j-invariant

In mathematics, Felix Klein's $j$ -invariant or $j$ function, regarded as a function of a complex variable $τ$ , is a modular function of weight zero for special linear group $SL(2, Z)$ defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic away from a simple pole at the cusp such that

Rational functions of $j$ are modular, and in fact give all modular functions. Classically, the $j$ -invariant was studied as a parameterization of elliptic curves over $\mathbb {C}$ , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).

If $τ$ is any point of the upper half plane whose corresponding elliptic curve has (that is, if $τ$ is any element of an imaginary quadratic field with positive imaginary part, so that $j$ is defined), then $j (τ)$ is an algebraic integer.^[3] These special values are called singular moduli.

complex multiplication

The field extension $Q [j (τ), τ]/ Q (τ)$ is abelian, that is, it has an abelian .

Galois group

Let $Λ$ be the lattice in $C$ generated by ${1, τ}.$ It is easy to see that all of the elements of $Q (τ)$ which fix $Λ$ under multiplication form a ring with units, called an . The other lattices with generators ${1, τ'},$ associated in like manner to the same order define the algebraic conjugates $j (τ')$ of $j (τ)$ over $Q (τ)$ . Ordered by inclusion, the unique maximal order in $Q (τ)$ is the ring of algebraic integers of $Q (τ)$ , and values of $τ$ having it as its associated order lead to unramified extensions of $Q (τ)$ .

order

The $j$ -invariant has many remarkable properties:

These classical results are the starting point for the theory of complex multiplication.

Apostol, Tom M.

; Chan, Heng Huat (1999), "Ramanujan and the modular j-invariant", Canadian Mathematical Bulletin, 42 (4): 427–440, doi:10.4153/CMB-1999-050-1, MR 1727340. Provides a variety of interesting algebraic identities, including the inverse as a hypergeometric series.

Berndt, Bruce C.

(1989), Primes of the Form x^2 + ny^2: Fermat, Class Field Theory, and Complex Multiplication, New York: Wiley-Interscience Publication, John Wiley & Sons Inc., MR 1028322 Introduces the j-invariant and discusses the related class field theory.

Cox, David A.

; Norton, Simon (1979), "Monstrous moonshine", Bulletin of the London Mathematical Society, 11 (3): 308–339, doi:10.1112/blms/11.3.308, MR 0554399. Includes a list of the 175 genus-zero modular functions.

Conway, John Horton

(1977), Modular forms and functions, Cambridge: Cambridge University Press, ISBN 978-0-521-21212-0, MR 0498390. Provides a short review in the context of modular forms.

Rankin, Robert A.

(1937), "Arithmetische Untersuchungen elliptischer Integrale", Math. Annalen, 113: 1–13, doi:10.1007/BF01571618, MR 1513075, S2CID 121073687.