The j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. We can express it in terms of Jacobi's theta functions, in which form it can very rapidly be computed. We have
The numerator and denominator above are in terms of
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and
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These have the properties that
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and possess the analytic properties making them modular forms. Δ is a modular form of weight twelve by the above, and one of weight four, so that its third power is also of weight twelve. The quotient is therefore a modular function of weight zero; this means j has the absolutely invariant property that
The two transformations and together generate a group, which we may identify with the group of two-by-two integral matrices of determinant one, where matrices of opposite sign are identified. By a suitable choice of transformation belonging to this group,
, with ad-bc=1, we may reduce τ to a value giving the same value for j, and lying in the fundamental region for j, which consists of values for τ satisfying the conditions
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As a Riemann surface, this has genus 0, and every (level one) modular function is a rational function in j; in other words the field of modular functions is .
The values of j are in a one-to-one relationship with values of τ lying in the fundamental region, and each value for j corresponds to the field of elliptic functions with periods 1 and τ, for the corresponding value of τ; this means that j is in a one-to-one relationship with isomorphism classes of elliptic curves.
The j-invariant has many remarkable properties. One of these is that if τ is any element of an imaginary quadratic number field with positive imaginary part (so that j is defined) then is an algebraic integer. The field extension is abelian, meaning with abelian Galois group. We have a lattice in the complex plane defined by 1 and τ, and it is easy to see that all of the elements of the
field which send lattice points to other lattice points under multiplication form a ring with units, called an order. The other lattices with generators 1 and τ' associated in like manner to the same order define the algebraic conjugates of over . The unique maximal order under inclusion of is the ring of algebraic integers of , and values of τ having it as its associated order lead to unramified extensions of . These classical results are the starting point for the theory of complex multiplication.
- The q-series and moonshine
Another remarkable property of j has to with what is called its ‘’q-series’’. If we fix the imaginary part of τ and vary the real part, we obtain a period complex function of a real variable with period one. The Fourier coefficients for these functions are extremely interesting. If we perform the substitution the Fourier series becomes a Laurent series in q, , where the values for for n < -1 are all zero, and where the are integers. The first few terms of it are
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as we may easily find by substituting q for in the definition for j with which we started. The coefficients for the positive exponents of q are the dimensions of the grade n part an infinite dimensional graded algebra representation of the Monster called the ‘’moonshine module’’, a fact which may be taken as the starting point for moonshine theory.
Still another remarkable property of the q-series for j is the product formula; if p and q are small enough we have
So far we have been considering j as a function of a complex variable. However, as an invariant for isomorphism classes of elliptic curves, it can be defined purely algebraically. Let
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be a plane elliptic curve in any field of characteristic neither 2 nor 3 in which the coefficients lie. Then we may define
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The j-invariant for the elliptic curve may now be defined as
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