1. CMB 2011 (vol 55 pp. 842)
2. CMB 2011 (vol 55 pp. 193)
 Ulas, Maciej

Rational Points in Arithmetic Progressions on $y^2=x^n+k$
Let $C$ be a hyperelliptic curve given by the equation $y^2=f(x)$
for $f\in\mathbb{Z}[x]$ without multiple roots. We say that points
$P_{i}=(x_{i}, y_{i})\in C(\mathbb{Q})$ for $i=1,2,\dots, m$ are in
arithmetic progression if the numbers $x_{i}$ for $i=1,2,\dots, m$
are in arithmetic progression.
In this paper we show that there exists a polynomial $k\in\mathbb{Z}[t]$
with the property that on the elliptic curve $\mathcal{E}':
y^2=x^3+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four
points in arithmetic progression that are independent in the group
of all $\mathbb{Q}(t)$rational points on the curve $\mathcal{E}'$. In
particular this result generalizes earlier results of Lee and
V\'{e}lez. We also show that if $n\in\mathbb{N}$ is odd,
then there are infinitely many $k$'s with the property that on
curves $y^2=x^n+k$ there are four rational points in arithmetic
progressions. In the case when $n$ is even we can find infinitely
many $k$'s such that on curves $y^2=x^n+k$ there are six rational
points in arithmetic progression.
Keywords:arithmetic progressions, elliptic curves, rational points on hyperelliptic curves Category:11G05 

3. CMB 2010 (vol 53 pp. 661)
4. 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 

5. CMB 2009 (vol 53 pp. 87)
6. CMB 2007 (vol 50 pp. 215)
 Kloosterman, Remke

Elliptic $K3$ Surfaces with Geometric MordellWeil Rank $15$
We prove that the elliptic surface
$y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric MordellWeil
rank $15$. This completes a list of Kuwata, who gave explicit examples
of elliptic $K3$surfaces with geometric MordellWeil ranks
$0,1,\dots, 14, 16, 17, 18$.
Categories:14J27, 14J28, 11G05 

7. CMB 2005 (vol 48 pp. 428)
8. CMB 2005 (vol 48 pp. 16)
 Cojocaru, Alina Carmen

On the Surjectivity of the Galois Representations Associated to NonCM Elliptic Curves
Let $ E $ be an elliptic curve defined over
$\Q,$ of conductor $N$ and without complex multiplication. For any
positive integer $l$, let $\phi_l$ be the Galois representation
associated to the $l$division points of~$E$. From a celebrated
1972 result of Serre we know that $\phi_l$ is surjective for any
sufficiently large prime $l$. In this paper we find conditional
and unconditional upper bounds in terms of $N$ for the primes $l$
for which $\phi_l$ is {\emph{not}} surjective.
Categories:11G05, 11N36, 11R45 

9. CMB 2003 (vol 46 pp. 157)
10. CMB 2002 (vol 45 pp. 337)
 Chen, Imin

Surjectivity of $\mod\ell$ Representations Attached to Elliptic Curves and Congruence Primes
For a modular elliptic curve $E/\mathbb{Q}$, we show a number of
links between the primes $\ell$ for which the mod $\ell$
representation of $E/\mathbb{Q}$ has projective dihedral image and
congruence primes for the newform associated to $E/\mathbb{Q}$.
Keywords:torsion points of elliptic curves, Galois representations, congruence primes, Serre tori, grossencharacters, nonsplit Cartan Categories:11G05, 11F80 

11. CMB 2001 (vol 44 pp. 313)
 Reverter, Amadeu; Vila, Núria

Images of mod $p$ Galois Representations Associated to Elliptic Curves
We give an explicit recipe for the determination of the images
associated to the Galois action on $p$torsion points of elliptic
curves. We present a table listing the image for all the elliptic
curves defined over $\QQ$ without complex multiplication with
conductor less than 200 and for each prime number~$p$.
Keywords:Galois groups, elliptic curves, Galois representation, isogeny Categories:11R32, 11G05, 12F10, 14K02 

12. CMB 2000 (vol 43 pp. 239)
 Yu, Gang

On the Number of Divisors of the Quadratic Form $m^2+n^2$
For an integer $n$, let $d(n)$ denote the ordinary divisor function.
This paper studies the asymptotic behavior of the sum
$$
S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2).
$$
It is proved in the paper that, as $x \to \infty$,
$$
S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 +
\epsilon}),
$$
where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any
fixed positive real number.
The result corrects a false formula given in a paper of Gafurov
concerning the same problem, and improves the error $O \bigl(
x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.
Keywords:divisor, large sieve, exponential sums Categories:11G05, 14H52 
