76. CMB 2010 (vol 53 pp. 654)
77. CMB 2010 (vol 53 pp. 661)
78. CMB 2010 (vol 53 pp. 385)
 Achter, Jeffrey D.

Exceptional Covers of Surfaces
Consider a finite morphism $f: X \rightarrow Y$ of smooth, projective varieties over a finite field $\mathbf{F}$. Suppose $X$ is the vanishing locus in $\mathbf{P}^N$ of $r$ forms of degree at most $d$. We show that there is a constant $C$ depending only on $(N,r,d)$ and $\deg(f)$ such that if ${\mathbf{F}}>C$, then $f(\mathbf{F}): X(\mathbf{F}) \rightarrow Y(\mathbf{F})$ is injective if and only if it is surjective.
Category:11G25 

79. CMB 2010 (vol 53 pp. 571)
 Trifković, Mak

Periods of Modular Forms and Imaginary Quadratic Base Change
Let $f$ be a classical newform of weight $2$ on the upper halfplane $\mathcal H^{(2)}$, $E$ the corresponding strong Weil curve, $K$ a class number one imaginary quadratic field, and $F$ the base change of $f$ to $K$. Under a mild hypothesis on the pair $(f,K)$, we prove that the period ratio $\Omega_E/(\sqrt{D}\Omega_F)$ is in $\mathbb Q$. Here $\Omega_F$ is the unique minimal positive period of $F$, and $\Omega_E$ the area of $E(\mathbb C)$. The claim is a specialization to base change forms of a conjecture proposed and numerically verified by Cremona and Whitley.
Category:11F67 

80. CMB 2009 (vol 53 pp. 140)
 Mukunda, Keshav

Pisot Numbers from $\{ 0, 1 \}$Polynomials
A \emph{Pisot number} is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a \emph{Salem number} is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial Â one with $\{0,1\}$coefficients Â and shows that they form a strictly increasing sequence with limit $(1+\sqrt{5}) / 2$. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.
Categories:11R06, 11R09, 11C08 

81. CMB 2009 (vol 53 pp. 102)
 Khan, Rizwanur

Spacings Between Integers Having Typically Many Prime Factors
We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\delta$ arbitrarily small and positive, the nearest neighbor spacings between integers n with $\omega(n)  log log n < (log log n)^{\delta}$ obey the Poisson distribution law.
Category:11K99 

82. CMB 2009 (vol 53 pp. 204)
83. CMB 2009 (vol 53 pp. 187)
84. CMB 2009 (vol 53 pp. 95)
 Ghioca, Dragos

Towards the Full MordellLang Conjecture for Drinfeld Modules
Let $\phi$ be a Drinfeld module of generic characteristic, and let X be a sufficiently generic affine subvariety of $\mathbb{G_a^g}$. We show that the intersection of X with a finite rank $\phi$submodule of $\mathbb{G_a^g}$ is finite.
Keywords:Drinfeld module, MordellLang conjecture Categories:11G09, 11G10 

85. CMB 2009 (vol 53 pp. 87)
86. CMB 2009 (vol 53 pp. 58)
 Dąbrowski, Andrzej; Jędrzejak, Tomasz

Ranks in Families of Jacobian Varieties of Twisted Fermat Curves
In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.
Keywords:Fermat curve, Jacobian variety, elliptic curve, canonical height Categories:11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15 

87. CMB 2009 (vol 52 pp. 511)
88. CMB 2009 (vol 52 pp. 583)
89. CMB 2009 (vol 52 pp. 481)
 Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S.

Some Infinite Products of Ramanujan Type
In his ``lost'' notebook, Ramanujan stated two results, which are equivalent to the identities
\[
\prod_{n=1}^{\infty} \frac{(1q^n)^5}{(1q^{5n})}
=15\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n
\]
and
\[
q\prod_{n=1}^{\infty} \frac{(1q^{5n})^5}{(1q^{n})}
=\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n.
\]
We give several more identities of this type.
Keywords:Power series expansions of certain infinite products Categories:11E25, 11F11, 11F27, 30B10 

90. CMB 2009 (vol 52 pp. 195)
91. CMB 2009 (vol 52 pp. 186)
 Broughan, Kevin A.

Extension of the Riemann $\xi$Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
If $K$ is a number field with $n_k=[k:\mathbb{Q}]$, and $\xi_k$
the symmetrized
Dedekind zeta function of the field, the inequality
$$\Re\,{\frac{ \xi_k'(\sigma + {\rm i} t)}{\xi_k(\sigma
+ {\rm i} t)}} > \frac{ \xi_k'(\sigma)}{\xi_k(\sigma)}$$ for $t\neq 0$ is
shown
to be true for $\sigma\ge 1+ 8/n_k^\frac{1}{3}$ improving the result of
Lagarias where the constant in the inequality was 9. In the case $k=\mathbb{Q}$
the
inequality is extended to $\si\ge 1$ for all $t$ sufficiently large or small
and to the region $\si\ge 1+1/(\log t 5)$ for all $t\neq 0$. This
answers positively a question posed by Lagarias.
Keywords:Riemann zeta function, xi function, zeta zeros Categories:11M26, 11R42 

92. CMB 2009 (vol 52 pp. 237)
93. CMB 2009 (vol 52 pp. 117)
 Poulakis, Dimitrios

On the Rational Points of the Curve $f(X,Y)^q = h(X)g(X,Y)$
Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with
integer coefficients. In this paper we deal with the curve
$f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable
conditions it is possible to determine all of its rational points.
Categories:11G30, 14G05, 14G25 

94. CMB 2009 (vol 52 pp. 3)
 Banks, W. D.

Carmichael Numbers with a Square Totient
Let $\varphi$ denote the Euler function. In this paper, we show that
for all large $x$ there are more than $x^{0.33}$ Carmichael numbers
$n\le x$ with the property that $\varphi(n)$ is a perfect square. We
also obtain similar results for higher powers.
Categories:11N25, 11A25 

95. CMB 2009 (vol 52 pp. 53)
 Cummins, C. J.

Cusp Forms Like $\Delta$
Let $f$ be a squarefree integer and denote by $\Gamma_0(f)^+$ the
normalizer of $\Gamma_0(f)$ in $\SL(2,\R)$. We find the analogues of
the cusp form $\Delta$ for the groups $\Gamma_0(f)^+$.
Categories:11F03, 11F22, 30F35 

96. CMB 2009 (vol 52 pp. 63)
 Dietmann, Rainer

Small Zeros of Quadratic Forms Avoiding a Finite Number of Prescribed Hyperplanes
We prove a new upper bound for the smallest zero $\mathbf{x}$
of a quadratic form over a number field with the additional
restriction that $\mathbf{x}$ does not lie in a finite number of $m$ prescribed
hyperplanes. Our bound is polynomial in the height of the quadratic
form, with an exponent depending only on the number of variables but
not on $m$.
Categories:11D09, 11E12, 11H46, 11H55 

97. CMB 2009 (vol 52 pp. 66)
 Dryden, Emily B.; Strohmaier, Alexander

Huber's Theorem for Hyperbolic Orbisurfaces
We show that for compact orientable hyperbolic orbisurfaces, the
Laplace spectrum determines the length spectrum as well as the
number of singular points of a given order. The converse also holds, giving
a full generalization of Huber's theorem to the setting of
compact orientable hyperbolic orbisurfaces.
Keywords:Huber's theorem, length spectrum, isospectral, orbisurfaces Categories:58J53, 11F72 

98. CMB 2008 (vol 51 pp. 627)
 Vidanovi\'{c}, Mirjana V.; Tri\v{c}kovi\'{c}, Slobodan B.; Stankovi\'{c}, Miomir S.

Summation of Series over Bourget Functions
In this paper we derive formulas for summation of series involving
J.~Bourget's generalization of Bessel functions of integer order, as
well as the analogous generalizations by H.~M.~Srivastava. These series are
expressed in terms of the Riemann $\z$ function and Dirichlet
functions $\eta$, $\la$, $\b$, and can be brought into closed form in
certain cases, which means that the infinite series are represented
by finite sums.
Keywords:Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functions Categories:33C10, 11M06, 65B10 

99. CMB 2008 (vol 51 pp. 497)
 Borwein, Peter; Choi, KwokKwong Stephen; Mercer, Idris

Expected Norms of ZeroOne Polynomials
Let $\cA_n = \big\{ a_0 + a_1 z + \cdots + a_{n1}z^{n1} : a_j \in \{0, 1 \
} \big\}$, whose elements are called \emf{zeroone polynomials}
and correspond naturally to the $2^n$ subsets of $[n] := \{ 0, 1,
\ldots, n1 \}$. We also let $\cA_{n,m} = \{ \alf(z) \in \cA_n :
\alf(1) = m \}$, whose elements correspond to the ${n \choose m}$
subsets of~$[n]$ of size~$m$, and let $\cB_n = \cA_{n+1} \setminus
\cA_n$, whose elements are the zeroone polynomials of degree
exactly~$n$.
Many researchers have studied norms of polynomials with restricted
coefficients. Using $\norm{\alf}_p$ to denote the usual $L_p$ norm
of~$\alf$ on the unit circle, one easily sees that $\alf(z) = a_0 +
a_1 z + \cdots + a_N z^N \in \bR[z]$ satisfies $\norm{\alf}_2^2 = c_0$
and $\norm{\alf}_4^4 = c_0^2 + 2(c_1^2 + \cdots + c_N^2)$, where $c_k
:= \sum_{j=0}^{Nk} a_j a_{j+k}$ for $0 \le k \le N$.
If $\alf(z) \in \cA_{n,m}$, say $\alf(z) = z^{\beta_1} + \cdots +
z^{\beta_m}$ where $\beta_1 < \cdots < \beta_m$, then $c_k$ is the
number of times $k$ appears as a difference $\beta_i  \beta_j$. The
condition that $\alf \in \cA_{n,m}$ satisfies $c_k \in \{0,1\}$ for $1
\le k \le n1$ is thus equivalent to the condition that $\{ \beta_1,
\ldots, \beta_m \}$ is a \emf{Sidon set} (meaning all differences of
pairs of elements are distinct).
In this paper, we find the average of~$\\alf\_4^4$ over $\alf \in
\cA_n$, $\alf \in \cB_n$, and $\alf \in \cA_{n,m}$. We further show
that our expression for the average of~$\\alf\_4^4$ over~$\cA_{n,m}$
yields a new proof of the known result: if $m = o(n^{1/4})$ and
$B(n,m)$ denotes the number of Sidon sets of size~$m$ in~$[n]$, then
almost all subsets of~$[n]$ of size~$m$ are Sidon, in the sense that
$\lim_{n \to \infty} B(n,m)/\binom{n}{m} = 1$.
Categories:11B83, 11C08, 30C10 

100. CMB 2008 (vol 51 pp. 561)