What does it mean for an elliptic curve to be modular?
Wiles proved that certain kinds of elliptic curves are modular — meaning that there is a particular modular form that corresponds to each curve — when the two variables and two coefficients involved in defining the curve are all rational numbers, values that can be written as fractions..
What is a modular form in math?
A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.
A modular form of weight k for the modular group SL(2,Z)={(abcd)a,b,c,d∈Z,ad−bc=1}Apr 6, 2016.
What is a modular function in complex analysis?
A modular form can equivalently be defined as a function F from the set of lattices in C to the set of complex numbers which satisfies certain conditions: If we consider the lattice Λ = Zα + Zz generated by a constant α and a variable z, then F(Λ) is an analytic function of z..
What is a modular function?
A modulus function is a function which gives the absolute value of a number or variable.
It produces the magnitude of the number of variables.
It is also termed as an absolute value function.
The outcome of this function is always positive, no matter what input has been given to the function..
What is an example of a modular form?
This writeup gives first examples of modular forms: Eisenstein series, the dis- criminant, and the j-function.
The modular group is the group of 2-by-2 matrices with integer entries and determinant 1, SL2(Z) = {[ a b c d ] : a, b, c, d ∈ Z, ad − bc = 1 } . c d ] (τ) = aτ + b cτ + d , τ ∈ u030.
- C
What is an example of a modular function?
Example: Consider the modulus function f(x) = x.
If x = − 3, then y = f(x) = f(−3) = −(−3) = 3, here x is less than 0..
What is the field of modular functions?
The field of modular functions for Γ0(N) is an extension of C(j) of degree n := [Γ(1) : Γ0(N)] generated by jN (τ)..
Why are modular forms useful?
Modular forms are very relevant to such algorithms.
Algebraic curves that are associated to modular forms are useful in constructing error-correcting codes.
This central open problem in arithmetic geometry relates arithmetic properties of elliptic curves (and abelian varieties) to special values of L-functions..
- A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.
A modular form of weight k for the modular group SL(2,Z)={(abcd)a,b,c,d∈Z,ad−bc=1}Apr 6, 2016 - Modular forms are used to solve the Lebesgue-Nagell equation.
Modular forms find the lattice with the densest sphere packing problem in dimensions 8 (E8 lattice) and in dimension 24 (Leech lattice).
Modular forms explain why eπ√163 is so close to an integer. - Wiles proved that certain kinds of elliptic curves are modular — meaning that there is a particular modular form that corresponds to each curve — when the two variables and two coefficients involved in defining the curve are all rational numbers, values that can be written as fractions.
