[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/siegel-modular-form-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki8\/siegel-modular-form-wikipedia\/","headline":"Siegel modular form – Wikipedia","name":"Siegel modular form – Wikipedia","description":"before-content-x4 Major type of automorphic form in mathematics after-content-x4 In mathematics, Siegel modular forms are a major type of automorphic","datePublished":"2016-06-07","dateModified":"2016-06-07","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki8\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bbfea1bd678909a322e644d6f9726958de6a2a1a","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bbfea1bd678909a322e644d6f9726958de6a2a1a","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki8\/siegel-modular-form-wikipedia\/","about":["Wiki"],"wordCount":7861,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Major type of automorphic form in mathematics       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.Siegel modular forms are holomorphic functions on the set of symmetric n \u00d7 n matrices with positive definite imaginary part; the forms must satisfy an automorphy condition. Siegel modular forms can be thought of as multivariable modular forms, i.e. as special functions of several complex variables.Siegel modular forms  were first investigated by Carl Ludwig Siegel\u00a0(1939) for the purpose of studying quadratic forms analytically. These primarily arise in various branches of number theory, such as arithmetic geometry and elliptic cohomology. Siegel modular forms have also been used in some areas of physics, such as conformal field theory and black hole thermodynamics in string theory.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsDefinition[edit]Preliminaries[edit]Siegel modular form[edit]Examples[edit]Level 1, small degree[edit]Level 1, small weight[edit]Table of dimensions of spaces of level 1 Siegel modular forms[edit]Koecher principle[edit]Applications to physics[edit]References[edit]Definition[edit]Preliminaries[edit]Let g,N\u2208N{displaystyle g,Nin mathbb {N} } and define       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Hg={\u03c4\u2208Mg\u00d7g(C)\u00a0|\u00a0\u03c4T=\u03c4,Im(\u03c4)\u00a0positive definite},{displaystyle {mathcal {H}}_{g}=left{tau in M_{gtimes g}(mathbb {C} ) {big |} tau ^{mathrm {T} }=tau ,{textrm {Im}}(tau ){text{  positive definite}}right},}the Siegel upper half-space. Define the symplectic group of level N{displaystyle N}, denoted by \u0393g(N),{displaystyle Gamma _{g}(N),} as\u0393g(N)={\u03b3\u2208GL2g(Z)\u00a0|\u00a0\u03b3T(0Ig\u2212Ig0)\u03b3=(0Ig\u2212Ig0),\u00a0\u03b3\u2261I2gmodN},{displaystyle Gamma _{g}(N)=left{gamma in GL_{2g}(mathbb {Z} ) {big |} gamma ^{mathrm {T} }{begin{pmatrix}0&I_{g}\\-I_{g}&0end{pmatrix}}gamma ={begin{pmatrix}0&I_{g}\\-I_{g}&0end{pmatrix}}, gamma equiv I_{2g}mod Nright},}where Ig{displaystyle I_{g}} is the g\u00d7g{displaystyle gtimes g} identity matrix. Finally, let\u03c1:GLg(C)\u2192GL(V){displaystyle rho :{textrm {GL}}_{g}(mathbb {C} )rightarrow {textrm {GL}}(V)}be a rational representation, where V{displaystyle V} is a finite-dimensional complex vector space.Siegel modular form[edit]Given\u03b3=(ABCD){displaystyle gamma ={begin{pmatrix}A&B\\C&Dend{pmatrix}}}and\u03b3\u2208\u0393g(N),{displaystyle gamma in Gamma _{g}(N),}define the notation(f|\u03b3)(\u03c4)=(\u03c1(C\u03c4+D))\u22121f(\u03b3\u03c4).{displaystyle (f{big |}gamma )(tau )=(rho (Ctau +D))^{-1}f(gamma tau ).}Then a holomorphic functionf:Hg\u2192V{displaystyle f:{mathcal {H}}_{g}rightarrow V}is a Siegel modular form of degree g{displaystyle g} (sometimes called the genus), weight \u03c1{displaystyle rho }, and level N{displaystyle N} if(f|\u03b3)=f{displaystyle (f{big |}gamma )=f}for all \u03b3\u2208\u0393g(N){displaystyle gamma in Gamma _{g}(N)}.In the case that g=1{displaystyle g=1}, we further require that f{displaystyle f} be holomorphic ‘at infinity’. This assumption is not necessary for 1″\/> due to the Koecher principle, explained below. Denote the space of weight \u03c1{displaystyle rho }, degree g{displaystyle g}, and level N{displaystyle N} Siegel modular forms byM\u03c1(\u0393g(N)).{displaystyle M_{rho }(Gamma _{g}(N)).}Examples[edit]Some methods for constructing Siegel modular forms include:Level 1, small degree[edit]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[E4,E6] in the (degree 1) Eisenstein series E4 and E6.For degree 2, (Igusa\u00a01962, 1967) showed that the ring of level 1 Siegel modular forms is generated by the (degree 2) Eisenstein series E4 and E6 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, Tsuyumine (1986) 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. 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 (Poor & Yuen 2007) harv error: no target: CITEREFPoorYuen2007 (help).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[edit]For small weights and level 1, Duke & Imamo\u1e21lu (1998) 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 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[edit]The following table combines the results above with information from Poor & Yuen (2006) harvtxt error: no target: CITEREFPoorYuen2006 (help) and Chenevier & Lannes (2014) and Ta\u00efbi (2014).Dimensions of spaces of level 1 Siegel cusp forms:  Siegel modular formsWeightdegree 0degree 1degree 2degree 3degree 4degree 5degree 6degree 7degree 8degree 9degree 10degree 11degree 1201: 10: 10: 10: 10: 10: 10: 10: 10: 10: 10: 10: 10: 121: 10: 00: 00: 00: 00: 00: 00: 00: 00: 00: 00: 00: 041: 10: 10: 10: 10: 10: 10: 10: 10: 10: 10: 10: 10: 161: 10: 10: 10: 10: 10: 10: 10: 10: 10: 00: 00: 00: 081: 10: 10\u00a0: 10 :11: 20: 20: 20: 20: 2101: 10: 11: 20\u00a0: 21: 30: 31: 40: 41:0:0:121: 11: 21: 31:  42: 62: 83: 113: 144: 182:202: 221: 231: 24141: 10: 11: 21: 33:63: 99: 189: 27161: 11: 22: 43: 77: 1413:2733:6083:143181: 11: 22: 44:812:2028: 48117: 163201: 11: 23: 56: 1122: 3376: 109486:595221: 11: 24\u00a0: 69:1538:53186:239241: 12: 35: 814: 22261: 11: 25: 717: 24281: 12: 37\u00a0: 1027: 37301: 12: 38: 1134: 45Koecher principle[edit]The theorem known as the Koecher principle states that if f{displaystyle f} is a Siegel modular form of weight \u03c1{displaystyle rho },  level 1, and degree 1″\/>, then f{displaystyle f} is bounded on subsets of Hg{displaystyle {mathcal {H}}_{g}} of the form"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki8\/siegel-modular-form-wikipedia\/#breadcrumbitem","name":"Siegel modular form – Wikipedia"}}]}]