Expand all Collapse all | Results 1 - 14 of 14 |
1. CJM Online first
Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials Assuming Lang's conjectured lower bound on the heights of non-torsion
points on an elliptic curve, we show that there exists an absolute
constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion
point $P \in E(\mathbb{Q})$, there is at most one integral multiple
$[n]P$ such that $n \gt C$. The proof is a modification of a proof
of Ingram giving an unconditional but not uniform bound. The
new ingredient is a collection of explicit formulae for the
sequence $v(\Psi_n)$ of valuations of the division polynomials.
For $P$ of non-singular reduction, such sequences are already
well described in most cases, but for $P$ of singular reduction,
we are led to define a new class of sequences called \emph{elliptic
troublemaker sequences}, which measure the failure of the NÃ©ron
local height to be quadratic. As a corollary in the spirit of
a conjecture of Lang and Hall, we obtain a uniform upper bound
on $\widehat{h}(P)/h(E)$ for integer points having two large
integral multiples.
Keywords:elliptic divisibility sequence, Lang's conjecture, height functions Categories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52 |
2. CJM 2011 (vol 64 pp. 1248)
Darmon's Points and Quaternionic Shimura Varieties In this paper, we generalize a conjecture due to Darmon and Logan in
an adelic setting. We study the relation between our construction and
Kudla's works on cycles on orthogonal Shimura varieties. This relation
allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's
points.
Keywords:elliptic curves, Stark-Heegner points, quaternionic Shimura varieties Categories:11G05, 14G35, 11F67, 11G40 |
3. CJM 2011 (vol 64 pp. 282)
Level Lowering Modulo Prime Powers and Twisted Fermat Equations We discuss a clean level lowering theorem modulo prime powers
for weight $2$ cusp forms.
Furthermore, we illustrate how this can be used to completely
solve certain twisted Fermat equations
$ax^n+by^n+cz^n=0$.
Keywords:modular forms, level lowering, Diophantine equations Categories:11D41, 11F33, 11F11, 11F80, 11G05 |
4. CJM 2011 (vol 64 pp. 151)
Moments of the Rank of Elliptic Curves Fix an elliptic curve $E/\mathbb{Q}$ and assume the Riemann Hypothesis
for the $L$-function $L(E_D, s)$ for every quadratic twist $E_D$ of
$E$ by $D\in\mathbb{Z}$. We combine Weil's
explicit formula with techniques of Heath-Brown to derive an asymptotic
upper bound for the weighted moments of the analytic rank of $E_D$. We
derive from this an upper bound for the density of low-lying zeros of
$L(E_D, s)$ that is compatible with the random matrix models of Katz and
Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbb{R}$,
the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer
conjecture)
the number of integral points of $E_D$ are less than $f(D)$
for almost all $D$.
Keywords:elliptic curve, explicit formula, integral point, low-lying zeros, quadratic twist, rank Categories:11G05, 11G40 |
5. CJM 2010 (vol 62 pp. 1060)
Heegner Points over Towers of Kummer Extensions
Let $E$ be an elliptic curve, and let $L_n$ be the Kummer extension
generated by a primitive $p^n$-th root of unity and a $p^n$-th root of
$a$ for a fixed $a\in \mathbb{Q}^\times-\{\pm 1\}$. A detailed case study
by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these
authors to predict unbounded and strikingly regular growth for the
rank of $E$ over $L_n$ in certain cases. The aim of this note is to
explain how some of these predictions might be accounted for by
Heegner points arising from a varying collection of Shimura curve
parametrisations.
Categories:11G05, 11R23, 11F46 |
6. CJM 2009 (vol 62 pp. 400)
On p-Adic Properties of Central L-Values of Quadratic Twists of an Elliptic Curve We study $p$-indivisibility of the central values $L(1,E_d)$ of
quadratic twists $E_d$ of a semi-stable elliptic curve $E$ of
conductor $N$. A consideration of the conjecture of Birch and
Swinnerton-Dyer shows that the set of quadratic discriminants $d$
splits naturally into several families $\mathcal{F}_S$, indexed by subsets $S$
of the primes dividing $N$. Let $\delta_S= \gcd_{d\in \mathcal{F}_S}
L(1,E_d)^{\operatorname{alg}}$, where $L(1,E_d)^{\operatorname{alg}}$ denotes the algebraic part
of the central $L$-value, $L(1,E_d)$. Our main theorem relates the
$p$-adic valuations of $\delta_S$ as $S$ varies. As a consequence we
present an application to a refined version of a question of
Kolyvagin. Finally we explain an intriguing (albeit speculative)
relation between Waldspurger packets on $\widetilde{\operatorname{SL}_2}$ and
congruences of modular forms of integral and half-integral weight. In
this context, we formulate a conjecture on congruences of
half-integral weight forms and explain its relevance to the problem of
$p$-indivisibility of $L$-values of quadratic twists.
Categories:11F40, 11F67, 11G05 |
7. CJM 2008 (vol 60 pp. 481)
Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields Let $k$ be a global field, $\overline{k}$ a separable
closure of $k$, and $G_k$ the absolute Galois group
$\Gal(\overline{k}/k)$ of $\overline{k}$ over $k$. For every
$\sigma\in G_k$, let $\ks$ be the fixed subfield of $\overline{k}$
under $\sigma$. Let $E/k$ be an elliptic curve over $k$. It is known
that the Mordell--Weil group $E(\ks)$ has infinite rank. We present a
new proof of this fact in the following two cases. First, when $k$
is a global function field of odd characteristic and $E$ is
parametrized by a Drinfeld modular curve, and secondly when $k$ is a
totally real number field and $E/k$ is parametrized by a Shimura
curve. In both cases our approach uses the non-triviality of a
sequence of Heegner points on $E$ defined over ring class fields.
Category:11G05 |
8. CJM 2006 (vol 58 pp. 796)
Mordell--Weil Groups and the Rank of Elliptic Curves over Large Fields Let $K$ be a number field, $\overline{K}$ an algebraic closure of
$K$ and $E/K$ an elliptic curve
defined over $K$. In this paper, we prove that if $E/K$ has a
$K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then
for each $\sigma\in \Gal(\overline{K}/K)$, the Mordell--Weil group
$E(\overline{K}^{\sigma})$ of $E$ over the fixed subfield of
$\overline{K}$ under $\sigma$ has infinite rank.
Category:11G05 |
9. CJM 2005 (vol 57 pp. 1155)
The Square Sieve and the Lang--Trotter Conjecture Let $E$ be an elliptic curve defined over $\Q$ and without
complex multiplication. Let $K$ be a fixed imaginary quadratic field.
We find nontrivial upper bounds for the number of ordinary primes $p \leq x$
for which $\Q(\pi_p) = K$, where $\pi_p$ denotes the Frobenius endomorphism
of $E$ at $p$. More precisely, under a generalized Riemann hypothesis
we show that this number is $O_{E}(x^{\slfrac{17}{18}}\log x)$, and unconditionally
we show that this number is $O_{E, K}\bigl(\frac{x(\log \log x)^{\slfrac{13}{12}}}
{(\log x)^{\slfrac{25}{24}}}\bigr)$. We also prove that the number of imaginary quadratic
fields $K$, with $-\disc K \leq x$ and of the form $K = \Q(\pi_p)$, is
$\gg_E\log\log\log x$ for $x\geq x_0(E)$. These results represent progress towards
a 1976 Lang--Trotter conjecture.
Keywords:Elliptic curves modulo $p$; Lang--Trotter conjecture;, applications of sieve methods Categories:11G05, 11N36, 11R45 |
10. CJM 2004 (vol 56 pp. 194)
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 |
11. CJM 2004 (vol 56 pp. 23)
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 |
12. CJM 2002 (vol 54 pp. 1202)
Octahedral Galois Representations Arising From $\mathbf{Q}$-Curves of Degree $2$ Generically, one can attach to a $\mathbf{Q}$-curve $C$ octahedral representations
$\rho\colon\Gal(\bar{\mathbf{Q}}/\mathbf{Q})\rightarrow\GL_2(\bar\mathbf{F}_3)$
coming from the Galois action on the $3$-torsion of those abelian varieties of
$\GL_2$-type whose building block is $C$. When $C$ is defined over a quadratic
field and has an isogeny of degree $2$ to its Galois conjugate, there exist
such representations $\rho$ having image into $\GL_2(\mathbf{F}_9)$. Going
the other way, we can ask which $\mod 3$ octahedral representations $\rho$ of
$\Gal(\bar\mathbf{Q}/\mathbf{Q})$ arise from $\mathbf{Q}$-curves in the above
sense. We characterize those arising from quadratic $\mathbf{Q}$-curves of
degree $2$. The approach makes use of Galois embedding techniques in
$\GL_2(\mathbf{F}_9)$, and the characterization can be given in terms of a
quartic polynomial defining the $\mathcal{S}_4$-extension of $\mathbf{Q}$
corresponding to the projective representation $\bar{\rho}$.
Categories:11G05, 11G10, 11R32 |
13. CJM 2001 (vol 53 pp. 449)
Descending Rational Points on Elliptic Curves to Smaller Fields In this paper, we study the Mordell-Weil group of an elliptic curve
as a Galois module. We consider an elliptic curve $E$ defined over a
number field $K$ whose Mordell-Weil rank over a Galois extension $F$ is
$1$, $2$ or $3$. We show that $E$ acquires a point (points) of
infinite order over a field whose Galois group is one of $C_n \times C_m$
($n= 1, 2, 3, 4, 6, m= 1, 2$), $D_n \times C_m$ ($n= 2, 3, 4, 6, m= 1, 2$),
$A_4 \times C_m$ ($m=1,2$), $S_4 \times C_m$ ($m=1,2$). Next, we consider
the case where $E$ has complex multiplication by the ring of integers $\o$
of an imaginary quadratic field $\k$ contained in $K$. Suppose that the
$\o$-rank over a Galois extension $F$ is $1$ or $2$. If $\k\neq\Q(\sqrt{-1})$
and $\Q(\sqrt{-3})$ and $h_{\k}$ (class number of $\k$) is odd, we show that
$E$ acquires positive $\o$-rank over a cyclic extension of $K$ or over a
field whose Galois group is one of $\SL_2(\Z/3\Z)$, an extension of
$\SL_2(\Z/3\Z)$ by $\Z/2\Z$, or a central extension by the dihedral group.
Finally, we discuss the relation of the above results to the vanishing of
$L$-functions.
Categories:11G05, 11G40, 11R32, 11R33 |
14. CJM 1997 (vol 49 pp. 749)
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 |