292 research outputs found
Points of Low Height on Elliptic Curves and Surfaces, I: Elliptic surfaces over P^1 with small d
For each of n=1,2,3 we find the minimal height h^(P) of a nontorsion point P
of an elliptic curve E over C(T) of discriminant degree d=12n (equivalently, of
arithmetic genus n), and exhibit all (E,P) attaining this minimum. The minimal
h^(P) was known to equal 1/30 for n=1 (Oguiso-Shioda) and 11/420 for n=2
(Nishiyama), but the formulas for the general (E,P) were not known, nor was the
fact that these are also the minima for an elliptic curve of discriminant
degree 12n over a function field of any genus. For n=3 both the minimal height
(23/840) and the explicit curves are new. These (E,P) also have the property
that that mP is an integral point (a point of naive height zero) for each
m=1,2,...,M, where M=6,8,9 for n=1,2,3; this, too, is maximal in each of the
three cases.Comment: 15 pages; some lines in the TeX source are commented out with "%" to
meet the 15-page limit for ANTS proceeding
Selmer Groups in Twist Families of Elliptic Curves
The aim of this article is to give some numerical data related to the order
of the Selmer groups in twist families of elliptic curves. To do this we assume
the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated
theorem of Waldspurger to get a fast algorithm to compute . Having
an extensive amount of data we compare the distribution of the order of the
Selmer groups by functions of type with small. We discuss how the
"best choice" of is depending on the conductor of the chosen elliptic
curves and the congruence classes of twist factors.Comment: to appear in Quaestiones Mathematicae. 16 page
The elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences
We define three hard problems in the theory of elliptic divisibility
sequences (EDS Association, EDS Residue and EDS Discrete Log), each of which is
solvable in sub-exponential time if and only if the elliptic curve discrete
logarithm problem is solvable in sub-exponential time. We also relate the
problem of EDS Association to the Tate pairing and the MOV, Frey-R\"{u}ck and
Shipsey EDS attacks on the elliptic curve discrete logarithm problem in the
cases where these apply.Comment: 18 pages; revised version includes some small mathematical
corrections, reformatte
Regulator constants and the parity conjecture
The p-parity conjecture for twists of elliptic curves relates multiplicities
of Artin representations in p-infinity Selmer groups to root numbers. In this
paper we prove this conjecture for a class of such twists. For example, if E/Q
is semistable at 2 and 3, K/Q is abelian and K^\infty is its maximal pro-p
extension, then the p-parity conjecture holds for twists of E by all orthogonal
Artin representations of Gal(K^\infty/Q). We also give analogous results when
K/Q is non-abelian, the base field is not Q and E is replaced by an abelian
variety. The heart of the paper is a study of relations between permutation
representations of finite groups, their "regulator constants", and
compatibility between local root numbers and local Tamagawa numbers of abelian
varieties in such relations.Comment: 50 pages; minor corrections; final version, to appear in Invent. Mat
Kernel density classification and boosting: an L2 sub analysis
Kernel density estimation is a commonly used approach to classification. However, most of the theoretical results for kernel methods apply to estimation per se and not necessarily to classification. In this paper we show that when estimating the difference between two densities, the optimal smoothing parameters are increasing functions of the sample size of the complementary group, and we provide a small simluation study which examines the relative performance of kernel density methods when the final goal is classification. A relative newcomer to the classification portfolio is âboostingâ, and this paper proposes an algorithm for boosting kernel density classifiers. We note that boosting is closely linked to a previously proposed method of bias reduction in kernel density estimation and indicate how it will enjoy similar properties for classification. We show that boosting kernel classifiers reduces the bias whilst only slightly increasing the variance, with an overall reduction in error. Numerical examples and simulations are used to illustrate the findings, and we also suggest further areas of research
On some notions of good reduction for endomorphisms of the projective line
Let be an endomorphism of \SR(\bar{\Q}), the projective line over
the algebraic closure of \Q, of degree defined over a number field
. Let be a non-archimedean valuation of . We say that has
critically good reduction at if any pair of distinct ramification points of
do not collide under reduction modulo and the same holds for any
pair of branch points. We say that has simple good reduction at if
the map , the reduction of modulo , has the same degree of
. We prove that if has critically good reduction at and the
reduction map is separable, then has simple good reduction at
.Comment: 15 page
Log Fano varieties over function fields of curves
Consider a smooth log Fano variety over the function field of a curve.
Suppose that the boundary has positive normal bundle. Choose an integral model
over the curve. Then integral points are Zariski dense, after removing an
explicit finite set of points on the base curve.Comment: 18 page
Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19
It is known that K3 surfaces S whose Picard number rho (= rank of the
Neron-Severi group of S) is at least 19 are parametrized by modular curves X,
and these modular curves X include various Shimura modular curves associated
with congruence subgroups of quaternion algebras over Q. In a family of such K3
surfaces, a surface has rho=20 if and only if it corresponds to a CM point on
X. We use this to compute equations for Shimura curves, natural maps between
them, and CM coordinates well beyond what could be done by working with the
curves directly as we did in ``Shimura Curve Computations'' (1998) =
Comment: 16 pages (1 figure drawn with the LaTeX picture environment); To
appear in the proceedings of ANTS-VIII, Banff, May 200
- âŠ