Modular form
The classic term of a module form is the generic term for a broad class of functions on the upper half-plane (elliptical module forms) and their higher-dimensional generalizations (e.g. Siegel modular forms ), which is considered in the mathematical sub-areas of function theory and number theory . The modern concept of a modular form is its comprehensive reformulation in terms of representation theory (automorphic representations) and arithmetic geometry ( p -adicModular forms). Classic modular forms are special cases of the so-called automorphic forms. In addition to applications in number theory, they also have important applications in string theory and algebraic topology, for example .
Module forms are complex-valued functions with certain symmetries (prescribed transformation behavior under the module group SL or its congruence subgroups). They are closely related to lattices in the complex plane , double-period functions ( elliptic functions ) and discrete groups .
history
The beginnings of the theory go back to Carl Friedrich Gauß , who considered transformations of special module forms under the module group as part of his theory of the arithmetic-geometrical mean in the complex (a fundamental domain can be found in his notes as early as 1805). The founders of the classical (purely analytical) theory of modular forms of the 19th century are Richard Dedekind , Felix Klein , Leopold Kronecker , Karl Weierstraß , Carl Gustav Jacobi , Gotthold Eisenstein and Henri Poincaré . A well-known example of the application of modular forms in number theory was Jacobi's theorem (number of representations of a number by four squares). The modern theory of modular forms arose in the first half of the twentieth century by Erich Hecke and Carl Ludwig Siegel , who pursued applications in number theory. The theory of the Hecke operators , which work in the space of the modular forms, and the Dirichlet series (Hecke L series) defined with them play a special role here. Module forms in terms of representation theory come from Robert Langlands ( Langlands program ). p-adic modular forms appear first in Nicholas Katz and Jean-Pierre Serre . Modular forms also played a central role in the proof of Fermat's conjecture ( modularity theorem ), which in turn is a special case of the Serre conjecture , proven in 2006 , which combines modular forms with Galois representations of the absolute Galois group of number fields. Both in the proof of the solution of the Gaussian class number problem by Kurt Heegner and the last part of the Weil conjectures (Riemann hypothesis) and the related Ramanujan conjecture by Pierre Deligne , modular forms played an important role, as did the proof by Maryna Viazovska (2016) that the E8-lattice in eight dimensions and the Leech-lattice in 24 dimensions provide the closest packing of spheres (the theta functions of these two lattices are modular forms, see below). Module forms often encode arithmetic information of the algebraic number fields, but are much more easily accessible by computer, sometimes even with computer algebra programs, and the number of linearly independent module forms of certain types is limited.
Elliptical modular shapes for SL _{2} (ℤ)
definition
Be it
the upper half-plane , ie the set of all complex numbers with a positive imaginary part.
For an integer , a holomorphic or meromorphic function on the upper half-plane is called a holomorphic or meromorphic elliptical modular form from the weight to the group (full module group) if it
- for everyone and with
- met and
- Is “holomorphic or meromorphic in infinity”. That means the function
- With
- (the Fourier or q expansion of f) is holomorphic or meromorphic at the point .
Is meromorphic and is called a module function. Module functions have a particularly simple behavior under the module group:
Holomorphic modular functions are of no interest since the only holomorphic modular functions are the constant functions ( Liouville's theorem ). The holomorphic modular forms are also called whole modular forms. If such a whole modular shape also disappears at infinity (in the tip, English cusp ), it is called a tip shape . More precisely, a tip shape disappears from the weight k for how . The j-function, on the other hand, is a holomorphic module function in the upper half-plane except for a simple pole in the tip, thus an example of meromorphism. From the definition it follows that a modular form for odd identical vanishes.
The special furniture transformations used in the definition of the module shape form the module group
The module group is also referred to as. The module forms are characterized by their simple transformation behavior compared to the module group. The module group maps the upper half-plane to itself and is generated by the generators (these are geometrically a reflection on a circle (inversion) and a translation):
The behavior of the modular form of the weight among these generators is:
and from the latter equation it follows that the modular shape is periodic. Therefore, the Fourier development of well defined and holomorphic or meromorphic. With the Fourier coefficients one has the Fourier series (also called q expansion):
Where m is called the order of the pole of in the tip (imaginary part of towards infinity). The module form is meromorphic at the top in negative Fourier terms. In the case of a tip shape disappears at ( ), i.e. the non- zero Fourier coefficients begin with a positive one , which is then called the order of the zero of in the tip.
In the complex plane, a module shape is defined by its values in the fundamental domain, which is colored gray in the adjacent figure. It is a triangle with a point at infinity. Each of the fundamental triangles bounded by straight lines or circles is created by applying operations of the module group to the fundamental domain. The application of the module group can be continued at will and results in an ever finer division, which was broken off at a certain point in the illustration.
The illustration shows the famous modular figure, which was artistically represented, for example, by MC Escher in several graphics.
Examples and connection with grids
The simplest examples for whole modular forms of weight are the so-called Eisenstein series , for a module function the j-function or absolute invariant and for a tip shape the discriminant .
The module group has the important property that it depicts lattices on the complex plane. These grids are of two complex numbers with spanned:
- .
They can be represented as parallelograms in the complex plane of numbers. Another basis of the lattice, given by two complex numbers , spans the same lattice if the two bases are transformed into one another by an element of the module group (this follows from the condition that the determinant is equal to 1 and the integer):
If one sets the above-mentioned transformation formula over a Möbius transformation.
Rows of iron stones are naturally defined on these grids:
Or with (i.e. one from the upper half level):
- .
Since the grid is invariant under the module group, this also applies to the Eisenstein rows. They are whole modular forms of weight , where is straight (otherwise the module form would disappear identically, since all grid points are summed up, including what is located below). For the series to converge, it must also be greater than 2.
Among the generators of the module group, the Eisenstein series transforms:
The connection with lattices also results in a connection of modular forms to elliptical functions , which are defined as double-periodic, meromorphic functions on such a lattice (if the sides of the lattice are identified with one another, a torus with topological gender results , the Riemann area of the elliptical functions). The easiest way to do this is to look at the Weierstrasse ℘ function . Meromorphic modular forms with weight 0 are defined on the isomorphism classes of the grids on which the elliptic functions are based. The j-invariant of an elliptic function characterizes these isomorphism classes, which are thus clearly parameterized by this function of the upper half-plane. It is a modular form of weight 0 and can be formed as a rational function from Eisenstein rows of weight 4 and 6, with the modular discriminant in the denominator, a module function of weight 12 (it is in turn related to Dedekind's η function ). The j-function has many interesting properties that make it important for number theory (construction of algebraic number fields) and group theory (the Fourier coefficients of its q expansion are related to the representation of the monster group, moonshine ).
The relationship between modular shapes and elliptic curves is also continued with elliptic curves defined over number fields, where the above-mentioned modularity theorem applies that all elliptic curves defined over number fields can be parameterized by module shapes (from this theorem follows the Fermat conjecture according to Andrew Wiles and others) .
Another example of modular forms is provided by theta functions that are defined on grids.
For example, the theta function for a straight, unimodular grid in :
a modular form of weight . To prove the behavior under inversion, Poisson's sum formula is used. Unimodular means that the grid discriminant equals 1 and even that the squares of the lengths of the vectors of the grid are all even. Examples of such lattices (whose dimension n must be divisible by 8) are the Leech lattice (n = 24, as one of 24 Niemeier lattices) and the lattice of the root system of the special Lie group (n = 8). In the case of n = 8 it is a modular form of weight 4, but there is only one of them, the Eisenstein series of weight 4.
Vector spaces of the modular forms
For odd is always , the following statements therefore apply to even .
Sums and products of modular forms are again modular forms. The modular forms of weight k form a vector space, as do all of the modular forms and the tip forms.
If one denotes these vector spaces with and , then the following applies:
The following applies to the dimension of these vector spaces (let k be a positive even integer):
Since the multiplication with the tip shape ( discriminant ) of weight 12 gives an isomorphism of to , the following applies
The modular spaces for are one-dimensional and are created by and for two-dimensional, created by , with the rows of iron stones . In general one can show that all elements of are generated by polynomials in :
with constants . However, it is often more useful to use bases of eigenmodes of the Hecke operators ( Atkin-Lehner theory ).
Hans Petersson introduced the Petersson scalar product in the space of the tip shapes and thus made them a Hilbert space . With the Riemann-Roch theorem one can make statements about the dimension of the vector spaces of the tip shapes. Eisenstein rows are orthogonal to the tip shapes with respect to the Petersson scalar product.
One reason for the usefulness of modular forms in the most varied of applications is that, although they often have different descriptions in the most varied of applications, one immediately finds connections among the modular forms, since the vector spaces are of relatively small dimensions.
Congruence subsets
Instead of for , module forms are also considered for discrete subgroups of this group, especially for (N is a positive integer) the so-called congruence subgroups of the module group:
The number is the level of the assigned module forms. is also called the main congruence group of the level . Each subgroup of which contains the main congruence group for a level N as a subgroup is called a congruence subgroup.
Sometimes the congruence subgroup is also considered:
which occupies a middle position between (mod N equivalent to the upper triangular matrix) and (mod N equivalent to the identity matrix). It applies and also applies .
The index of the congruence subgroups as subgroups of is finite and can be specified explicitly. So is:
The module forms for the congruence subgroup and have Fourier expansions in ; that of for not necessarily, since the matrix (a = d = 1, b = 1, c = 0) does not belong in the transformation matrix (they have a Fourier expansion in ). However, it can always be assigned to a modular form for such for (which has a Fourier expansion in ). There is also no such simple criterion for peak shapes for congruence subgroups (the constant Fourier term does not necessarily have to disappear as with the full module group). In addition to module forms with transformation behavior as discussed in the full module group, those with extended transformation behavior (multiplication with a Dirichlet character) are also considered.
With these congruence subgroups one can form the quotient spaces as , which are compacted by adding a finite number of points (cusp, tips of the congruence subgroup) in the extended upper half-plane , the corresponding compactified quotient space is then called . Accordingly, we speak in the congruence subgroup of or and of . After compacting, compact Riemann surfaces of different topological sex are obtained . The different ones are also called module curves.
For example, the Riemann sphere (gender 0) has 12 points that are arranged like the icosahedron . is the small quartic with gender 3 and 24 tips. is the classic module curve and is often referred to simply as the module curve.
Module curves parameterize equivalence classes of elliptical curves depending on the type of congruence subgroup and can be defined purely algebraically and thus also viewed over other fields . They are important in arithmetic geometry.
Generalizations, automorphic forms
Module functions can be generalized by expanding the type of transformation behavior and for groups other than the module group.
Initially, only module forms with integer weight k were considered above, but there are also those with rational values that also play a role in number theory, so Jerrold Tunnell used module forms for weight when solving the problem of congruent numbers .
For example, one can consider functions that transform themselves through multiplication with an automorphic factor:
with the automorphic factor , where . These are examples of automorphic functions. One example is the Dedekind etafunction. In algebraic number theory, modular functions for the congruence subgroup are often considered with an automorphic factor that is formed with the Dirichlet character (modular forms of weight , secondary type and level ):
They are defined for in the upper half-plane and holomorphic in the apex.
Automorphic forms are defined for topological groups ( Lie group ) and their discrete subgroups . In the case of the module forms, this corresponds to the module group itself as a discrete subgroup of the Lie group or the congruence subgroups as a discrete subgroup of the module group. The transformation law is generally defined here with automorphism factors. Automorphic forms are eigenfunctions of certain Casimir operators of (in the case of the modular functions this corresponds to the fact that these are analytic functions in two dimensions that satisfy the Laplace equation, which corresponds to the Casimir operator for ) and, like the modular forms, satisfy certain growth conditions. They were already considered in the 19th century for Fuchsian groups (discrete subgroups of ) by Henri Poincaré and in number theory at the beginning of the 20th century by David Hilbert (Hilbert's modular forms for totally real number fields for the general linear group over the ring of whole numbers of the number field , defined as the modular form on the -fold product of the upper half-plane, with the degree of over the rational numbers).
Another example of automorphic forms in several complex variables are Siegel modular forms, which are defined in Siegel's half-space and are automorphic forms belonging to the symplectic group . They play a similar role for the parameterization of Abelian varieties as module forms for the parameterization of elliptical functions (as respective module spaces) and were originally considered by Carl Ludwig Siegel in the theory of quadratic forms.
Also Jacobi forms are automorphic functions in several variables to them as part of the Weierstrass ℘ function and the Jacobi theta function .
Automorphic forms play an essential role in the Langlands program, where algebraic groups are considered in a number-theoretic context (as algebraic groups over the noble ring of an algebraic number field) and their representation theory plays a special role.
Further examples of extensions to the concept of modular forms are the mock theta functions by S. Ramanujan or mock modular forms. They are not modular forms themselves, but can be completed to a modular form by adding a non-holomorphic component (called the shadow of the mock module form) and found spectacular application in the theory of partitions by Ken Ono , Jan Hendrik Bruinier and Kathrin Bringmann . According to Sander Zwegers, they are related to Maaß forms or Maaß wave forms by Hans Maaß , non-analytic automorphic forms that are eigenfunctions of the invariant (hyperbolic) Laplace operator for weight . Mock modular forms are the holomorphic part of a weak dimensional form, whereby the weak refers to the required growth conditions.
literature
- Eberhard Freitag , Rolf Busam: Function theory 1st 4th edition, Springer, Berlin (2006), ISBN 3-540-31764-3 .
- Max Koecher , Aloys Krieg : Elliptical functions and modular forms. 2nd edition, Springer, Berlin (2007) ISBN 978-3-540-49324-2 .
- Jean-Pierre Serre : A course in arithmetic. Springer, 1973.
- Don Zagier : Introduction to Modular Forms. In: M. Waldschmidt, P. Moussa, J.-M. Luck, C. Itzykson: From Number Theory to Physics. Springer, 1995, Chapter 4, pp. 238-291, Online, PDF.
- Serge Lang : Introduction to Modular Forms. Springer, Basic Teachings of Mathematical Sciences, 1976.
- Neal Koblitz : Introduction to Elliptic Curves and Modular Forms. Springer, 1984.
- LJP Kilford: Modular forms, a classical and computational introduction. Imperial College Press, London 2008.
- T. Miyake: Modular forms. Springer, 1989.
Web links
- Gabor Wiese, various lectures on module forms
- Igor Dolgachev, Lectures on Modular Forms, PDF
- The L-functions and modular forms data base
References and comments
- ↑ Topological modular form by Michael J. Hopkins et al., Topological modular form, ncatLab
- ↑ Houzel: Elliptic Functions and Abelian Integrals. In: Dieudonné: History of Mathematics. Vieweg, 1985, p. 486 f.
- ↑ Some authors also refer to the projective special linear group PSL (2, Z) as a module group in which matrices A and −A are identified. It is the quotient of SL (2, Z) with its center Z = (1, −1).
- ↑ For example Gabriele Nebe: Grids and modular forms. Annual report DMV, Volume 104, 2002, pp. 123-142, Arxiv.
- ↑ Kilford: Modular Forms. 2008, p. 48.
- ↑ Kilford: Modular forms. 2008, p. 70. Sometimes this is also used to define the Eisenstein series.
- ^ Zagier: Introduction to modular forms. P. 240.
- ↑ The extended upper half level consists of , and . The rational numbers appear because the orbit passes through the infinite through the action of the congruence subgroups in the infinite.
- ↑ Generated as an extension of the rational numbers by the adjunction of a root of an integer polynomial with real roots.
- ↑ Amanda Folsom: What is a mock theta modular form? Notices AMS, December 2010, PDF.