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Min Max Min Max Re Im Let and be periods of a doubly periodic function, with the half-period ratio a number with . Then Klein's absolute invariant (also called Klein's modular function) is defined as(1)where and are the invariants of the Weierstrass elliptic function with modular discriminant(2)(Klein 1877). If , where is the upper half-plane, then(3)is a function of the ratio only, as are , , and . Furthermore, , , , and are analytic in (Apostol 1997, p. 15).Klein's absolute invariant is implemented in the WolframLanguage as KleinInvariantJ[tau].The function is the same as the j-function, modulo a constant multiplicative factor.Every rational function of is a modular function, and every modular function can be expressed as a rational function of (Apostol 1997, p. 40).Klein's invariant can be given explicitly by(4)(5)(Klein 1878-1879, Cohn 1994), where is the elliptic lambda function(6) is a Jacobi theta function, the are..

Petersson considered the absolutely converging DirichletL-series(1)Writing the denominator as(2)where(3)and(4)Petersson conjectured that and are always complex conjugate, which implies(5)and(6)This conjecture was proven by Deligne (1974), which also proved the tau conjecture as a special case. Deligne was awarded the Fields medal for his proof.

Let , , and(1)where(2)(3)Then if either and , or and ,(4)where is a Bernoulli polynomial, and the second term on the right side can be written explicitly as(5)

A family of operators mapping each space of modular forms onto itself. For a fixed integer and any positive integer , the Hecke operator is defined on the set of entire modular forms of weight by(1)For a prime , the operator collapses to(2)If has the Fourier series(3)then has Fourier series(4)where(5)(Apostol 1997, p. 121).If , the Hecke operators obey the composition property(6)Any two Hecke operators and on commute with each other, and moreover(7)(Apostol 1997, pp. 126-127).Each Hecke operator has eigenforms when the dimension of is 1, so for , 6, 8, 10, and 14, the eigenforms are the Eisenstein series , , , , and , respectively. Similarly, each has eigenforms when the dimension of the set of cusp forms is 1, so for , 16, 18, 20, 22, and 26, the eigenforms are , , , , , and , respectively, where is the modular discriminant of the Weierstrass elliptic function (Apostol 1997, p. 130)...

A function is said to be modular (or "elliptic modular") if it satisfies: 1. is meromorphic in the upper half-plane , 2. for every matrix in the modular group Gamma, 3. The Laurent series of has the form(Apostol 1997, p. 34). Every rational function of Klein's absolute invariant is a modular function, and every modular function can be expressed as a rational function of (Apostol 1997, p. 40). Modular functions are special cases of modular forms, but not vice versa.An important property of modular functions is that if is modular and not identically 0, then the number of zeros of is equal to the number of poles of in the closure of the fundamental region (Apostol 1997, p. 34).

Let be a subgroup of the modular group Gamma. Then an open subset of the upper half-plane is called a fundamental region of if 1. No two distinct points of are equivalent under , 2. If , then there is a point in the closure of such that is equivalent to under . A fundamental region of the modular group Gamma is given by such that and , illustrated above, where is the complex conjugate of (Apostol 1997, p. 31). Borwein and Borwein (1987, p. 113) define the boundaries of the region slightly differently by including the boundary points with .

A function is said to be an entire modular form of weight if it satisfies 1. is analytic in the upper half-plane , 2. whenever is a member of the modular group Gamma, 3. The Fourier series of has the form(1)Care must be taken when consulting the literature because some authors use the term "dimension " or "degree " instead of "weight ," and others write instead of (Apostol 1997, pp. 114-115). More general types of modular forms (which are not "entire") can also be defined which allow poles in or at . Since Klein's absolute invariant , which is a modular function, has a pole at , it is a nonentire modular form of weight 0.The set of all entire forms of weight is denoted , which is a linear space over the complex field. The dimension of is 1 for , 6, 8, 10, and 14 (Apostol 1997, p. 119). is the value of at , and if , the function is called a cusp form. The smallest such that is called the order of the zero of at . An estimate..

The modular equation of degree gives an algebraic connection of the form(1)between the transcendental complete elliptic integrals of the first kind with moduli and . When and satisfy a modular equation, a relationship of the form(2)exists, and is called the multiplier. In general, if is an odd prime, then the modular equation is given by(3)where(4)(5) is a elliptic lambda function, and(6)(Borwein and Borwein 1987, p. 126), where is the half-period ratio. An elliptic integral identity gives(7)so the modular equation of degree 2 is(8)which can be written as(9)A few low order modular equations written in terms of and are(10)(11)(12)In terms of and ,(13)(14)(15)(16)where(17)and(18)Here, are Jacobi theta functions.A modular equation of degree for can be obtained by iterating the equation for . Modular equations for prime from 3 to 23 are given in Borwein and Borwein (1987).Quadratic modular identities include(19)Cubic identities..

A cusp form is a modular form for which the coefficient in the Fourier series(1)(Apostol 1997, p. 114). The only entire cusp form of weight is the zero function (Apostol 1997, p. 116). The set of all cusp forms in (all modular forms of weight ) is a linear subspace of which is denoted . The dimension of is 1 for , 16, 18, 20, 22, and 26 (Apostol 1997, p. 119). For a cusp form ,(2)(Apostol 1997, p. 135) or, more precisely,(3)for every (Selberg 1965; Apostol 1997, p. 136). It is conjectured that the in the exponent can be reduced to (Apostol 1997, p. 136).

The conjecture that the Artin L-function of any -dimensional complex representation of the Galois group of a finite extension of the rational numbers is an Artin L-function obtained from the general linear group .

The set of linear Möbius transformations which satisfywhere and are odd and and are even. is a subgroup of the modular group Gamma, and is also called the theta subgroup. The fundamental region of the modular lambda group is illustrated above.

The group of all Möbius transformations of the form(1)where , , , and are integers with . The group can be represented by the matrix(2)where . Every can be expressed in the form(3)where(4)(5)although the representation is not unique (Apostol 1997, pp. 28-29).

An Eisenstein series with half-period ratio and index is defined by(1)where the sum excludes , , and is an integer (Apostol 1997, p. 12).The Eisenstein series satisfies the remarkable property(2)if the matrix is in the special linear group (Serre 1973, pp. 79 and 83). Therefore, is a modular form of weight (Serre 1973, p. 83).Furthermore, each Eisenstein series is expressible as a polynomial of the elliptic invariants and of the Weierstrass elliptic function with positive rational coefficients (Apostol 1997).The Eisenstein series satisfy(3)where is the Riemann zeta function and is the divisor function (Apostol 1997, pp. 24 and 69). Writing the nome as(4)where is a complete elliptic integral of the first kind, , is the elliptic modulus, and defining(5)we have(6)(7)where(8)(9)(10)where is a Bernoulli number. For , 2, ..., the first few values of are , 240, , 480, -264, , ... (OEIS A006863 and A001067).The first..

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