In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way...

of vector multiplication or changing the sign of complex numbers. In arithmetic, multiplication is often written using the multiplication sign (either...

some sense with loss of explicit information (as is typical of several complex variables). The Manin–Mumford conjecture of Yuri Manin and David Mumford...

SL2(L) an additional structure can be imposed of a building with complex multiplication. These were first introduced by Martin L. Brown (Brown 2004). These...

complex structure J {\displaystyle J} (different multiplication by i {\displaystyle i} ). If V {\displaystyle V} and W {\displaystyle W} are complex vector...

multiplication is commutative, associative and distributes over addition. Just as for complex numbers, one can define the notion of a split-complex conjugate...

A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient...

subring in its endomorphism ring End(A). The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth...

arithmetic geometry. He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the...

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...

"Tate-Shafarevich groups andL-functions of elliptic curves with complex multiplication". Inventiones Mathematicae. 89 (3): 527–559. doi:10.1007/BF01388984...

hence their coefficients, like quaternions. Multiplication of octonions is more complex. Multiplication is distributive over addition, so the product...

conjecture was first proved by Deuring (1941) for elliptic curves with complex multiplication. It was subsequently shown to be true for all elliptic curves over...

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...

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...

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...

The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers...

nonreal complex solutions − 1 + 3 i {\displaystyle -1+3i} and − 1 − 3 i {\displaystyle -1-3i} . Addition, subtraction and multiplication of complex numbers...

image for ⁠ n = 6 {\displaystyle n=6} ⁠. The group operation is multiplication of complex numbers. In the picture, multiplying with z {\displaystyle z}...

complex multiplications (again, ignoring simplifications of multiplications by 1 and similar) and n log 2 ⁡ ( n ) {\textstyle n\log _{2}(n)} complex additions...

the multiplication theorem for the gamma function follows from the Chowla–Selberg formula, which follows from the theory of complex multiplication. The...

Lang, Taniyama's eleventh problem deals with elliptic curves with complex multiplication, but is unrelated to Taniyama's twelfth and thirteenth problems...

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on...

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...

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...

generalized this example by showing that whenever the variety has complex multiplication by an imaginary quadratic field, then Hdg2(X) is not generated by...

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...

Benedict Gross's Harvard University homepage "Benedict Gross "Complex Multiplication: Past, Present, Future" Lecture 1". YouTube. January 30, 2019. Archived...

+i\sin \theta .} (See Euler's formula.) Under the complex multiplication operation, the unit complex numbers form a group called the circle group, usually...

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...