Siegel Modular Form
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Siegel modular forms are a major type of
automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
. These generalize conventional ''elliptic''
modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
s which are closely related to
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
for abelian varieties (with some extra
level structure In the mathematical subfield of graph theory a level structure of an undirected graph is a partition of the vertices into subsets that have the same distance from a given root vertex.. Definition and construction Given a connected graph ''G'' ...
) should be and are constructed as quotients of the
Siegel upper half-space In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It ...
rather than the
upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...
by
discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...
s. Siegel modular forms are
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s on the set of
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
''n'' × ''n'' matrices with
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
imaginary part; the forms must satisfy an automorphy condition. Siegel modular forms can be thought of as multivariable modular forms, i.e. as
special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
s of
several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
. Siegel modular forms were first investigated by for the purpose of studying
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
s analytically. These primarily arise in various branches of
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, such as
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and
elliptic cohomology In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms. History and motivation Historically, elliptic cohomology arose from the study of elliptic genera. ...
. Siegel modular forms have also been used in some areas of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, such as
conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes ...
and
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in
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
.


Definition


Preliminaries

Let g, N \in \mathbb and define :\mathcal_g=\left\, the
Siegel upper half-space In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It ...
. Define the
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
of level N, denoted by \Gamma_g(N), as :\Gamma_g(N)=\left\, where I_g is the g \times g
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
. Finally, let :\rho:\textrm_g(\mathbb) \rightarrow \textrm(V) be a
rational representation In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map In mathematics, in particu ...
, where V is a finite-dimensional complex
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
.


Siegel modular form

Given :\gamma=\begin A & B \\ C & D \end and :\gamma \in \Gamma_g(N), define the notation :(f\big, \gamma)(\tau)=(\rho(C\tau+D))^f(\gamma\tau). Then a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
:f:\mathcal_g \rightarrow V is a ''Siegel modular form'' of degree g (sometimes called the genus), weight \rho, and level N if :(f\big, \gamma)=f for all \gamma \in \Gamma_g(N). In the case that g=1, we further require that f be holomorphic 'at infinity'. This assumption is not necessary for g>1 due to the Koecher principle, explained below. Denote the space of weight \rho, degree g, and level N Siegel modular forms by :M_(\Gamma_g(N)).


Examples

Some methods for constructing Siegel modular forms include: *
Eisenstein series Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
*Theta functions of lattices (possibly with a pluri-harmonic polynomial) *
Saito–Kurokawa lift In mathematics, the Saito–Kurokawa lift (or lifting) takes elliptic modular forms to Siegel modular forms of degree 2. The existence of this lifting was conjectured in 1977 independently by Hiroshi Saito and . Its existence was almost proved by , ...
for degree 2 *
Ikeda lift In mathematics, the Ikeda lift is a lifting of modular forms to Siegel modular forms. The existence of the lifting was conjectured by W. Duke and Ö. Imamoḡlu and also by T. Ibukiyama, and the lifting was constructed by . It generalized the S ...
*
Miyawaki lift The Miyawaki lift or Ikeda–Miyawaki lift or Miyawaki–Ikeda lift, is a mathematical lift that takes two Siegel modular form In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' m ...
*Products of Siegel modular forms.


Level 1, small degree

For degree 1, the level 1 Siegel modular forms are the same as level 1 modular forms. The ring of such forms is a polynomial ring C 'E''4,''E''6in the (degree 1) Eisenstein series ''E''4 and ''E''6. For degree 2, showed that the ring of level 1 Siegel modular forms is generated by the (degree 2) Eisenstein series ''E''4 and ''E''6 and 3 more forms of weights 10, 12, and 35. The ideal of relations between them is generated by the square of the weight 35 form minus a certain polynomial in the others. For degree 3, described the ring of level 1 Siegel modular forms, giving a set of 34 generators. For degree 4, the level 1 Siegel modular forms of small weights have been found. There are no cusp forms of weights 2, 4, or 6. The space of cusp forms of weight 8 is 1-dimensional, spanned by the
Schottky form In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form ''J'' of degree 4 and weight 8, introduced by as a degree 16 polynomial in the Thetanullwerte of genus 4. He showed that it vanished at all Jacobian points (the po ...
. The space of cusp forms of weight 10 has dimension 1, the space of cusp forms of weight 12 has dimension 2, the space of cusp forms of weight 14 has dimension 3, and the space of cusp forms of weight 16 has dimension 7 . For degree 5, the space of cusp forms has dimension 0 for weight 10, dimension 2 for weight 12. The space of forms of weight 12 has dimension 5. For degree 6, there are no cusp forms of weights 0, 2, 4, 6, 8. The space of Siegel modular forms of weight 2 has dimension 0, and those of weights 4 or 6 both have dimension 1.


Level 1, small weight

For small weights and level 1, give the following results (for any positive degree): *Weight 0: The space of forms is 1-dimensional, spanned by 1. *Weight 1: The only Siegel modular form is 0. *Weight 2: The only Siegel modular form is 0. *Weight 3: The only Siegel modular form is 0. *Weight 4: For any degree, the space of forms of weight 4 is 1-dimensional, spanned by the theta function of the E8 lattice (of appropriate degree). The only cusp form is 0. *Weight 5: The only Siegel modular form is 0. *Weight 6: The space of forms of weight 6 has dimension 1 if the degree is at most 8, and dimension 0 if the degree is at least 9. The only cusp form is 0. *Weight 7: The space of cusp forms vanishes if the degree is 4 or 7. *Weight 8:In genus 4, the space of cusp forms is 1-dimensional, spanned by the
Schottky form In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form ''J'' of degree 4 and weight 8, introduced by as a degree 16 polynomial in the Thetanullwerte of genus 4. He showed that it vanished at all Jacobian points (the po ...
and the space of forms is 2-dimensional. There are no cusp forms if the genus is 8. *There are no cusp forms if the genus is greater than twice the weight.


Table of dimensions of spaces of level 1 Siegel modular forms

The following table combines the results above with information from and and .


Koecher principle

The theorem known as the ''Koecher principle'' states that if f is a Siegel modular form of weight \rho, level 1, and degree g>1, then f is bounded on subsets of \mathcal_g of the form :\left\, where \epsilon>0. Corollary to this theorem is the fact that Siegel modular forms of degree g>1 have
Fourier expansion A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
s and are thus holomorphic at infinity.


Applications to physics

In the D1D5P system of supersymmetric black holes in string theory, the function that naturally captures the microstates of black hole entropy is a Siegel modular form. In general, Siegel modular forms have been described as having the potential to describe black holes or other gravitational systems. Siegel modular forms also have uses as generating functions for families of CFT2 with increasing central charge in
conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes ...
, particularly the hypothetical
AdS/CFT correspondence In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter s ...
.


References

* * * * * * * * *{{citation, mr=0853217 , last=Tsuyumine, first= Shigeaki , title=On Siegel modular forms of degree three , journal=Amer. J. Math. , volume=108 , year=1986, issue= 4, pages= 755–862, jstor=2374517, doi=10.2307/2374517 Modular forms Automorphic forms