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href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> <h4 id="complex_geometry">Complex geometry</h4> <div class="hide"><div> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>, <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, <a class="existingWikiWord" href="/nlab/show/complex+line">complex line</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> </ul> <h3 id="structures">Structures</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h3 id="examples">Examples</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim = 1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, <a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim = 2</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></p> </li> </ul> </div></div> <h4 id="elliptic_cohomology">Elliptic cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a>, <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented</a><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <a class="existingWikiWord" href="/nlab/show/chromatic+level">chromatic level</a> 2</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supersingular+elliptic+curve">supersingular elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+elliptic+curve">derived elliptic curve</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+form">modular form</a>, <a class="existingWikiWord" href="/nlab/show/Jacobi+form">Jacobi form</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eisenstein+series">Eisenstein series</a>, <a class="existingWikiWord" href="/nlab/show/j-invariant">j-invariant</a>, <a class="existingWikiWord" href="/nlab/show/Weierstrass+sigma-function">Weierstrass sigma-function</a>, <a class="existingWikiWord" href="/nlab/show/Dedekind+eta+function">Dedekind eta function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+modular+form">topological modular form</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a></li> </ul> </li> </ul></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#crossratios'>Cross-ratios</a></li> <li><a href='#action_on_hyperbolic_space'>Action on hyperbolic space</a></li> </ul> <li><a href='#ModularGroup'>Modular group</a></li> <ul> <li><a href='#abstract_structure_of_modular_group'>Abstract structure of modular group</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/field">field</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{P}^1(k)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/projective+line">projective line</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. A <em>Möbius transformation</em> (also called a <em>homography</em>, a <em>linear fractional transformation</em>, or a <em>fractional linear transformation</em>) is a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f: \mathbb{P}^1(k) \to \mathbb{P}^1(k)</annotation></semantics></math> defined by the rule</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><mrow><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex"> f(x) = \frac{a x + b}{c x + d} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>∈</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">a, b, c, d \in k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>d</mi><mo>−</mo><mi>b</mi><mi>c</mi><mo>∈</mo><msup><mi>k</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">a d - b c \in k^\times</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>). Möbius transformations form a <a class="existingWikiWord" href="/nlab/show/group">group</a> under <a class="existingWikiWord" href="/nlab/show/composition">composition</a>, <a class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/projective+linear+group">projective linear group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>PGL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>GL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">{</mo><mi>λ</mi><mi>I</mi><mo>:</mo><mi>λ</mi><mo>∈</mo><msup><mi>k</mi> <mo>×</mo></msup><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PGL_2(k) \coloneqq GL_2(k)/\{\lambda I: \lambda \in k^\times\} \,. </annotation></semantics></math></div> <p>If, as in the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">k = \mathbb{C}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>), each element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">k^\times</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/square+root">square root</a>, then this group is identified with the <a class="existingWikiWord" href="/nlab/show/projective+special+linear+group">projective special linear group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo>±</mo><mi>I</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PSL_2(k) \coloneqq SL_2(k)/\pm I \,. </annotation></semantics></math></div> <p>Alternatively, a fractional linear transformation can be considered as synonymous with an <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> of the <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/rational+functions">rational functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k(x)</annotation></semantics></math> as a field over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/transcendence+degree">transcendence degree</a> 1.</p> <p>In <a class="existingWikiWord" href="/nlab/show/complex+analysis">complex analysis</a> (which is the usual context when one speaks of Möbius transformations; otherwise one usually calls them by some combination of “linear” and “fractional”), Möbius transformations are precisely the <a class="existingWikiWord" href="/nlab/show/biholomorphisms">biholomorphisms</a> of the <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a>, hence exactly its <a class="existingWikiWord" href="/nlab/show/bijection">bijective</a> <a class="existingWikiWord" href="/nlab/show/conformal+transformations">conformal transformations</a>.</p> <p>Often, and particularly when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is the the <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>, one considers a <em><a class="existingWikiWord" href="/nlab/show/modular+group">modular group</a></em> where the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">a, b, c, d</annotation></semantics></math> are assumed to lie in an <a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>d</mi><mo>−</mo><mi>b</mi><mi>c</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a d - b c = 1</annotation></semantics></math>. (The <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> of the <a class="existingWikiWord" href="/nlab/show/upper+half-plane">upper half-plane</a> by the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PGL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PGL_2(\mathbb{Z})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a> over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>.)</p> <h2 id="properties">Properties</h2> <h3 id="crossratios">Cross-ratios</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PGL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>×</mo><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PGL_2(k) \times \mathbb{P}^1(k) \to \mathbb{P}^1(k)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/transitive+action">3-transitive</a>, i.e., any triplet of distinct points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, b, c)</annotation></semantics></math> may be mapped to any other triplet of distinct points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>b</mi><mo>′</mo><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a', b', c')</annotation></semantics></math> by applying a group element.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>It suffices to consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>′</mo><mo>=</mo><mn>0</mn><mo>,</mo><mi>b</mi><mo>′</mo><mo>=</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>′</mo><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">a' = 0, b' = 1, c' = \infty</annotation></semantics></math> where one applies the transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">x \mapsto \frac{(x-a)(b-c)}{(x-c)(b-a)}</annotation></semantics></math>.</p> </div> <p>This motivates the following definition: given a 4-tuple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, b, c, d)</annotation></semantics></math> of distinct points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{P}^1(k)</annotation></semantics></math>, its <em>cross-ratio</em> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex">\gamma(a, b, c, d) = \frac{(d-a)(b-c)}{(d-c)(b-a)}.</annotation></semantics></math></div> <p>It is not hard to see that the group action preserves the cross-ratio, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo stretchy="false">(</mo><mi>g</mi><mo>⋅</mo><mi>a</mi><mo>,</mo><mi>g</mi><mo>⋅</mo><mi>b</mi><mo>,</mo><mi>g</mi><mo>⋅</mo><mi>c</mi><mo>,</mo><mi>g</mi><mo>⋅</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma(g \cdot a, g \cdot b, g \cdot c, g \cdot d) = \gamma(a, b, c, d)</annotation></semantics></math>. Moreover, the group action is transitive on each cross-ratio-equivalence class of 4-tuples.</p> <p>In the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">k = \mathbb{C}</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℙ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{P}^1</annotation></semantics></math> is interpreted as the Riemann sphere, it turns out that the cross-ratio of a 4-tuple is a <a class="existingWikiWord" href="/nlab/show/real+number">real number</a> if and only if the four points lie on a <a class="existingWikiWord" href="/nlab/show/circle">circle</a> (or a line which is a circle passing through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>). Hence Möbius = conformal transformations take circles to circles.</p> <h3 id="action_on_hyperbolic_space">Action on hyperbolic space</h3> <p>As explained at <a class="existingWikiWord" href="/nlab/show/Poincare+group">Poincare group</a>, the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSL_2(\mathbb{C})</annotation></semantics></math> can be identified with those <a class="existingWikiWord" href="/nlab/show/linear+transformations">linear transformations</a> of Minkowski space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> that preserve the Minkowski <a class="existingWikiWord" href="/nlab/show/quadratic+form">form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>, are <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>-preserving, and take the forward <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>v</mi><mo>=</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{v = (\vec{x}, t): Q(v) = 0, t \gt 0\}</annotation></semantics></math> to itself. It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSL_2(\mathbb{C})</annotation></semantics></math> acts on the hyperboloid sheet</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup><mo>=</mo><mo stretchy="false">{</mo><mi>v</mi><mo>=</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">H^3 = \{v = (\vec{x}, t): Q(v) = 1, t \gt 0\}</annotation></semantics></math></div> <p>which is naturally identified with hyperbolic 3-space. There is a Poincaré disk model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">H^3</annotation></semantics></math>; consider the disk <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">D^3</annotation></semantics></math> that is the intersection of the <a class="existingWikiWord" href="/nlab/show/future+cone">future cone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>v</mi><mo>=</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{v = (\vec{x}, t): Q(v) \geq 0, t \gt 0\}</annotation></semantics></math> with the hyperplane <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">t = 1</annotation></semantics></math>. Its interior is an open 3-disk <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>int</mi><mo stretchy="false">(</mo><msup><mi>D</mi> <mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">int(D^3)</annotation></semantics></math> which can be placed in perspective with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">H^3</annotation></semantics></math> by considering lines through the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math>: each line that passes through a unique point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">H^3</annotation></semantics></math> passes through a unique point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>int</mi><mo stretchy="false">(</mo><msup><mi>D</mi> <mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">int(D^3)</annotation></semantics></math>. In this way, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">D^3</annotation></semantics></math> is viewed as a natural compactification of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">H^3</annotation></semantics></math>, and the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSL_2(\mathbb{C})</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">H^3</annotation></semantics></math> induces an action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSL_2(\mathbb{C})</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">D^3</annotation></semantics></math>. The restriction of this action to the boundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>=</mo><mo>∂</mo><msup><mi>D</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^2 = \partial D^3</annotation></semantics></math> (“the heavenly sphere”) coincides with the action on the Riemann sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^2 = \mathbb{P}^1(\mathbb{C})</annotation></semantics></math>.</p> <h2 id="ModularGroup">Modular group</h2> <p>The <em><a class="existingWikiWord" href="/nlab/show/modular+group">modular group</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is the subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSL_2(\mathbb{Z}) \hookrightarrow PSL_2(\mathbb{C})</annotation></semantics></math> consisting of Möbius transformations with <a class="existingWikiWord" href="/nlab/show/integer">integer</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>, hence maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>↦</mo><mfrac><mrow><mi>a</mi><mi>z</mi><mo>+</mo><mi>b</mi></mrow><mrow><mi>c</mi><mi>z</mi><mo>+</mo><mi>d</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">z \mapsto \frac{a z + b}{c z + d}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a, b, c, d \in \mathbb{Z}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>d</mi><mo>−</mo><mi>b</mi><mi>c</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a d - b c = 1</annotation></semantics></math>.</p> <p>The group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSL_2(\mathbb{R})</annotation></semantics></math> acts on the <a class="existingWikiWord" href="/nlab/show/upper+half-plane">upper half-plane</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><mo stretchy="false">{</mo><mi>z</mi><mo>∈</mo><mi>ℂ</mi><mo>:</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">H = \{z \in \mathbb{C}: Im(z) \geq 0\}</annotation></semantics></math> (or rather <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">H \cup \{\infty\}</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of the <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a>), by restriction of the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSL_2(\mathbb{C})</annotation></semantics></math> on the Riemann sphere. Indeed, the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSL_2(\mathbb{R})</annotation></semantics></math> takes the real line <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{R} \cup \{\infty\}</annotation></semantics></math> to itself, and any element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mi>z</mi><mo>+</mo><mi>b</mi></mrow><mrow><mi>c</mi><mi>z</mi><mo>+</mo><mi>d</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">f(z) = \frac{a z + b}{c z + d}</annotation></semantics></math> takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>a</mi><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mi>c</mi><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>d</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(b + a i)(d - c i)/(c^2 + d^2)</annotation></semantics></math>, whose imaginary part <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ad</mi><mo>−</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>d</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>d</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(ad - b c)/(c^2 + d^2) = 1/(c^2 + d^2)</annotation></semantics></math> is positive. By continuity it follows that the action preserves the sign of the imaginary part, hence takes the upper-half plane <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> to itself.</p> <p>It is illuminating to think of complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Im</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Im(\tau) \gt 0</annotation></semantics></math> as representing <a class="existingWikiWord" href="/nlab/show/elliptic+curves">elliptic curves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. Indeed, the <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/meromorphic+functions">meromorphic functions</a> on an elliptic curve (i.e., a complex projective curve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/genus">genus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>, or a <a class="existingWikiWord" href="/nlab/show/torus">torus</a> equipped with a structure of complex analytic 1-manifold) can be identified with a field of doubly periodic <a class="existingWikiWord" href="/nlab/show/holomorphic+functions">holomorphic functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">/</mo><mi>L</mi><mo>→</mo><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}/L \to \mathbb{P}^1(\mathbb{C})</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">L = \langle \omega_1, \omega_2\rangle</annotation></semantics></math> is a fundamental lattice (a discrete cocompact subgroup of the additive <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>) attached to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. In essence, this field is generated by <a class="existingWikiWord" href="/nlab/show/Weierstrass+elliptic+functions">Weierstrass elliptic functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>℘</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo><mi>℘</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>:</mo><mi>ℂ</mi><mo stretchy="false">/</mo><mi>L</mi><mo>→</mo><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\wp(z), \wp'(z): \mathbb{C}/L \to \mathbb{P}^1(\mathbb{C})</annotation></semantics></math> (here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>℘</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\wp'</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/derivative">derivative</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>℘</mi></mrow><annotation encoding="application/x-tex">\wp</annotation></semantics></math>) which satisfy a <a class="existingWikiWord" href="/nlab/show/cubic+curve">cubic algebraic relation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>℘</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mi>℘</mi> <mn>3</mn></msup><mo>−</mo><msub><mi>g</mi> <mn>2</mn></msub><mi>℘</mi><mo>−</mo><msub><mi>g</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">(\wp')^2 = 4\wp^3 - g_2\wp - g_3</annotation></semantics></math></div> <p>where the constants <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">g_2, g_3</annotation></semantics></math> are expressed as certain <a class="existingWikiWord" href="/nlab/show/Eisenstein+series">Eisenstein series</a> in the fundamental periods <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\omega_1, \omega_2</annotation></semantics></math>. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\omega_1, \omega_2</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/lattice">lattice</a> may be arranged so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>=</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tau = \omega_2/\omega_1</annotation></semantics></math> has positive imaginary part. Of course, if there is a <a class="existingWikiWord" href="/nlab/show/homothety">homothety</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>↦</mo><mi>λ</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">z \mapsto \lambda z</annotation></semantics></math> that takes a lattice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> to a lattice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">L'</annotation></semantics></math>, then the elliptic curves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>ℂ</mi><mo stretchy="false">/</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">E = \mathbb{C}/L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>′</mo><mo>=</mo><mi>ℂ</mi><mo stretchy="false">/</mo><mi>L</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">E' = \mathbb{C}/L'</annotation></semantics></math> are analytically isomorphic, so the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>↦</mo><mi>ℂ</mi><mo stretchy="false">/</mo><mo stretchy="false">⟨</mo><mn>1</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\tau \mapsto \mathbb{C}/\langle 1, \tau\rangle</annotation></semantics></math></div> <p>gives a surjection from complex numbers with positive imaginary part to isomorphism classes of smooth elliptic curves. Thus we may restrict attention to lattices of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>=</mo><mo stretchy="false">⟨</mo><mn>1</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">L = \langle 1, \tau \rangle</annotation></semantics></math>.</p> <p>Of course, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> admits more than one such basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, \tau)</annotation></semantics></math>, but for any other <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>τ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, \tau')</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/linear+transformation">linear transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>∈</mo><mi>Γ</mi><mo>≔</mo><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma \in \Gamma \coloneqq PSL_2(\mathbb{Z})</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>τ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\gamma(\tau) = \tau'</annotation></semantics></math>. In summary, the <a class="existingWikiWord" href="/nlab/show/orbit+space">orbit space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>τ</mi><mo>∈</mo><mi>ℂ</mi><mo>:</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">/</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\{\tau \in \mathbb{C}: Im(\tau) \gt 0\}/\Gamma</annotation></semantics></math></div> <p>is a coarse <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> for elliptic curves. In this context, one often says that elliptic curves are paramatrized by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/j-invariant">invariant</a>, a certain <a class="existingWikiWord" href="/nlab/show/modular+form">modular form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j(\tau)</annotation></semantics></math> defined on the upper half-plane such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>j</mi><mo stretchy="false">(</mo><mi>τ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j(\tau) = j(\tau')</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>′</mo><mo>=</mo><mi>γ</mi><mo>⋅</mo><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau' = \gamma \cdot \tau</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>∈</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\gamma \in \Gamma</annotation></semantics></math>.</p> <p>Of course, in some cases there may be more than one <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>∈</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\gamma \in \Gamma</annotation></semantics></math> that fixes a given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math>; this is notably the case when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> is a fourth root of unity or a sixth root of unity. A more refined approach then is to consider, instead of the coarse orbit space, the (compactified) <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">(H \cup \{\infty\})//\Gamma</annotation></semantics></math> for elliptic curves, as a central geometric object of interest.</p> <h3 id="abstract_structure_of_modular_group">Abstract structure of modular group</h3> <p>As an abstract <a class="existingWikiWord" href="/nlab/show/group">group</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>=</mo><msub><mi>PSL</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma = PSL_2(\mathbb{Z})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/free+product">free product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>*</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}/(2) \ast \mathbb{Z}/(3)</annotation></semantics></math>; explicitly, we may take the generator of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math> to be given by the Moebius transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">z \mapsto -1/z</annotation></semantics></math>, and the generator of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math> to be given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">z \mapsto (z-1)/z</annotation></semantics></math>.</p> <p>This <a class="existingWikiWord" href="/nlab/show/concrete+group">concrete group</a> and certain of its subgroups, such as <a class="existingWikiWord" href="/nlab/show/congruence+subgroups">congruence subgroups</a>, are fairly ubiquitous; for example the modular group appears in the theory of <a href="http://ncatlab.org/nlab/show/continued+fraction#rational_tangles">rational tangles</a> and of <a class="existingWikiWord" href="/nlab/show/combinatorial+map">trivalent maps</a>, and these groups frequently crop up in the theory of buildings (stuff on hyperbolic buildings to be filled in).</p> <p>It is also worth pointing out that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is a quotient of the <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">B_3</annotation></semantics></math>. Consider the standard Artin presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">B_3</annotation></semantics></math>, with two generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_2</annotation></semantics></math> subject to the relation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>1</mn></msub><msub><mi>σ</mi> <mn>2</mn></msub><msub><mi>σ</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>σ</mi> <mn>2</mn></msub><msub><mi>σ</mi> <mn>1</mn></msub><msub><mi>σ</mi> <mn>2</mn></msub><mo>.</mo></mrow><annotation encoding="application/x-tex">\sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2.</annotation></semantics></math></div> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>σ</mi> <mn>1</mn></msub><msub><mi>σ</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">z \coloneqq (\sigma_1 \sigma_2)^3</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/center">central element</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">B_3</annotation></semantics></math>, and there is a central extension</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mi>ℤ</mi><mover><mo>→</mo><mrow><mn>1</mn><mo>↦</mo><mi>z</mi></mrow></mover><msub><mi>B</mi> <mn>3</mn></msub><mover><mo>→</mo><mi>q</mi></mover><mi>Γ</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \to \mathbb{Z} \stackrel{1 \mapsto z}{\to} B_3 \stackrel{q}{\to} \Gamma \to 1</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> is the unique homomorphism mapping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>1</mn></msub><msub><mi>σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_1\sigma_2</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>z</mi><mo>.</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">\lambda z. (z-1)/z</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>1</mn></msub><msub><mi>σ</mi> <mn>2</mn></msub><msub><mi>σ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_1\sigma_2\sigma_1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>z</mi><mo>.</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mi>z</mi></mfrac></mrow><annotation encoding="application/x-tex">\lambda z. \frac{-1}{z}</annotation></semantics></math>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/SL%282%2CZ%29">SL(2,Z)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+group">conformal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kleinian+group">Kleinian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/congruence+subgroup">congruence subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/level+n+subgroup">level n subgroup</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+equivariant+elliptic+cohomology">modular equivariant elliptic cohomology</a></p> </li> </ul> <h2 id="references">References</h2> <p>Named after <em><a class="existingWikiWord" href="/nlab/show/August+M%C3%B6bius">August Möbius</a></em>.</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Serre">Jean-Pierre Serre</a>, §VII.1 of: <em>A Course in Arithmetic</em>, Graduate Texts in Mathematics <strong>7</strong>, Springer (1973) [<a href="https://doi.org/10.1007/978-1-4684-9884-4">doi:10.1007/978-1-4684-9884-4</a>, <a href="https://www.math.purdue.edu/~jlipman/MA598/Serre-Course%20in%20Arithmetic.pdf">pdf</a>]</li> </ul> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.m.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 13, 2025 at 14:01:23. 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