Modular equation
Encyclopedia
In mathematics
, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. That is, given a number of functions on a moduli space
, a modular equation is an equation holding between them, or in other words an identity
for moduli.
The most frequent use of the term modular equation is in relation with the moduli problems for elliptic curve
s. In that case the moduli space itself is of dimension 1. That implies that any two rational function
s F and G, in the function field
of the modular curve, will satisfy a modular equation P(F,G) = 0 with P a non-zero polynomial
of two variables over the complex number
s. For suitable non-degenerate choice of F and G, the equation P(X,Y) = 0 will actually define the modular curve.
One should qualify that by saying that P, in the worst case, will be of high degree and the plane curve it defines will have singular points
; and the coefficient
s of P may be very large numbers. Further, the 'cusps' of the moduli problem, which are the points of the modular curve not corresponding to honest elliptic curves but degenerate cases, may be difficult to read off from knowledge of P.
That all being said, in that sense a modular equation becomes the equation of a modular curve. Such equations first arose in the theory of multiplication of elliptic function
s (geometrically, the n2-fold covering map
from a 2-torus
to itself given by the mapping x → n·x on the underlying group) expressed in terms of complex analysis
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. That is, given a number of functions on a moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
, a modular equation is an equation holding between them, or in other words an identity
Identity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
for moduli.
The most frequent use of the term modular equation is in relation with the moduli problems for elliptic curve
Elliptic curve
In 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 in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s. In that case the moduli space itself is of dimension 1. That implies that any two rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s F and G, in the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
of the modular curve, will satisfy a modular equation P(F,G) = 0 with P a non-zero polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
of two variables over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. For suitable non-degenerate choice of F and G, the equation P(X,Y) = 0 will actually define the modular curve.
One should qualify that by saying that P, in the worst case, will be of high degree and the plane curve it defines will have singular points
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
; and the coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s of P may be very large numbers. Further, the 'cusps' of the moduli problem, which are the points of the modular curve not corresponding to honest elliptic curves but degenerate cases, may be difficult to read off from knowledge of P.
That all being said, in that sense a modular equation becomes the equation of a modular curve. Such equations first arose in the theory of multiplication of elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
s (geometrically, the n2-fold covering map
Covering map
In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p...
from a 2-torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
to itself given by the mapping x → n·x on the underlying group) expressed in terms of complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
.