ of characteristic zero, and let $n$ be a nonnegative integer. Suppose that $\sigma \in \Irr \bigl(\OO (V)\bigr)$ and $\pi \in \Irr \bigl(\Sp (n,F)\bigr)$ correspond under the theta correspondence. Assuming that $\sigma$ is tempered, we investigate the problem of determining the Langlands quotient data for $\pi$. Categories:11F27, 22E50

213. CJM 1998 (vol 50 pp. 794)

Louboutin, Stéphane
 Upper bounds on $|L(1,\chi)|$ and applications We give upper bounds on the modulus of the values at $s=1$ of Artin $L$-functions of abelian extensions unramified at all the infinite places. We also explain how we can compute better upper bounds and explain how useful such computed bounds are when dealing with class number problems for $\CM$-fields. For example, we will reduce the determination of all the non-abelian normal $\CM$-fields of degree $24$ with Galois group $\SL_2(F_3)$ (the special linear group over the finite field with three elements) which have class number one to the computation of the class numbers of $23$ such $\CM$-fields. Keywords:Dedekind zeta function, Dirichlet series, $\CM$-field, relative class numberCategories:11M20, 11R42, 11Y35, 11R29

214. CJM 1998 (vol 50 pp. 465)

Balog, Antal
 Six primes and an almost prime in four linear equations There are infinitely many triplets of primes $p,q,r$ such that the arithmetic means of any two of them, ${p+q\over2}$, ${p+r\over2}$, ${q+r\over2}$ are also primes. We give an asymptotic formula for the number of such triplets up to a limit. The more involved problem of asking that in addition to the above the arithmetic mean of all three of them, ${p+q+r\over3}$ is also prime seems to be out of reach. We show by combining the Hardy-Littlewood method with the sieve method that there are quite a few triplets for which six of the seven entries are primes and the last is almost prime.} Categories:11P32, 11N36

215. CJM 1998 (vol 50 pp. 563)

Goldston, D. A.; Yildirim, C. Y.
 Primes in short segments of arithmetic progressions Consider the variance for the number of primes that are both in the interval $[y,y+h]$ for $y \in [x,2x]$ and in an arithmetic progression of modulus $q$. We study the total variance obtained by adding these variances over all the reduced residue classes modulo $q$. Assuming a strong form of the twin prime conjecture and the Riemann Hypothesis one can obtain an asymptotic formula for the total variance in the range when $1 \leq h/q \leq x^{1/2-\epsilon}$, for any $\epsilon >0$. We show that one can still obtain some weaker asymptotic results assuming the Generalized Riemann Hypothesis (GRH) in place of the twin prime conjecture. In their simplest form, our results are that on GRH the same asymptotic formula obtained with the twin prime conjecture is true for almost all'' $q$ in the range $1 \leq h/q \leq h^{1/4-\epsilon}$, that on averaging over $q$ one obtains an asymptotic formula in the extended range $1 \leq h/q \leq h^{1/2-\epsilon}$, and that there are lower bounds with the correct order of magnitude for all $q$ in the range $1 \leq h/q \leq x^{1/3-\epsilon}$. Category:11M26

216. CJM 1998 (vol 50 pp. 412)

McIntosh, Richard J.
 Asymptotic transformations of $q$-series For the $q$-series $\sum_{n=0}^\infty a^nq^{bn^2+cn}/(q)_n$ we construct a companion $q$-series such that the asymptotic expansions of their logarithms as $q\to 1^{\scriptscriptstyle -}$ differ only in the dominant few terms. The asymptotic expansion of their quotient then has a simple closed form; this gives rise to a new $q$-hypergeometric identity. We give an asymptotic expansion of a general class of $q$-series containing some of Ramanujan's mock theta functions and Selberg's identities. Categories:11B65, 33D10, 34E05, 41A60

217. CJM 1998 (vol 50 pp. 74)

Flicker, Yuval Z.
 Elementary proof of the fundamental lemma for a unitary group The fundamental lemma in the theory of automorphic forms is proven for the (quasi-split) unitary group $U(3)$ in three variables associated with a quadratic extension of $p$-adic fields, and its endoscopic group $U(2)$, by means of a new, elementary technique. This lemma is a prerequisite for an application of the trace formula to classify the automorphic and admissible representations of $U(3)$ in terms of those of $U(2)$ and base change to $\GL(3)$. It compares the (unstable) orbital integral of the characteristic function of the standard maximal compact subgroup $K$ of $U(3)$ at a regular element (whose centralizer $T$ is a torus), with an analogous (stable) orbital integral on the endoscopic group $U(2)$. The technique is based on computing the sum over the double coset space $T\bs G/K$ which describes the integral, by means of an intermediate double coset space $H\bs G/K$ for a subgroup $H$ of $G=U(3)$ containing $T$. Such an argument originates from Weissauer's work on the symplectic group. The lemma is proven for both ramified and unramified regular elements, for which endoscopy occurs (the stable conjugacy class is not a single orbit). Categories:22E35, 11F70, 11F85, 11S37

218. CJM 1997 (vol 49 pp. 1139)

Kraus, Alain
 Majorations effectives pour l'Ã©quation de Fermat gÃ©nÃ©ralisÃ©e Soient $A$, $B$ et $C$ trois entiers non nuls premiers entre eux deux \a deux, et $p$ un nombre premier. Comme cons\'equence des travaux de A. Wiles et F. Diamond sur la conjecture de Taniyama-Weil, on explicite une constante $f(A,B,C)$ telle que, sous certaines conditions portant sur $A$, $B$ et $C$, l'\'equation $Ax^p+By^p+Cz^p=0$ n'a aucune solution non triviale dans $\Z$, si $p$ est $>f(A,B,C)$. On d\'emontre par ailleurs, sans condition suppl\'ementaire portant sur $A$, $B$ et $C$, que cette \'equation n'a aucune solution non triviale dans $\Z$, si $p$ divise $xyz$, et si $p$ est $>f(A,B,C)$. Category:11G

219. CJM 1997 (vol 49 pp. 1265)

Snaith, V. P.
 Hecke algebras and class-group invariant Let $G$ be a finite group. To a set of subgroups of order two we associate a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its units to the class-group of ${\bf Z}[G]$. We show that this homomorphism takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of Chinburg invariants arising from the Galois module structure of higher-dimensional algebraic $K$-groups of rings of algebraic integers often differ by elements in the image of $\psi$. As an application we show that two such constructions coincide. Categories:16S34, 19A99, 11R65

220. CJM 1997 (vol 49 pp. 887)

Borwein, Peter; Pinner, Christopher
 Polynomials with $\{ 0, +1, -1\}$ coefficients and a root close to a given point For a fixed algebraic number $\alpha$ we discuss how closely $\alpha$ can be approximated by a root of a $\{0,+1,-1\}$ polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, $k$, of the polynomial at $\alpha$. In particular we obtain the following. Let ${\cal B}_{N}$ denote the set of roots of all $\{0,+1,-1\}$ polynomials of degree at most $N$ and ${\cal B}_{N}(\alpha,k)$ the roots of those polynomials that have a root of order at most $k$ at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$ we show that $\min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} |\alpha -\beta| \asymp \frac{1}{\alpha^{N}},$ and for a root of unity $\alpha$ that $\min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}} |\alpha -\beta|\asymp \frac{1}{N^{(k+1) \left\lceil \frac{1}{2}\phi (d)\right\rceil +1}}.$ We study in detail the case of $\alpha=1$, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When $k=0$ or 1 we can describe the extremal polynomials explicitly. Keywords:Mahler measure, zero one polynomials, Pisot numbers, root separationCategories:11J68, 30C10

221. CJM 1997 (vol 49 pp. 641)

Burris, Stanley; Compton, Kevin; Odlyzko, Andrew; Richmond, Bruce
 Fine spectra and limit laws II First-order 0--1 laws. Using Feferman-Vaught techniques a condition on the fine spectrum of an admissible class of structures is found which leads to a first-order 0--1 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 0--1 law. If the condition is satisfied (and hence we have a first-order %% 0--1 law) Categories:03N45, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81

222. 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

223. 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

224. 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

225. 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
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