This volume contains the proceedings of the AMS Special Session on Higher Genus Curves and Fibrations in Mathematical Physics and Arithmetic Geometry, held on January 8, 2016, in Seattle, Washington. ...Algebraic curves and their fibrations have played a major role in both mathematical physics and arithmetic geometry. This volume focuses on the role of higher genus curves; in particular, hyperelliptic and superelliptic curves in algebraic geometry and mathematical physics. The articles in this volume investigate the automorphism groups of curves and superelliptic curves and results regarding integral points on curves and their applications in mirror symmetry. Moreover, geometric subjects are addressed, such as elliptic K3 surfaces over the rationals, the birational type of Hurwitz spaces, and links between projective geometry and abelian functions.
The congruent number elliptic curves are defined by
$E_{d}:y^{2}=x^{3}-d^{2}x$
, where
$d\in \mathbb{N}$
. We give a simple proof of a formula for
$L(\operatorname{Sym}^{2}(E_{d}),3)$
in terms of the ...determinant of the elliptic trilogarithm evaluated at some degree zero divisors supported on the torsion points on
$E_{d}(\overline{\mathbb{Q}})$
.
This volume contains the proceedings of the AMS Special Session on Higher Genus Curves and Fibrations in Mathematical Physics and Arithmetic Geometry, held on January 8, 2016, in Seattle, ...Washington.Algebraic curves and their fibrations have played a major role in both mathematical physics and arithmetic geometry. This volume focuses on the role of higher genus curves; in particular, hyperelliptic and superelliptic curves in algebraic geometry and mathematical physics.The articles in this volume investigate the automorphism groups of curves and superelliptic curves and results regarding integral points on curves and their applications in mirror symmetry. Moreover, geometric subjects are addressed, such as elliptic $K$3 surfaces over the rationals, the birational type of Hurwitz spaces, and links between projective geometry and abelian functions.
There are many studies on regular rectifying curves in classical differential geometry, and important results have been obtained. However, these studies are limited for a smooth curve with singular ...points. To examine such curves and surfaces, the concept of framed curve, which is the general form of regular and Legendre curves, is used. Framed curves are defined as curves that have a moving frame with singular points in Euclidean space. We investigate framed rectifying curves via the dilation of framed curves on S2 in
ℝ3. Moreover, the result of dilation of framed curves is the framed rectifying curve or not. We give some classifications for the dilation of framed curves. Finally, we give some related examples with their figures.
Abstract
Extending the idea of Dabrowski ‘On the proportion of rank 0 twists of elliptic curves’,
C. R. Acad. Sci. Paris, Ser. I
346
(2008), 483–486 and using the 2-descent method, we provide three ...general families of elliptic curves over
$\mathbb{Q}$
such that a positive proportion of prime-twists of such elliptic curves have rank zero simultaneously.
Let
$Y$
be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of
$\mathbb{Q}$
. In 2008, Skorobogatov and Zarhin showed that the Brauer group ...modulo algebraic classes
$\text{Br}\,Y/\text{Br}_{1}\,Y$
is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron–Severi lattices. Over a field of characteristic
$0$
, we prove that the existence of a strong uniform bound on the size of the odd torsion of
$\text{Br}Y/\text{Br}_{1}Y$
is equivalent to the existence of a strong uniform bound on integers
$n$
for which there exist non-CM elliptic curves with abelian
$n$
-division fields. Using the same methods we show that, for a fixed prime
$\ell$
, a number field
$k$
of fixed degree
$r$
, and a fixed discriminant of the geometric Néron–Severi lattice,
$\#(\text{Br}Y/\text{Br}_{1}Y)\ell ^{\infty }$
is bounded by a constant that depends only on
$\ell$
,
$r$
, and the discriminant.
In this paper, we study inextensible ows of curves in 3-dimensional pseudo- Galilean space. We give necessary and sufficient conditions for inextensible ows of curves according to equiform geometry ...in pseudo-Galilean space.
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