Expand all Collapse all | Results 1 - 8 of 8 |
1. CJM Online first
The frequency of elliptic curve groups over prime finite fields Letting $p$ vary over all primes and $E$ vary over all elliptic
curves over the finite field $\mathbb{F}_p$, we study the frequency to
which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$.
It is well-known that the only permissible groups are of the
form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$.
Given such a candidate group, we let $M(G_{m,k})$ be the frequency
to which the group $G_{m,k}$ arises in this way.
Previously, the second and fourth named authors determined an
asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes
in short arithmetic progressions. In this paper, we prove several
unconditional bounds for $M(G_{m,k})$, pointwise and on average. In
particular, we show that $M(G_{m,k})$ is bounded above by a constant
multiple of the expected quantity when $m\le k^A$ and that the
conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups
$G_{m,k}$ when $m\le k^{1/4-\epsilon}$.
We also apply our methods to study the frequency to which a given
integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.
Keywords:average order, elliptic curves, primes in short intervals Categories:11G07, 11N45, 11N13, 11N36 |
2. CJM 2013 (vol 66 pp. 826)
Signed-Selmer Groups over the $\mathbb{Z}_p^2$-extension of an Imaginary Quadratic Field Let $E$ be an elliptic curve over $\mathbb Q$ which has good supersingular
reduction at $p\gt 3$. We construct what we call the $\pm/\pm$-Selmer
groups of $E$ over the $\mathbb Z_p^2$-extension of an imaginary quadratic
field $K$ when the prime $p$ splits completely over $K/\mathbb Q$, and
prove they enjoy a property analogous to Mazur's control theorem.
Furthermore, we propose a conjectural connection between the
$\pm/\pm$-Selmer groups and Loeffler's two-variable $\pm/\pm$-$p$-adic
$L$-functions of elliptic curves.
Keywords:elliptic curves, Iwasawa theory Category:11Gxx |
3. CJM 2013 (vol 66 pp. 924)
Twists of Shimura Curves Consider a Shimura curve $X^D_0(N)$ over the rational
numbers. We determine criteria for the twist by an Atkin-Lehner
involution to have points over a local field. As a corollary we give a
new proof of the theorem of Jordan-LivnÃ© on $\mathbf{Q}_p$ points
when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$
points when $p\mid N$. We also give congruence conditions for roots
modulo $p$ of Hilbert class polynomials.
Keywords:Shimura curves, complex multiplication, modular curves, elliptic curves Categories:11G18, 14G35, 11G15, 11G10 |
4. 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 |
5. CJM 2011 (vol 64 pp. 81)
Pseudoprime Reductions of Elliptic Curves
Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication,
and for each prime
$p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer
$b$, we consider elliptic pseudoprimes to the base
$b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq
x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E,
b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such
that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called
elliptic curve pseudoprimes). Motivated by cryptography applications,
we address the problem of finding upper bounds for
$Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$,
generalising some of the literature for the classical pseudoprimes
to this new setting.
Keywords:Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes Categories:11N36, 14H52 |
6. CJM 2011 (vol 63 pp. 992)
The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians
In this paper we study genus $2$ curves whose Jacobians admit a
polarized $(4,4)$-isogeny to a product of elliptic curves. We consider
base fields of characteristic different from $2$ and $3$, which we do
not assume to be algebraically closed.
We obtain a full classification of all principally polarized abelian
surfaces that can arise from gluing two elliptic curves along their
$4$-torsion, and we derive the relation their absolute invariants
satisfy.
As an intermediate step, we give a general description of Richelot
isogenies between Jacobians of genus $2$ curves, where previously only
Richelot isogenies with kernels that are pointwise defined over the
base field were considered.
Our main tool is a Galois theoretic characterization of genus $2$
curves admitting multiple Richelot isogenies.
Keywords:Genus 2 curves, isogenies, split Jacobians, elliptic curves Categories:11G30, 14H40 |
7. CJM 2008 (vol 60 pp. 1267)
Nonadjacent Radix-$\tau$ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields In his seminal papers, Koblitz proposed curves
for cryptographic use. For fast operations on these curves,
these papers also
initiated a study of the radix-$\tau$ expansion of integers in the number
fields $\Q(\sqrt{-3})$ and $\Q(\sqrt{-7})$. The (window)
nonadjacent form of $\tau$-expansion of integers in
$\Q(\sqrt{-7})$ was first investigated by Solinas.
For integers in $\Q(\sqrt{-3})$, the nonadjacent form
and the window nonadjacent form of the $\tau$-expansion were
studied. These are used for efficient
point multiplications on Koblitz curves.
In this paper, we complete
the picture by producing the (window)
nonadjacent radix-$\tau$ expansions
for integers in all Euclidean imaginary quadratic number fields.
Keywords:algebraic integer, radix expression, window nonadjacent expansion, algorithm, point multiplication of elliptic curves, cryptography Categories:11A63, 11R04, 11Y16, 11Y40, 14G50 |
8. 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 |