AI & ChatGPT searches , social queries for COMPLEX MULTIPLICATION
Search references for COMPLEX MULTIPLICATION. Phrases containing COMPLEX MULTIPLICATION
See searches and references containing COMPLEX MULTIPLICATION!
Search references for COMPLEX MULTIPLICATION. Phrases containing COMPLEX MULTIPLICATION
See searches and references containing COMPLEX MULTIPLICATION!COMPLEX MULTIPLICATION
Theory of a class of elliptic curves
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way
Complex_multiplication
Arithmetical operation
of vector multiplication or changing the sign of complex numbers. In arithmetic, multiplication is often written using the multiplication sign (either
Multiplication
Mathematical structure
SL2(L) an additional structure can be imposed of a building with complex multiplication. These were first introduced by Martin L. Brown. These buildings
Building_(mathematics)
Prime number with a certain relationship to an elliptic curve
order in an imaginary quadratic field. When E {\displaystyle E} has complex multiplication (CM) by an order in an imaginary quadratic field K {\displaystyle
Supersingular prime (algebraic number theory)
Supersingular_prime_(algebraic_number_theory)
Japanese mathematician (1930–2019)
arithmetic geometry. He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the
Goro_Shimura
Number with a real and an imaginary part
Every nonzero complex number has a multiplicative inverse, allowing division by complex numbers other than zero. This makes the complex numbers a field
Complex_number
subring in its endomorphism ring End(A). The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth
Complex multiplication of abelian varieties
Complex_multiplication_of_abelian_varieties
Methods to test or prove primality
construct a curve E where the number of points is easy to compute. Complex multiplication is key in the construction of the curve. Now, given an N for which
Elliptic_curve_primality
Hypercomplex number system
hence their coefficients, like quaternions. Multiplication of octonions is more complex. Multiplication is distributive over addition, so the product
Octonion
Reals with an extra square root of +1 adjoined
the ordinary complex ones. The collection of all such z is called the split-complex plane. Addition and multiplication of split-complex numbers are defined
Split-complex_number
Algorithm to multiply two numbers
A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient
Multiplication_algorithm
Identity obeyed by many special functions related to the gamma function
the multiplication theorem for the gamma function follows from the Chowla–Selberg formula, which follows from the theory of complex multiplication. The
Multiplication_theorem
Potential counterexample to the generalized Riemann hypothesis
an elliptic curve E D / C {\textstyle E_{D}/\mathbb {C} } with complex multiplication by Z [ τ D ] {\textstyle \mathbb {Z} [\tau _{D}]} , we have − 2
Siegel_zero
Complex multiplication field
number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field. The abbreviation "CM" was introduced
CM-field
Unproved conjecture in mathematics
the whole complex plane.[citation needed] This conjecture was first proved by Max Deuring for elliptic curves with complex multiplication. It was subsequently
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
American mathematician (1950–2025)
Benedict Gross's Harvard University homepage "Benedict Gross "Complex Multiplication: Past, Present, Future" Lecture 1". YouTube. January 30, 2019. Archived
Benedict_Gross
Algorithm for integer multiplication
The Karatsuba algorithm is a fast multiplication algorithm for integers. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. It is a
Karatsuba_algorithm
Algebraic structure with addition, multiplication, and division
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on
Field_(mathematics)
Circle with radius of one
+i\sin \theta .} (See Euler's formula.) Under the complex multiplication operation, the unit complex numbers form a group called the circle group, usually
Unit_circle
36 mathematical problems stated in 1955
Lang, Taniyama's eleventh problem deals with elliptic curves with complex multiplication, but is unrelated to Taniyama's twelfth and thirteenth problems
Taniyama's_problems
Number which when multiplied by x equals 1
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1 x {\displaystyle {\tfrac {1}{x}}} or x−1, is a number which when multiplied
Multiplicative_inverse
Algorithm for modelling sequential data
real numbers, not the complex numbers, but since complex multiplication can be implemented as real 2-by-2 matrix multiplication, this is a mere notational
Transformer_(deep_learning)
some sense with loss of explicit information (as is typical of several complex variables). The Manin–Mumford conjecture of Yuri Manin and David Mumford
Arithmetic of abelian varieties
Arithmetic_of_abelian_varieties
Modular function in mathematics
of the upper half plane whose corresponding elliptic curve has complex multiplication (that is, if τ is any element of an imaginary quadratic field with
J-invariant
Mathematical form
in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or
Product_(mathematics)
Geometric representation of the complex numbers
The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers
Complex_plane
Evaluates a certain product of values of the Gamma function at rational values
The origin of such formulae is now seen to be in the theory of complex multiplication, and in particular in the theory of periods of an abelian variety
Chowla–Selberg_formula
Problem about mathematical number fields
cyclotomic fields and their subfields. Leopold Kronecker described the complex multiplication issue as his liebster Jugendtraum, or "dearest dream of his youth"
Hilbert's_twelfth_problem
Mathematics concept
define multiplication by complex scalars in a canonical fashion so as to regard V {\displaystyle V} as a complex vector space. Every complex vector space
Linear_complex_structure
Mathematical conjecture about elliptic curves
be an elliptic curve defined over the rational numbers without complex multiplication. For a prime number p, define θp as the solution to the equation
Sato–Tate_conjecture
Fast method for calculating the digits of π
Chudnovsky, David; Chudnovsky, Gregory (1988), Approximation and complex multiplication according to Ramanujan, Ramanujan revisited: proceedings of the
Chudnovsky_algorithm
Four-dimensional number system
number system similar to the complex numbers, with the usual arithmetical operations of addition, subtraction, multiplication, and division, but with four
Quaternion
Principal square root of minus 1
numbers with the imaginary unit using addition and multiplication, a new number system known as the complex numbers is formed; it consists of all numbers of
Imaginary_unit
Mathematical concept
j-invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the
Supersingular_elliptic_curve
Branch of algebraic number theory concerned with abelian extensions
extensions of Q {\displaystyle \mathbb {Q} } , and the theory of complex multiplication to construct abelian extensions of CM-fields. There are three main
Class_field_theory
Mathematical concept
introduced by Goro Shimura in the course of his generalization of the complex multiplication theory. Shimura showed that while initially defined analytically
Shimura_variety
Discrete Fourier transform algorithm
complex multiplications (again, ignoring simplifications of multiplications by 1 and similar) and n log 2 ( n ) {\textstyle n\log _{2}(n)} complex additions
Fast_Fourier_transform
Set with associative invertible operation
image for n = 6 {\displaystyle n=6} . The group operation is multiplication of complex numbers. In the picture, multiplying with z {\displaystyle z}
Group_(mathematics)
American mathematician (1925–2019)
the use of formal groups, creating the Lubin–Tate local theory of complex multiplication. He has also made a number of individual and important contributions
John_Tate_(mathematician)
Australian mathematician (1945–2022)
the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication. In 1977, Coates moved back to Australia, becoming a professor at
John_H._Coates
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, exponentials, and logarithms with arbitrary base, typically
CORDIC
Type of character in number theory
a} of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all Archimedean completions of K {\displaystyle
Hecke_character
Mathematical operation in linear algebra
linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns
Matrix_multiplication
Algebraic operation
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract
Scalar_multiplication
Group that is also a differentiable manifold with group operations that are smooth
is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that
Lie_group
Unsolved problem in geometry
generalized this example by showing that whenever the variety has complex multiplication by an imaginary quadratic field, then Hdg2(X) is not generated by
Hodge_conjecture
Mathematics concept
complex structure J {\displaystyle J} (different multiplication by i {\displaystyle i} ). If V {\displaystyle V} and W {\displaystyle W} are complex vector
Complex conjugate of a vector space
Complex_conjugate_of_a_vector_space
Array of numbers
and columns, usually satisfying certain properties of addition and multiplication. For example, [ 1 9 − 13 20 5 − 6 ] {\displaystyle
Matrix_(mathematics)
Japanese mathematician
12:08 minutes in. BBC. Shimura, Goro; Taniyama, Yutaka (1961), Complex multiplication of abelian varieties and its applications to number theory, Publications
Yutaka_Taniyama
Algorithmic runtime requirements for common math procedures
Chudnovsky, David; Chudnovsky, Gregory (1988). "Approximations and complex multiplication according to Ramanujan". Ramanujan revisited: Proceedings of the
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Method for producing composition algebras
to generalize the multiplication and conjugation operations. Form ordered pairs (a, b) of complex numbers a and b, with multiplication defined by ( a ,
Cayley–Dickson_construction
American mathematician
"Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication". Inventiones Mathematicae. 89 (3): 527–559. doi:10.1007/BF01388984
Karl_Rubin
Fundamental operation on complex numbers
{\displaystyle a+bi} . For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division: z + w ¯ = z ¯ + w
Complex_conjugate
Concept in mathematics
module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization
Drinfeld_module
Coefficient used in fast Fourier transform (FFT) algorithms
specifically, "twiddle factors" originally referred to the root-of-unity complex multiplicative constants in the butterfly operations of the Cooley–Tukey FFT algorithm
Twiddle_factor
Complex number whose mapping on a coordinate plane produces a triangular lattice
(1997-05-08). Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication (PDF). Wiley. p. 77. ISBN 0-471-19079-9. " X 2 + X + 1 {\displaystyle
Eisenstein_integer
Linear operator scaling by a fixed function
multiplication operator is a linear operator Tf defined on some vector space of functions and whose value at a function φ is given by multiplication by
Multiplication_operator
Topic in mathematics
the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers
Complexification
Method in signal processing
(log2(N) + 1) complex multiplications for the FFT, product of arrays, and IFFT. Each iteration produces N-M+1 output samples, so the number of complex multiplications
Overlap–add_method
Branch of elementary mathematics
mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction
Arithmetic
Vector space equipped with a bilinear product
consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms
Algebra_over_a_field
Approach to public-key cryptography
points and generate a curve with this number of points using the complex multiplication technique. Several classes of curves are weak and should be avoided:
Elliptic-curve_cryptography
Gauss sum on an elliptic curve
is an analog of a Gauss sum depending on an elliptic curve with complex multiplication. The quadratic residue symbol in a Gauss sum is replaced by a higher
Elliptic_Gauss_sum
Field in algebraic number theory
{\displaystyle K} , but ramified at both real places. By the theory of complex multiplication, the Hilbert class field of an imaginary quadratic field is generated
Hilbert_class_field
American mathematician
(2013) on the Galois representations of elliptic curves without complex multiplication. Computations by Galbraith (2002) and Baran (2014) had previously
Jennifer_Balakrishnan
Plane algebraic curve
Gaussian integers. For this reason the case of elliptic functions with complex multiplication by √−1 is called the lemniscatic case in some sources. Using the
Lemniscate_of_Bernoulli
Descriptive vector graphics language
equations. It is mathematically oriented (e.g. rotation of vectors by complex multiplication), and uses the simplex method and deferred drawing to solve overall
Asymptote (vector graphics language)
Asymptote_(vector_graphics_language)
Mathematical formal group law
Tate (1965) to isolate the local field part of the classical theory of complex multiplication of elliptic functions. In particular it can be used to construct
Lubin–Tate_formal_group_law
formula Gauss–Newton algorithm Gauss–Legendre algorithm Gauss's complex multiplication algorithm Gauss's theorem may refer to the divergence theorem, which
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Number-theoretic concept
sums as Hecke characters. This was to become important once the complex multiplication of abelian varieties became established. The Hecke characters in
Jacobi_sum
Richard Dedekind, Leopold Kronecker describes his Jugendtraum, to use complex multiplication theory to generate abelian extensions of imaginary quadratic fields
Timeline_of_abelian_varieties
Group in arithmetic geometry
Rubin proved this for some elliptic curves of rank at most 1 with complex multiplication. Victor A. Kolyvagin extended this to modular elliptic curves over
Tate–Shafarevich_group
American mathematician
{\displaystyle x^{2}+n\cdot y^{2}} : Fermat, class field theory, and complex multiplication, Wiley 1989 With John Little, Henry Schenck: Toric Varieties, American
David_A._Cox
Arithmetic operation
When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases: b n
Exponentiation
British mathematician who proved Fermat's Last Theorem
Together they worked on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory. He further worked with Barry Mazur
Andrew_Wiles
Concept in mathematics
effort to isolate the local field part of the classical theory of complex multiplication of elliptic functions. It is also a major ingredient in some approaches
Formal_group_law
Manifold
bundle, on which multiplication by complex numbers makes sense (even if we started with a real manifold). The eigenvalues of an almost complex structure are
Complex_manifold
Algorithm to multiply matrices
Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms
Matrix multiplication algorithm
Matrix_multiplication_algorithm
Electrical engineering concept
equivalent formulation that replaces the modulo 2π operation with a complex multiplication is: φ [ n ] = φ [ n − 1 ] + arg { s a [ n ] s a ∗ [ n − 1 ] }
Instantaneous phase and frequency
Instantaneous_phase_and_frequency
Method in signal processing
(log2(N) + 1) complex multiplications for the FFT, product of arrays, and IFFT. Each iteration produces N-M+1 output samples, so the number of complex multiplications
Overlap–save_method
Recursive algorithm for matrix multiplication
Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for large matrices, with a better
Strassen_algorithm
Isomorphism of differentiable manifolds
of a complex number of a particular type. When (dx, dy) is also interpreted as that type of complex number, the action is of complex multiplication in the
Diffeomorphism
Japanese mathematician (born 1952)
1977, and a Ph.D. in 1980 with thesis On Abelian Varieties with Complex Multiplication as Factors of the Jacobians of Shimura Curves, although he left
Haruzo_Hida
Complex exponential in terms of sine and cosine
the mathematics when used in multiplication or powers of complex numbers. Any complex number z = x + iy, and its complex conjugate, z = x − iy, can be
Euler's_formula
Multiplication is a mathematical practice that can be applied to music. The operation multiplies the numeric value of musical parameters like notes or
Multiplication_(music)
Algebraic curve
It is isogenous to a product of simple abelian varieties with complex multiplication. The Fermat curve also has gonality: n − 1. {\displaystyle n-1
Fermat_curve
Modular unit in mathematics
elliptic units may be constructed for an elliptic curve E with complex multiplication by the ring of integers R of an imaginary quadratic field F. For
Elliptic_unit
Mathematics of varieties with integer coordinates
now includes Diophantine geometry along with class field theory, complex multiplication, local zeta-functions and L-functions. Paul Vojta wrote: While others
Diophantine_geometry
Belgian mathematician
geometry. There is a Gross–Deligne conjecture in the theory of complex multiplication. There is a Deligne conjecture on monodromy, also known as the weight
Pierre_Deligne
Mathematical operation on points on an elliptic curve
Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic
Elliptic curve point multiplication
Elliptic_curve_point_multiplication
Concept in algebraic number theory
complex multiplication and the q-expansion of the j-invariant. In what follows, j ( z ) {\displaystyle j(z)} denotes the j-invariant of the complex number
Heegner_number
Study of objects of arithmetic interest over infinite towers of number fields
de Shalit, Ehud (1987), Iwasawa theory of elliptic curves with complex multiplication. p-adic L functions, Perspectives in Mathematics, vol. 3, Boston
Iwasawa_theory
Mathematical concept
has complex multiplication by Z [ ζ 3 ] {\displaystyle \mathbb {Z} [\zeta _{3}]} , and j = 1728 {\displaystyle j=1728} has complex multiplication by Z
J-line
Algebraic structure in linear algebra
vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector spaces and complex vector spaces are kinds
Vector_space
Result about when a matrix can be diagonalized
may be complex-valued. There is also a formulation of the spectral theorem in terms of direct integrals. It is similar to the multiplication-operator
Spectral_theorem
Proposed lower bound on the Mahler measure for polynomials with integer coefficients
: K ] {\displaystyle D=[K(Q):K]} . If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's result holds: h ^ E ( Q ) ≥
Lehmer's_conjecture
Sequence of program instructions invokable by other software
one can designate subroutine A as division and subroutine B as complex multiplication and subroutine C as the evaluation of a standard error of a sequence
Function (computer programming)
Function_(computer_programming)
Concept in military and political science
The expression military–industrial complex (MIC) describes the relationship between a country's military and the defense industry that supplies it, seen
Military–industrial_complex
Israeli mathematician (born 1955)
De Shalit, Ehud (1987). Iwasawa theory of elliptic curves with complex multiplication. Perspectives in Mathematics. Boston: Academic Press. ISBN 978-0-12-210255-4
Ehud_de_Shalit
Algebraic structure with addition and multiplication
multiplication, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they
Ring_(mathematics)
COMPLEX MULTIPLICATION
COMPLEX MULTIPLICATION
Boy/Male
Tamil
Complete
Surname or Lastname
English
English : habitational name from Coppull in Lancashire, recorded in the 13th century as Cophill, from Old English copp ‘peak’ + hyll ‘hill’.English : nickname from Old French curt peil ‘short hair’.Probably an Americanized spelling of German and Jewish Koppel or German and Dutch Kappel.
Girl/Female
Tamil
Complete
Boy/Male
Indian
Complete
Girl/Female
Tamil
Complete
Girl/Female
Arabic, Muslim
Complex; Zigzag; Curling
Boy/Male
Tamil
Complete
Girl/Female
Tamil
Complete
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : habitational name from any of various places called Copley, for example in County Durham, Staffordshire, and Yorkshire, from the Old English personal name Coppa (apparently a byname for a tall man) or from copp ‘hilltop’ + lēah ‘woodland clearing’.
Girl/Female
Tamil
Complete
Girl/Female
Hindu, Indian
Complex
Girl/Female
Muslim
Complex, Zigzag, Curling
Girl/Female
Tamil
Shesha Harani | ஷேஷ ஹரணீÂ
Complete
Shesha Harani | ஷேஷ ஹரணீÂ
Girl/Female
Bengali, Indian
Good Complex
Surname or Lastname
English
English : habitational name, probably from Comley in Shropshire or Combley on the Isle of Wight; both are named with Old English cumb ‘valley’ + lēah ‘woodland clearing’.
Boy/Male
Tamil
Poornan | பூரà¯à®¨à®¾à®¨
Complete
Poornan | பூரà¯à®¨à®¾à®¨
Boy/Male
Tamil
Complete
Girl/Female
Tamil
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Complete
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Surname or Lastname
English
English : unexplained.Americanized form of German Koppler.
Boy/Male
Indian
Complete
COMPLEX MULTIPLICATION
COMPLEX MULTIPLICATION
Girl/Female
Arabic, Australian, Indian, Muslim, Sikh, Swedish
Skillful; Radiance; Elegance; Brilliant; Praise
Female
Hebrew
(×ֲבִיטַל) Hebrew name ABIYTAL means "my father is dew." In the bible, this is the name of one of David's wives.Â
Girl/Female
American, Australian
Female Version of Daniel
Boy/Male
Arabic, Urdu
Bright
Girl/Female
Hebrew
Rejoice.
Boy/Male
Hindu
Traveler
Boy/Male
Indian
Shastra
Girl/Female
Australian, Chinese, French, Greek, Japanese, Latin
Grain
Male
English
Unisex pet form of English Lauren and Laurence, both LAURIE means "of Laurentum."
Girl/Female
Tamil
Shivatmika | ஷீவாதà¯à®®à¯€à®•à®¾
Goddess Lakshmi
COMPLEX MULTIPLICATION
COMPLEX MULTIPLICATION
COMPLEX MULTIPLICATION
COMPLEX MULTIPLICATION
COMPLEX MULTIPLICATION
a.
One of the pairs of plates of two metals which compose a voltaic battery; -- called a voltaic couple or galvanic couple.
a.
That which joins or links two things together; a bond or tie; a coupler.
a.
Complex, complicated.
a.
Intricate; entangled; complicated; complex.
n.
Two taken together; a pair or couple; especially two lines of verse that rhyme with each other.
n.
A complex; an aggregate of parts; a complication.
imp. & p. p.
of Couple
n.
One who couples; that which couples, as a link, ring, or shackle, to connect cars.
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
a.
Repeatedly compound; made up of complex constituents.
a.
Not complex; uncompounded; simple.
a.
See Couple-close.
pl.
of Couple-close
v. t.
To bring to a state in which there is no deficiency; to perfect; to consummate; to accomplish; to fulfill; to finish; as, to complete a task, or a poem; to complete a course of education.
imp. & p. p.
of Compile
imp. & p. p.
of Comply
n.
One who compiles; esp., one who makes books by compilation.
adv.
In a complex manner; not simply.
n.
One who complies, yields, or obeys; one of an easy, yielding temper.
a.
Finished; ended; concluded; completed; as, the edifice is complete.