In mathematics, a modular form is a holomorphic function on the complex upper half-plane,
Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory.
Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group
[1] The term "modular form", as a systematic description, is usually attributed to Erich Hecke.
of finite index (called an arithmetic group), a modular form of level
are sections of a line bundle on the moduli stack of elliptic curves.
satisfying the following three conditions: Remarks: 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: The key idea in proving the equivalence of the two definitions is that such a function F is determined, because of the second condition, by its values on lattices of the form Z + Zτ, where τ ∈ H. I. Eisenstein series The simplest examples from this point of view are the Eisenstein series.
The so-called theta function converges when Im(z) > 0, and as a consequence of the Poisson summation formula can be shown to be a modular form of weight n/2.
Because there is only one modular form of weight 8 up to scalar multiplication, even though the lattices L8 × L8 and L16 are not similar.
John Milnor observed that the 16-dimensional tori obtained by dividing R16 by these two lattices are consequently examples of compact Riemannian manifolds which are isospectral but not isometric (see Hearing the shape of a drum.)
The modular discriminant The Dedekind eta function is defined as where q is the square of the nome.
A celebrated conjecture of Ramanujan asserted that when Δ(z) is expanded as a power series in q, the coefficient of qp for any prime p has absolute value ≤ 2p11/2.
This was confirmed by the work of Eichler, Shimura, Kuga, Ihara, and Pierre Deligne as a result of Deligne's proof of the Weil conjectures, which were shown to imply Ramanujan's conjecture.
When the weight k is zero, it can be shown using Liouville's theorem that the only modular forms are constant functions.
However, relaxing the requirement that f be holomorphic leads to the notion of modular functions.
A function f : H → C is called modular if it satisfies the following properties: It is often written in terms of
[note 2] Sometimes a weaker definition of modular functions is used – under the alternative definition, it is sufficient that f be meromorphic in the open upper half-plane and that f be invariant with respect to a sub-group of the modular group of finite index.
Typically it is not compact, but can be compactified by adding a finite number of points called cusps.
What is more, it can be endowed with the structure of a Riemann surface, which allows one to speak of holo- and meromorphic functions.
Important examples are, for any positive integer N, either one of the congruence subgroups For G = Γ0(N) or Γ(N), the spaces G\H and G\H∗ are denoted Y0(N) and X0(N) and Y(N), X(N), respectively.
The theory of Riemann surfaces can be applied to G\H∗ to obtain further information about modular forms and functions.
Alternatively, we can stick with polynomials and loosen the dependence on c, letting F(cv) = ckF(v).
The solutions are then the homogeneous polynomials of degree k. On the one hand, these form a finite dimensional vector space for each k, and on the other, if we let k vary, we can find the numerators and denominators for constructing all the rational functions which are really functions on the underlying projective space P(V).
The algebro-geometric answer is that they are sections of a sheaf (one could also say a line bundle in this case).
Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.
Rings of modular forms of congruence subgroups of SL(2, Z) are finitely generated due to a result of Pierre Deligne and Michael Rapoport.
More generally, there are formulas for bounds on the weights of generators of the ring of modular forms and its relations for arbitrary Fuchsian groups.
The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's mock theta functions.
Hilbert modular forms are functions in n variables, each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in a totally real number field.
In 2013 elliptic curves were proven to be modular over real quadratic fields.