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Mathematical concept
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic
Elliptic_surface
Type of surface singularity used in algebraic geometry
In algebraic geometry, an elliptic singularity of a surface, introduced by Philip Wagreich in 1970, is a surface singularity such that the arithmetic genus
Elliptic_singularity
One-dimensional complex manifold
Weierstrass elliptic function. Likewise, genus g {\displaystyle g} surfaces have Riemann surface structures, as (compactifications of) hyperelliptic surfaces y
Riemann_surface
Mathematical algorithm
studying elliptic surfaces. It determines whether a given set of sections of an elliptic surface provides a basis, up to torsion, for the surface's Mordell–Weil
Cox–Zucker_machine
Three-dimensional solid
hyperbola is called an elliptic cylinder, parabolic cylinder and hyperbolic cylinder, respectively. These are degenerate quadric surfaces. When the principal
Cylinder
Mathematical classification of surfaces
surfaces, all hyperelliptic surfaces, all Kodaira surfaces, some K3 surfaces, some abelian surfaces, and some rational surfaces are elliptic surfaces
Enriques–Kodaira classification
Enriques–Kodaira_classification
hyperelliptic surface, or bi-elliptic surface, is a minimal surface whose Albanese morphism is an elliptic fibration without singular fibres. Any such surface can
Hyperelliptic_surface
Quadric surface with one axis of symmetry and no center of symmetry
plane). A paraboloid is either elliptic or hyperbolic. Equivalently, a paraboloid may be defined as a quadric surface that is not a cylinder, and has
Paraboloid
Algebraic surface with special triviality properties
trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex numbers) and are elliptic surfaces of genus 0. Over fields
Enriques_surface
Algebraic curve in mathematics
mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined
Elliptic_curve
Type of algebraic surface
named for Michel Raynaud (1978). To be precise, a Raynaud surface is a quasi-elliptic surface over an algebraic curve of genus g greater than 1, such that
Raynaud_surface
American mathematician (1949–2019)
provides a basis (up to torsion) for the Mordell–Weil group of an elliptic surface E → S {\displaystyle E\to S} , where S {\displaystyle S} is isomorphic
Steven_Zucker
modular surface is one of the elliptic surfaces studied by Shioda (1972). Barth, Wolf; Hulek, Klaus (1985), "Projective models of Shioda modular surfaces",
Shioda_modular_surface
Non-Euclidean geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel
Elliptic_geometry
Type of smooth complex surface of kodaira dimension 0
general elliptic K3 surface has exactly 24 singular fibers, each of type I 1 {\displaystyle I_{1}} (a nodal cubic curve). Whether a K3 surface is elliptic can
K3_surface
Concept in algebraic geometry
1 (an abelian surface) has Kodaira dimension 0; the product of a curve of genus 1 with a curve of genus at least 2 (an elliptic surface) has Kodaira dimension
Kodaira_dimension
Smooth closed surface with g holes
A genus one orientable surface is the ordinary torus. A non-orientable surface of genus one is the projective plane. Elliptic curves over the complex
Genus_g_surface
Critical point on a surface graph which is not a local extremum
hyperbolic paraboloid shape. Saddle surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian
Saddle_point
Cone with an elliptical base
lateral conic surface, a quadric called conical quadric or quadratic cone. In a three-dimensional Cartesian coordinate system, an elliptic cone is the locus
Elliptic_cone
Humbert surfaces Picard modular surfaces Shioda modular surfaces Elliptic surfaces, surfaces with an elliptic fibration; quasielliptic surfaces constitute
List of complex and algebraic surfaces
List_of_complex_and_algebraic_surfaces
In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by Igor Dolgachev (1981). They can be used to give examples
Dolgachev_surface
American mathematician
Amherst College. He studies, among other things, étale homotopy theory, elliptic surfaces, computer-based algebraic geometry (such as Gröbner basis), Torelli
David_A._Cox
Mathematical classification
simple groups, Monster, Baby and Fischer 24', cf. monstrous moonshine. Elliptic surface Catastrophe theory § Arnold's notation, an ADE classification (Proctor
ADE_classification
Concept in algebraic geometry
In mathematics, an abelian surface is a 2-dimensional abelian variety. One-dimensional complex tori are just elliptic curves and are all algebraic, but
Abelian_surface
Russian mathematician
1-dimensional global fields to 2-dimensional arithmetic surfaces such as proper regular models of elliptic curves over global fields. His theory led to three
Ivan_Fesenko
fourth order elliptic partial differential equation can be formulated. Biharmonic Bézier surfaces are related to minimal surfaces. i.e. surfaces that minimise
Biharmonic_Bézier_surface
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
that every Riemann surface admits a Riemannian metric of constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic
Uniformization_theorem
Mathematical concept
crystalline cohomology are all n/2. For special classes of varieties such as elliptic curves, it is common to use various ad hoc definitions of "supersingular"
Supersingular_variety
Kepler orbit with an eccentricity of less than one
In astrodynamics or celestial mechanics, an elliptical orbit or eccentric orbit is an orbit with an eccentricity of less than 1;[citation needed] this
Elliptic_orbit
Smooth 4-manifold homeomorphic yet not diffeomorphic to Euclidean space
complex surface C P 2 # 9 C P 9 ¯ {\displaystyle \mathbb {C} P^{2}\#9{\overline {\mathbb {C} P^{9}}}} , the simplest example of an elliptic surface known
Exotic_R4
ISSN 0002-9939, MR 2390492, S2CID 35276117 Ishida, Masa-Nori (1988), "An elliptic surface covered by Mumford's fake projective plane", The Tohoku Mathematical
Bogomolov–Miyaoka–Yau inequality
Bogomolov–Miyaoka–Yau_inequality
Surface drawn by a moving line passing through a fixed point
point not on the plane of C {\displaystyle C} , one obtains an elliptic cone. A conical surface S {\displaystyle S} can be described parametrically as S (
Conical_surface
algebraic surfaces Ruled surface Cubic surface Veronese surface Del Pezzo surface Rational surface Enriques surface K3 surface Hodge index theorem Elliptic surface
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Algebraic variety of dimension two
hyperelliptic surfaces κ = 1: elliptic surfaces κ = 2: surfaces of general type. For more examples see the list of algebraic surfaces. The first five examples
Algebraic_surface
The idea goes further. Thus elliptic surfaces over the complex numbers, also, have some quite strict analogies with elliptic curves over number fields.
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
definition. For elliptic curves, potential good reduction is equivalent to the j-invariant being an algebraic integer. Elliptic surface Serre, Jean-Pierre;
Potential_good_reduction
Number of "holes" of a surface
applied to the Riemann surface of X {\displaystyle X} (its manifold of complex points). For example, the definition of elliptic curve from algebraic geometry
Genus_(mathematics)
Polarization of electromagnetic radiation
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an
Elliptical_polarization
Wilson. The PDE method involves generating a surface for some boundary by means of solving an elliptic partial differential equation of the form ( ∂
PDE_surface
Third planet from the Sun
being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all of Earth's water is contained in its ocean, which covers
Earth
Elliptical galaxy smaller than normal ones
Dwarf elliptical galaxies have blue absolute magnitudes within the range −18 < MV < −14 : fainter than ordinary elliptical galaxies. The surface brightness
Dwarf_elliptical_galaxy
Type of vehicle suspension
laminated or carriage spring, and sometimes referred to as a semi-elliptical spring, elliptical spring, or cart spring, it is one of the oldest forms of vehicle
Leaf_spring
Algorithm in the theory of elliptic curves
fibers given by the Kodaira symbol or Néron symbol, for which, see elliptic surfaces: in turn this determines the exponent fp of the conductor E. Tate's
Tate's_algorithm
the historical design. The vane tips radially contact the cam ring elliptic surface and cause a significant friction as the rotor and vanes turn. This
Hydristor
Distinguished surfaces of dynamic trajectories
members of nested families of elliptic LCSs. Two- and three-dimensional examples of elliptic LCS revealed by tubular level surfaces of the PRA are shown in
Lagrangian_coherent_structure
Geometric shape
base, it is called a frustum. An elliptical cone is a cone with an elliptical base. A generalized cone is the surface created by the set of lines passing
Cone
Algorithm for integer factorization
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
general fibers have the Kodaira dimension 0 i.e. elliptic curve. Therefore, S is the elliptic surface. These fact can be generalized to the general n.
Iitaka_dimension
Surface containing a line through every point
S. Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid,
Ruled_surface
Mathematical concept
new surfaces and open conjectures in topology. The Costa surface can be described using the Weierstrass zeta function and the Weierstrass elliptic function
Costa's_minimal_surface
Locus of the zeros of a polynomial of degree two
In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space,
Quadric
Conjecture on zeros of the zeta function
(2011), "Positivity of certain functions associated with analysis on elliptic surfaces", Journal of Number Theory, 131 (10): 1770–1796, doi:10.1016/j.jnt
Riemann_hypothesis
Stability conditions for triangulated cateogires
Hokuto (2015-11-18). "Autoequivalences of derived categories of elliptic surfaces with non-zero Kodaira dimension". pp. 10–12. arXiv:1501.06657 [math
Bridgeland stability condition
Bridgeland_stability_condition
Lines with constant perpendicular distance between them
In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The
Clifford_parallel
Parameter in the gravitational two-body problem
R, then the additional specific energy of an elliptic orbit compared to being stationary at the surface is − μ 2 a + μ R = μ ( 2 a − R ) 2 a R {\displaystyle
Specific_orbital_energy
Special functions of several complex variables
when arranged in a lattice?" geometry: "what are the shape properties of elliptic curves?" and others, including abelian varieties, moduli spaces, quadratic
Theta_function
Special point on a modular curve in mathematics
arXiv:math.NT/0506325v2. Brown, Mark (1994), "On a conjecture of Tate for elliptic surfaces over finite fields", Proc. London Math. Soc., 69 (3): 489–514, doi:10
Heegner_point
analogies with elliptic fibrations. Given two distinct irreducible divisors and a closed point on the special fiber of an arithmetic surface, we can define
Arithmetic_surface
Electromagnetic method of non-destructive testing of conductive materials
effects as flaws become more distant from the inspection surface. Flaw area: For semi-elliptical surface flaws, the flaw area A is determined by the flaw depth
Eddy-current_testing
Astrodynamic equation
only slightly larger than the potential energy at the surface of the Earth, then the orbit is elliptic with eccentricity close to 1 and one end of the ellipse
Orbit_equation
These surfaces contain an elliptic curve (the image of the x-axis) and if λ = 0 {\displaystyle \lambda =0} the image of the y-axis is a second elliptic curve
Hopf_surface
Area of mathematics
hyperbolic umbilic catastrophe modeled the breaking of a wave and the elliptical umbilic modeled the creation of hair-like structures. V = x 3 + y 3 +
Catastrophe_theory
American mathematician (born 1937)
These are the elliptic and quasi-elliptic surfaces not contained in the last two groups. Kodaira dimension 2. These are the surfaces of general type
David_Mumford
French mathematician
elliptic curves. With David Harbater and following the work of Jean-Pierre Serre, Raynaud proved Abhyankar's conjecture in 1994. The Raynaud surface was
Michel_Raynaud
Algebraic surface in mathematics
algebraic surfaces. Most of them are surfaces of general type, but several are rational surfaces or blown up K3 surfaces or elliptic surfaces. van der
Hilbert_modular_variety
Type of non-Euclidean geometry
name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry (spherical geometry), parabolic geometry (Euclidean geometry)
Hyperbolic_geometry
Dutch mathematician (1942–2010)
interested in algebraic geometry and wrote a book on QRT maps and elliptic surfaces. Duistermaat contributed also to applied mathematics. He was a consultant
Hans_Duistermaat
In mathematics, the nodal line conjecture is a statement posed in 1967 by Lawrence E. Payne about the Laplacian partial differential equation. The original
Nodal_line_conjecture
Gauss curvature Elliptic point Types of surfaces Minimal surface Ruled surface Conical surface Developable surface Nadirashvili surface See also multivariable
List of differential geometry topics
List_of_differential_geometry_topics
Locally spherical point on a mathematical surface
isolated points in the elliptical region of the surface; that is, where the Gaussian curvature is positive. The sphere is the only surface with non-zero curvature
Umbilical_point
Surface with vanishing affine mean curvature
points of an affine area functional. Affine maximal surfaces are often studied under nonlinear elliptic PDE theory via the linearized Monge–Ampère equation
Affine_maximal_surface
unduloid, or onduloid, is a surface with constant nonzero mean curvature obtained as a surface of revolution of an elliptic catenary: that is, by rolling
Unduloid
Mathematics of smooth surfaces
of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have
Differential geometry of surfaces
Differential_geometry_of_surfaces
Type of orbital maneuver
In astronautics and aerospace engineering, the bi-elliptic transfer is an orbital maneuver that moves a spacecraft from one orbit to another and may, in
Bi-elliptic_transfer
American mathematician
Morgan and Kieran G. O'Grady. Differential topology of complex surfaces. Elliptic surfaces with pg = 1: smooth classification. With the collaboration of
John_Morgan_(mathematician)
Mathematical surface
that the elliptic modular surface of level 4 (the universal generalized elliptic curve E(4) → X(4)) in characteristic 3 mod 4 is a K3 surface with Picard
Supersingular_K3_surface
Nonlinear and exact periodic wave solution of the Korteweg–de Vries equation
are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves. They are used to describe surface gravity waves of fairly long
Cnoidal_wave
In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions (doubly periodic meromorphic functions on the complex plane) that map
Dixon_elliptic_functions
2D surface which extends indefinitely
intersect, so that every pair of lines intersects in exactly one point. The elliptic plane may be further defined by adding a metric to the real projective
Plane_(mathematics)
Result concerning properties of Galois representations associated with modular forms
Ribet's theorem shows that if the Galois representation associated with an elliptic curve has certain properties, then that curve cannot be modular (in the
Ribet's_theorem
Hyperelliptic curve cryptography is similar to elliptic curve cryptography (ECC) insofar as the Jacobian of a hyperelliptic curve is an abelian group in
Hyperelliptic curve cryptography
Hyperelliptic_curve_cryptography
Concept in gravitational orbital mechanics
lost in the system while the work is being done. For any Keplerian orbit (elliptic, parabolic, hyperbolic, or radial), the vis-viva equation is as follows:
Vis-viva_equation
Surface a body of energy cannot cross
masses in elliptic orbits, the general planar three-body problem, the four-body problem with solar wind drag, or in rings. The zero-velocity surface is also
Zero-velocity_surface
Chinese-American mathematician
University for her graduate studies; her dissertation, Modularity of Elliptic Surfaces, she worked on with Noriko Yui, visiting from Queen's University,
Ling_Long_(mathematician)
Term used in the theories of Riemann surfaces and algebraic curves
when integrated along paths, give rise to integrals that generalise the elliptic integrals to all curves over the complex numbers. They include for example
Differential of the first kind
Differential_of_the_first_kind
In mathematics, a Riemann surface
maximized in genus 2 for the Bolza surface. The Jacobian variety of the Bolza curve is the product of two copies of the elliptic curve C / Z [ − 2 ] {\displaystyle
Bolza_surface
Product of the principal curvatures of a surface
curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome like, locally lying on one side of its tangent
Gaussian_curvature
Set of bivariate correlations among galaxies
between the effective radius, average surface brightness and central velocity dispersion of normal elliptical galaxies. Any one of the three parameters
Fundamental plane (elliptical galaxies)
Fundamental_plane_(elliptical_galaxies)
Asymptotically stable in the sense of geometric invariant theory
is the same. Note that for (1) the types of singularities found in elliptic surfaces can be completely classified. One classical example of a family of
Stable_curve
Two-dimensional manifold
non-isomorphic compact Riemann surfaces of genus 1 (the elliptic curves). Complex structures on a closed oriented surface correspond to conformal equivalence
Surface_(topology)
British mathematician
the relation between algebraic groups and del Pezzo surfaces; the period map for elliptic surfaces.[citation needed] In 2008, with the number theorists
Nicholas_Shepherd-Barron
Italian mathematician (born 1957)
September 2016. (Wednesday, May 5th, 2010) "Unlikely Intersections in Elliptic Surfaces and Problems of Masser - Umberto Zannier". YouTube. 1 September 2016
Umberto_Zannier
into a blown-up Hopf surface. Parabolic Inoue surfaces contain a cycle of rational curves with 0 self-intersection and an elliptic curve. They are a particular
Inoue_surface
Curves in algebraic geometric
In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Bernstein, Birkner
Twisted_Edwards_curve
Galaxy which is less bright than the ambient night sky
A low-surface-brightness galaxy, or LSB galaxy, is a diffuse galaxy with a surface brightness that, when viewed from Earth, is at least one magnitude lower
Low_surface_brightness_galaxy
Two geometries based on axioms closely related to those specifying Euclidean geometry
metric geometry. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When isotropic quadratic
Non-Euclidean_geometry
Swiss mathematician
S2CID 119435006. Habegger, Philipp (2013). "Special points on fibered powers of elliptic surfaces". Journal für die reine und angewandte Mathematik. 2013 (685). arXiv:1110
Philipp_Habegger
Relates rational elliptic curves to modular forms
In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way
Modularity_theorem
In mathematics, the conductor of an elliptic curve over the field of rational numbers (or more generally a local or global field) is an integral ideal
Conductor of an elliptic curve
Conductor_of_an_elliptic_curve
British mathematician who proved Fermat's Last Theorem
graduating from there in 1974, he worked on unifying Galois representations, elliptic curves and modular forms, starting with Barry Mazur's generalizations of
Andrew_Wiles
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