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126. CJM 2005 (vol 57 pp. 328)

Kuo, Wentang; Murty, M. Ram
 On a Conjecture of Birch and Swinnerton-Dyer Let $$E/\mathbb{Q}$$ be an elliptic curve defined by the equation $$y^2=x^3 +ax +b$$. For a prime $$p, \linebreak p \nmid\Delta =-16(4a^3+27b^2)\neq 0$$, define $N_p = p+1 -a_p = |E(\mathbb{F}_p)|.$ As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant $c$, $\prod_{p \leq x, p \nmid\Delta } \frac{N_p}{p} \sim c (\log x)^r, \quad x \to \infty.$ Let $$\alpha _p$$ and $$\beta _p$$ be the eigenvalues of the Frobenius at $$p$$. Define $\tilde{c}_n = \begin{cases} \frac{\alpha_p^k + \beta_p^k}{k}& n =p^k, p \textrm{ is a prime, k is a natural number, p\nmid \Delta} . \\ 0 & \text{otherwise}. \end{cases}.$ and $$\tilde{C}(x)= \sum_{n\leq x} \tilde{c}_n$$. In this paper, we establish the equivalence between the conjecture and the condition $$\tilde{C}(x)=\mathbf{o}(x)$$. The asymptotic condition is indeed much deeper than what we know so far or what we can know under the analogue of the Riemann hypothesis. In addition, we provide an oscillation theorem and an $$\Omega$$ theorem which relate to the constant $c$ in the conjecture. Categories:11M41, 11M06

127. CJM 2005 (vol 57 pp. 267)

Conrad, Keith
 Partial Euler Products on the Critical Line The initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve $L$-function at $s = 1$. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the $L$-function and that the constant in the asymptotics has an unexpected factor of $\sqrt{2}$. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical $L$-function along its critical line. The general $\sqrt{2}$ phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seems much deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally. Keywords:Euler product, explicit formula, second momentCategories:11M41, 11S40

128. CJM 2005 (vol 57 pp. 298)

Kumchev, Angel V.
 On the Waring--Goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers We investigate exceptional sets in the Waring--Goldbach problem. For example, in the cubic case, we show that all but $O(N^{79/84+\epsilon})$ integers subject to the necessary local conditions can be represented as the sum of five cubes of primes. Furthermore, we develop a new device that leads easily to similar estimates for exceptional sets for sums of fourth and higher powers of primes. Categories:11P32, 11L15, 11L20, 11N36, 11P55

129. CJM 2005 (vol 57 pp. 180)

Somodi, Marius
 On the Size of the Wild Set To every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the {\it minimum number of wild primes}. Earlier investigations have established lower bounds for this number. In this paper an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered to be an exact formula for the minimum number of wild primes. Categories:11E12, 11E81, 19F15, 11R29

130. CJM 2004 (vol 56 pp. 897)

Borwein, Jonathan M.; Borwein, David; Galway, William F.
 Finding and Excluding $b$-ary Machin-Type Individual Digit Formulae Constants with formulae of the form treated by D.~Bailey, P.~Borwein, and S.~Plouffe (\emph{BBP formulae} to a given base $b$) have interesting computational properties, such as allowing single digits in their base $b$ expansion to be independently computed, and there are hints that they should be \emph{normal} numbers, {\em i.e.,} that their base $b$ digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call \emph{Machin-type BBP formulae}, for which it is relatively easy to determine whether or not a given constant $\kappa$ has a Machin-type BBP formula. In particular, given $b \in \mathbb{N}$, $b>2$, $b$ not a proper power, a $b$-ary Machin-type BBP arctangent formula for $\kappa$ is a formula of the form $\kappa = \sum_{m} a_m \arctan(-b^{-m})$, $a_m \in \mathbb{Q}$, while when $b=2$, we also allow terms of the form $a_m \arctan(1/(1-2^m))$. Of particular interest, we show that $\pi$ has no Machin-type BBP arctangent formula when $b \neq 2$. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant. Keywords:BBP formulae, Machin-type formulae, arctangents,, logarithms, normality, Mersenne primes, Bang's theorem,, Zsigmondy's theorem, primitive prime factors, $p$-adic analysisCategories:11Y99, 11A51, 11Y50, 11K36, 33B10

131. CJM 2004 (vol 56 pp. 673)

Cali, Élie
 DÃ©faut de semi-stabilitÃ© des courbes elliptiques dans le cas non ramifiÃ© Let $\overline {\Q_2}$ be an algebraic closure of $\Q_2$ and $K$ be an unramified finite extension of $\Q_2$ included in $\overline {\Q_2}$. Let $E$ be an elliptic curve defined over $K$ with additive reduction over $K$, and having an integral modular invariant. Let us denote by $K_{nr}$ the maximal unramified extension of $K$ contained in $\overline {\Q_2}$. There exists a smallest finite extension $L$ of $K_{nr}$ over which $E$ has good reduction. We determine in this paper the degree of the extension $L/K_{nr}$. Category:11G07

132. CJM 2004 (vol 56 pp. 612)

Pál, Ambrus
 Solvable Points on Projective Algebraic Curves We examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus $g$, when $g$ is at least $40$, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus $0$, $2$, $3$ or $4$ defined over any field has a solvable point. Finally we prove that every genus $1$ curve defined over a local field of characteristic zero with residue field of characteristic $p$ has a divisor of degree prime to $6p$ defined over a solvable extension. Categories:14H25, 11D88

133. CJM 2004 (vol 56 pp. 356)

Murty, M. Ram; Saidak, Filip
 Non-Abelian Generalizations of the Erd\H os-Kac Theorem Let $a$ be a natural number greater than $1$. Let $f_a(n)$ be the order of $a$ mod $n$. Denote by $\omega(n)$ the number of distinct prime factors of $n$. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erd\"os and Pomerance: The number of $n\leq x$ coprime to $a$ satisfying $$\alpha \leq \frac{\omega(f_a(n)) - (\log \log n)^2/2 }{ (\log \log n)^{3/2}/\sqrt{3}} \leq \beta$$ is asymptotic to $$\left(\frac{ 1 }{ \sqrt{2\pi}} \int_{\alpha}^{\beta} e^{-t^2/2}dt\right) \frac{x\phi(a) }{ a},$$ as $x$ tends to infinity. Keywords:Tur{\' a}n's theorem, Erd{\H o}s-Kac theorem, Chebotarev density theorem,, Erd{\H o}s-Pomerance conjectureCategories:11K36, 11K99

134. CJM 2004 (vol 56 pp. 373)

Orton, Louisa
 An Elementary Proof of a Weak Exceptional Zero Conjecture In this paper we extend Darmon's theory of integration on $\uh_p\times \uh$'' to cusp forms $f$ of higher even weight. This enables us to prove a weak exceptional zero conjecture'': that when the $p$-adic $L$-function of $f$ has an exceptional zero at the central point, the $\mathcal{L}$-invariant arising is independent of a twist by certain Dirichlet characters. Categories:11F11, 11F67

135. CJM 2004 (vol 56 pp. 406)

Pál, Ambrus
 Theta Series, Eisenstein Series and PoincarÃ© Series over Function Fields We construct analogues of theta series, Eisenstein series and Poincar\'e series for function fields of one variable over finite fields, and prove their basic properties. Category:11F12

136. CJM 2004 (vol 56 pp. 23)

Bennett, Michael A.; Skinner, Chris M.
 Ternary Diophantine Equations via Galois Representations and Modular Forms In this paper, we develop techniques for solving ternary Diophantine equations of the shape $Ax^n + By^n = Cz^2$, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters $A$, $B$ and $C$. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan--Nagell type. Categories:11D41, 11F11, 11G05

137. CJM 2004 (vol 56 pp. 168)

Pogge, James Todd
 On a Certain Residual Spectrum of $\Sp_8$ Let $G=\Sp_{2n}$ be the symplectic group defined over a number field $F$. Let $\mathbb{A}$ be the ring of adeles. A fundamental problem in the theory of automorphic forms is to decompose the right regular representation of $G(\mathbb{A})$ acting on the Hilbert space $L^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)$. Main contributions have been made by Langlands. He described, using his theory of Eisenstein series, an orthogonal decomposition of this space of the form: $L_{\dis}^2 \bigl( G(F)\setminus G(\mathbb{A}) \bigr)=\bigoplus_{(M,\pi)} L_{\dis}^2(G(F) \setminus G(\mathbb{A}) \bigr)_{(M,\pi)}$, where $(M,\pi)$ is a Levi subgroup with a cuspidal automorphic representation $\pi$ taken modulo conjugacy (Here we normalize $\pi$ so that the action of the maximal split torus in the center of $G$ at the archimedean places is trivial.) and $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$ is a space of residues of Eisenstein series associated to $(M,\pi)$. In this paper, we will completely determine the space $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$, when $M\simeq\GL_2\times\GL_2$. This is the first result on the residual spectrum for non-maximal, non-Borel parabolic subgroups, other than $\GL_n$. Categories:11F70, 22E55

138. CJM 2004 (vol 56 pp. 194)

Saikia, A.
 Selmer Groups of Elliptic Curves with Complex Multiplication Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in $K$. Let $p$ be a rational prime that splits as $\mathfrak{p}_{1}\mathfrak{p}_{2}$ in $K$. Let $E_{p^{n}}$ denote the $p^{n}$-division points on $E$. Assume that $F(E_{p^{n}})$ is abelian over $K$ for all $n\geq 0$. This paper proves that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty}$-Selmer group of $E$ over $F(E_{p^{\infty}})$ is a finitely generated free $\Lambda$-module, where $\Lambda$ is the Iwasawa algebra of $\Gal\bigl(F(E_{p^{\infty}})/ F(E_{\mathfrak{p}_{1}^{\infty}\mathfrak{p}_{2}})\bigr)$. It also gives a simple formula for the rank of the Pontrjagin dual as a $\Lambda$-module. Categories:11R23, 11G05

139. CJM 2004 (vol 56 pp. 71)

Harper, Malcolm; Murty, M. Ram
 Euclidean Rings of Algebraic Integers Let $K$ be a finite Galois extension of the field of rational numbers with unit rank greater than~3. We prove that the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally. Categories:11R04, 11R27, 11R32, 11R42, 11N36

140. CJM 2004 (vol 56 pp. 55)

Harper, Malcolm
 $\mathbb{Z}[\sqrt{14}]$ is Euclidean We provide the first unconditional proof that the ring $\mathbb{Z} [\sqrt{14}]$ is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of $\mathbb{Q}$. It is proved that if $K$ is a real quadratic field (modulo the existence of two special primes of $K$) or if $K$ is a cyclotomic extension of $\mathbb{Q}$ then: \begin{center} \emph{% the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain.} \end{center} The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when $K$ is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property. Categories:11R04, 11R11

141. CJM 2003 (vol 55 pp. 1191)

Granville, Andrew; Soundararajan, K.
 Decay of Mean Values of Multiplicative Functions For given multiplicative function $f$, with $|f(n)| \leq 1$ for all $n$, we are interested in how fast its mean value $(1/x) \sum_{n\leq x} f(n)$ converges. Hal\'asz showed that this depends on the minimum $M$ (over $y\in \mathbb{R}$) of $\sum_{p\leq x} \bigl( 1 - \Re (f(p) p^{-iy}) \bigr) / p$, and subsequent authors gave the upper bound $\ll (1+M) e^{-M}$. For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Hal\'asz-Montgomery lemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of $y$ that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the $k$-th powers mod~$p$. Categories:11N60, 11N56, 10K20, 11N37

142. CJM 2003 (vol 55 pp. 933)

Beineke, Jennifer; Bump, Daniel
 Renormalized Periods on $\GL(3)$ A theory of renormalization of divergent integrals over torus periods on $\GL(3)$ is given, based on a relative truncation. It is shown that the renormalized periods of Eisenstein series have unexpected functional equations. Categories:11F12, 11F55

143. CJM 2003 (vol 55 pp. 897)

Archinard, Natália
 Hypergeometric Abelian Varieties In this paper, we construct abelian varieties associated to Gauss' and Appell--Lauricella hypergeometric series. Abelian varieties of this kind and the algebraic curves we define to construct them were considered by several authors in settings ranging from monodromy groups (Deligne, Mostow), exceptional sets (Cohen, Wolfart, W\"ustholz), modular embeddings (Cohen, Wolfart) to CM-type (Cohen, Shiga, Wolfart) and modularity (Darmon). Our contribution is to provide a complete, explicit and self-contained geometric construction. Categories:11, 14

144. CJM 2003 (vol 55 pp. 711)

Broughan, Kevin A.
 Adic Topologies for the Rational Integers A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers. Keywords:$p$-adic, metrizable, quasi-valuation, topological ring,, completion, inverse limit, diophantine equation, prime integers,, Fermat numbers, Fibonacci numbersCategories:11B05, 11B25, 11B50, 13J10, 13B35

145. CJM 2003 (vol 55 pp. 673)

Anderson, Greg W.; Ouyang, Yi
 A Note on Cyclotomic Euler Systems and the Double Complex Method Let $\FF$ be a finite real abelian extension of $\QQ$. Let $M$ be an odd positive integer. For every squarefree positive integer $r$ the prime factors of which are congruent to $1$ modulo $M$ and split completely in $\FF$, the corresponding Kolyvagin class $\kappa_r\in\FF^{\times}/ \FF^{\times M}$ satisfies a remarkable and crucial recursion which for each prime number $\ell$ dividing $r$ determines the order of vanishing of $\kappa_r$ at each place of $\FF$ above $\ell$ in terms of $\kappa_{r/\ell}$. In this note we give the recursion a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satisfied by Kolyvagin classes is the specialization of a universal recursion independent of $\FF$ satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution {\it \a la\/} Kubert tensored with $\ZZ/M\ZZ$. Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang also satisfy the universal recursion. Categories:11R18, 11R23, 11R34

146. CJM 2003 (vol 55 pp. 432)

Zaharescu, Alexandru
 Pair Correlation of Squares in $p$-Adic Fields Let $p$ be an odd prime number, $K$ a $p$-adic field of degree $r$ over $\mathbf{Q}_p$, $O$ the ring of integers in $K$, $B = \{\beta_1,\dots, \beta_r\}$ an integral basis of $K$ over $\mathbf{Q}_p$, $u$ a unit in $O$ and consider sets of the form $\mathcal{N}=\{n_1\beta_1+\cdots+n_r\beta_r: 1\leq n_j\leq N_j, 1\leq j\leq r\}$. We show under certain growth conditions that the pair correlation of $\{uz^2:z\in\mathcal{N}\}$ becomes Poissonian. Categories:11S99, 11K06, 1134

147. CJM 2003 (vol 55 pp. 292)

Pitman, Jim; Yor, Marc
 Infinitely Divisible Laws Associated with Hyperbolic Functions The infinitely divisible distributions on $\mathbb{R}^+$ of random variables $C_t$, $S_t$ and $T_t$ with Laplace transforms $$\left( \frac{1}{\cosh \sqrt{2\lambda}} \right)^t, \quad \left( \frac{\sqrt{2\lambda}}{\sinh \sqrt{2\lambda}} \right)^t, \quad \text{and} \quad \left( \frac{\tanh \sqrt{2\lambda}}{\sqrt{2\lambda}} \right)^t$$ respectively are characterized for various $t>0$ in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their L\'evy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for $t=1$ or $2$ in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of $C_1$ and $S_2$ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet $L$-function associated with the quadratic character modulo~4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from $S_t$ and $C_t$ by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways. Keywords:Riemann zeta function, Mellin transform, characterization of distributions, Brownian motion, Bessel process, LÃ©vy process, gamma process, Meixner processCategories:11M06, 60J65, 60E07

148. CJM 2003 (vol 55 pp. 353)

Silberger, Allan J.; Zink, Ernst-Wilhelm
 Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of $\mathfrak{p}$-Adic Simple Algebras Let $F$ be a $p$-adic local field and let $A_i^\times$ be the unit group of a central simple $F$-algebra $A_i$ of reduced degree $n>1$ ($i=1,2$). Let $\mathcal{R}^2 (A_i^\times)$ denote the set of irreducible discrete series representations of $A_i^\times$. The Abstract Matching Theorem'' asserts the existence of a bijection, the Jacquet-Langlands'' map, $\mathcal{J} \mathcal{L}_{A_2,A_1} \colon \mathcal{R}^2 (A_1^\times) \to \mathcal{R}^2 (A_2^\times)$ which, up to known sign, preserves character values for regular elliptic elements. This paper addresses the question of explicitly describing the map $\mathcal{J} \mathcal{L}$, but only for `level zero'' representations. We prove that the restriction $\mathcal{J} \mathcal{L}_{A_2,A_1} \colon \mathcal{R}_0^2 (A_1^\times) \to \mathcal{R}_0^2 (A_2^\times)$ is a bijection of level zero discrete series (Proposition~3.2) and we give a parameterization of the set of unramified twist classes of level zero discrete series which does not depend upon the algebra $A_i$ and is invariant under $\mathcal{J} \mathcal{L}_{A_2,A_1}$ (Theorem~4.1). Categories:22E50, 11R39

149. CJM 2003 (vol 55 pp. 331)

Savitt, David
 The Maximum Number of Points on a Curve of Genus $4$ over $\mathbb{F}_8$ is $25$ We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions. Categories:11G20, 14H25

150. CJM 2003 (vol 55 pp. 225)

Banks, William D.; Harcharras, Asma; Shparlinski, Igor E.
 Short Kloosterman Sums for Polynomials over Finite Fields We extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring $\mathbb{F}_q[x]/M(x)$ for collections of polynomials either of the form $f^{-1}g^{-1}$ or of the form $f^{-1}g^{-1}+afg$, where $f$ and $g$ are polynomials coprime to $M$ and of very small degree relative to $M$, and $a$ is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields. Categories:11T23, 11T06
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