Expand all Collapse all | Results 1 - 12 of 12 |
1. CMB 2011 (vol 55 pp. 842)
The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture |
The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture Frey and Jarden asked if
any abelian variety over a number field $K$
has the infinite Mordell-Weil rank over
the maximal abelian extension $K^{\operatorname{ab}}$.
In this paper,
we give an affirmative answer to their conjecture
for the Jacobian variety
of any smooth projective curve $C$
over $K$
such that $\sharp C(K^{\operatorname{ab}})=\infty$
and for any abelian variety of $\operatorname{GL}_2$-type with trivial character.
Keywords:Mordell-Weil rank, Jacobian varieties, Frey-Jarden conjecture, abelian points Categories:11G05, 11D25, 14G25, 14K07 |
2. CMB 2011 (vol 55 pp. 193)
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)
Congruent Number Elliptic Curves with Rank at Least Three
We give an infinite family of congruent number elliptic curves each with rank
at least three.
Keywords:congruent number, elliptic curve, rank Category:11G05 |
4. CMB 2009 (vol 53 pp. 58)
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)
Elliptic Curves over the Perfect Closure of a Function Field We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.
Keywords:elliptic curves, heights Categories:11G50, 11G05 |
6. CMB 2007 (vol 50 pp. 215)
Elliptic $K3$ Surfaces with Geometric Mordell--Weil 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 Mordell--Weil
rank $15$. This completes a list of Kuwata, who gave explicit examples
of elliptic $K3$-surfaces with geometric Mordell--Weil ranks
$0,1,\dots, 14, 16, 17, 18$.
Categories:14J27, 14J28, 11G05 |
7. CMB 2005 (vol 48 pp. 428)
Reduction of Elliptic Curves in Equal Characteristic~3 (and~2) and fibre type for elliptic curves
over discrete valued fields of equal characteristic~3.
Along the same lines, partial results are obtained
in equal characteristic~2.
Categories:14H52, 14K15, 11G07, 11G05, 12J10 |
8. CMB 2005 (vol 48 pp. 16)
On the Surjectivity of the Galois Representations Associated to Non-CM 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)
Torsion Points on Certain Families of Elliptic Curves Fix an elliptic curve $y^2 = x^3+Ax+B$, satisfying $A,B \in \ZZ$,
$A\geq |B| > 0$. We prove that the $\QQ$-torsion subgroup is one of
$(0)$, $\ZZ/3\ZZ$, $\ZZ/9\ZZ$. Related numerical calculations are
discussed.
Category:11G05 |
10. CMB 2002 (vol 45 pp. 337)
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, non-split Cartan Categories:11G05, 11F80 |
11. CMB 2001 (vol 44 pp. 313)
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)
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 |