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class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </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> </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='#function_field_analogy'>Function field analogy</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>The <em>genus</em> of a <a class="existingWikiWord" href="/nlab/show/surface">surface</a> (2-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>) is an <a class="existingWikiWord" href="/nlab/show/integer">integer</a> that helps classify the surface’s <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> class. It gives the number of handles or ‘cross caps’ that are in a sum decomposition of the surface.</p> <p>In <a class="existingWikiWord" href="/nlab/show/complex+analytic+geometry">complex analytic geometry</a> a surface, being complex 1-dimensional, is called a <a class="existingWikiWord" href="/nlab/show/complex+curves">complex curves</a> and so one speaks of the <a class="existingWikiWord" href="/nlab/show/genus+of+a+curve">genus of a curve</a>. This holds also for <a class="existingWikiWord" href="/nlab/show/algebraic+curves">algebraic curves</a>. Even in <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>, there is <em><a class="existingWikiWord" href="/nlab/show/genus+of+a+number+field">genus of a number field</a></em>.</p> <h2 id="properties">Properties</h2> <h3 id="function_field_analogy">Function field analogy</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a></strong></p> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/number+fields">number fields</a> (“<a class="existingWikiWord" href="/nlab/show/function+fields">function fields</a> of <a class="existingWikiWord" href="/nlab/show/curves">curves</a> over <a class="existingWikiWord" href="/nlab/show/F1">F1</a>”)</th><th><a class="existingWikiWord" href="/nlab/show/function+fields">function fields</a> of <a class="existingWikiWord" href="/nlab/show/curves">curves</a> over <a class="existingWikiWord" href="/nlab/show/finite+fields">finite fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_q</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/arithmetic+curves">arithmetic curves</a>)</th><th><a class="existingWikiWord" href="/nlab/show/Riemann+surfaces">Riemann surfaces</a>/<a class="existingWikiWord" href="/nlab/show/complex+curves">complex curves</a></th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/affine+line">affine</a> and <a class="existingWikiWord" href="/nlab/show/projective+line">projective line</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><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/integers">integers</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q[z]</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a>, <a class="existingWikiWord" href="/nlab/show/polynomial+algebra">polynomial algebra</a> on <a class="existingWikiWord" href="/nlab/show/affine+line">affine line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔸</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1_{\mathbb{F}_q}</annotation></semantics></math>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{\mathbb{C}}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/holomorphic+functions">holomorphic functions</a> on <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><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">\mathbb{Q}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q(z)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/rational+fractions">rational fractions</a>/<a class="existingWikiWord" href="/nlab/show/rational+function+on+an+affine+variety">rational function on affine line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔸</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1_{\mathbb{F}_q}</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/meromorphic+functions">meromorphic functions</a> on <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/prime+number">prime number</a>/non-archimedean <a class="existingWikiWord" href="/nlab/show/place">place</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x \in \mathbb{F}_p</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>−</mo><mi>x</mi><mo>∈</mo><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">z - x \in \mathbb{F}_q[z]</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/irreducible+element">irreducible</a> <a class="existingWikiWord" href="/nlab/show/monic+polynomial">monic polynomial</a> of <a class="existingWikiWord" href="/nlab/show/degree+of+a+polynomial">degree</a> one</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{C}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>−</mo><mi>x</mi><mo>∈</mo><msub><mi>𝒪</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">z - x \in \mathcal{O}_{\mathbb{C}}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/function">function</a> which <a class="existingWikiWord" href="/nlab/show/subtracts">subtracts</a> the <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/variable">variable</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">z</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><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> (<a class="existingWikiWord" href="/nlab/show/place+at+infinity">place at infinity</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><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></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(\mathbb{Z})</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Spec%28Z%29">Spec(Z)</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔸</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1_{\mathbb{F}_q}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/affine+line">affine line</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>∪</mo><msub><mi>place</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">Spec(\mathbb{Z}) \cup place_{\infty}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℙ</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{P}_{\mathbb{F}_q}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/projective+line">projective line</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>p</mi></msub><mo>≔</mo><mfrac><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>p</mi></msup><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mi>p</mi></mfrac></mrow><annotation encoding="application/x-tex">\partial_p \coloneqq \frac{(-)^p - (-)}{p}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Fermat+quotient">Fermat quotient</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial}{\partial z}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a> <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>)</td><td style="text-align: left;">“</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/genus+of+a+number+field">genus of the rational numbers</a> = 0</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of the Riemann sphere</a> = 0</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/formal+neighbourhoods">formal neighbourhoods</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><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><msup><mi>p</mi> <mi>n</mi></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}/(p^n \mathbb{Z})</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/prime+power+local+ring">prime power local ring</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-th order univariate <a class="existingWikiWord" href="/nlab/show/local+Artinian+ring">local Artinian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>𝔽</mi> <mi>q</mi></msub> </mrow> <annotation encoding="application/x-tex">\mathbb{F}_q</annotation> </semantics> </math>-algebra</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mi>ℂ</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-th order univariate <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil <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>-algebra</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q[ [ z -x ] ]</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/power+series">power series</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}[ [z-x] ]</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/holomorphic+functions">holomorphic functions</a> on <a class="existingWikiWord" href="/nlab/show/formal+disk">formal disk</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spf</mi><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mi>p</mi></msub><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow></munder><mi>X</mi></mrow><annotation encoding="application/x-tex">Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X</annotation></semantics></math> (“<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic jet space</a>” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+disks">formal disks</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℚ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Q}_p</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/p-adic+numbers">p-adic numbers</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q((z-x))</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Laurent+series">Laurent series</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}((z-x))</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/holomorphic+functions">holomorphic functions</a> on punctured <a class="existingWikiWord" href="/nlab/show/formal+disk">formal disk</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔸</mi> <mi>ℚ</mi></msub><mo>=</mo><munder><mrow><msup><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><mrow><mi>p</mi><mspace width="thickmathspace"></mspace><mi>place</mi></mrow></munder><msub><mi>ℚ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/ring+of+adeles">ring of adeles</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔸</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{A}_{\mathbb{F}_q((t))}</annotation></semantics></math> ( <a href="ring%20of%20adeles#ForAGlobalField">adeles of function field</a> )</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mrow><msup><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><mrow><mi>x</mi><mo>∈</mo><mi>ℂ</mi></mrow></munder><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/restricted+product">restricted product</a> of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕀</mi> <mi>ℚ</mi></msub><mo>=</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔸</mi> <mi>ℚ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/group+of+ideles">group of ideles</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕀</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{I}_{\mathbb{F}_q((t))}</annotation></semantics></math> ( <a class="existingWikiWord" href="/nlab/show/group+of+ideles">ideles of function field</a> )</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mrow><msup><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><mrow><mi>x</mi><mo>∈</mo><mi>ℂ</mi></mrow></munder><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/theta+functions">theta functions</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Jacobi+theta+function">Jacobi theta function</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/zeta+functions">zeta functions</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Riemann+zeta+function">Riemann zeta function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Goss+zeta+function">Goss zeta function</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/branched+covering">branched covering</a> curves</strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><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 <a class="existingWikiWord" href="/nlab/show/number+field">number field</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mo>↪</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q} \hookrightarrow K</annotation></semantics></math> a possibly <a class="existingWikiWord" href="/nlab/show/ramified">ramified</a> <a class="existingWikiWord" href="/nlab/show/finite+set">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/field+extension">field extension</a>)</td><td style="text-align: left;"><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 <a class="existingWikiWord" href="/nlab/show/function+field">function field</a> of an <a class="existingWikiWord" href="/nlab/show/algebraic+curve">algebraic curve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_p</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">K_\Sigma</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/sheaf+of+rational+functions">sheaf of rational functions</a> on <a class="existingWikiWord" href="/nlab/show/complex+curve">complex curve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_K</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/ring+of+integers">ring of integers</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{\Sigma}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Spec</mi> <mi>an</mi></msub><mo stretchy="false">(</mo><msub><mi>𝒪</mi> <mi>K</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/spectrum+of+a+commutative+ring">spectrum</a> with archimedean <a class="existingWikiWord" href="/nlab/show/places">places</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/arithmetic+curve">arithmetic curve</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Sigma \to \mathbb{C}P^1</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/complex+curve">complex curve</a> being <a class="existingWikiWord" href="/nlab/show/branched+cover+of+Riemann+sphere">branched cover of Riemann sphere</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>p</mi></msup><mo>−</mo><mi>Φ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mi>p</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{(-)^p - \Phi(-)}{p}</annotation></semantics></math> (lift of <a class="existingWikiWord" href="/nlab/show/Frobenius+morphism">Frobenius morphism</a>/<a class="existingWikiWord" href="/nlab/show/Lambda-ring">Lambda-ring</a> structure)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial}{\partial z}</annotation></semantics></math></td><td style="text-align: left;">“</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/genus+of+a+number+field">genus of a number field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/genus+of+an+algebraic+curve">genus of an algebraic curve</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/formal+neighbourhoods">formal neighbourhoods</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> prime ideal in <a class="existingWikiWord" href="/nlab/show/ring+of+integers">ring of integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_K</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">x \in \Sigma</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">x \in \Sigma</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>v</mi></msub></mrow><annotation encoding="application/x-tex">K_v</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}((z_x))</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a> on punctured <a class="existingWikiWord" href="/nlab/show/formal+disk">formal disk</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mrow><msub><mi>K</mi> <mi>v</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{K_v}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/ring+of+integers">ring of integers</a> of <a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>z</mi> <mi>x</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}[ [ z_x ] ]</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a> on <a class="existingWikiWord" href="/nlab/show/formal+disk">formal disk</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔸</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{A}_K</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/ring+of+adeles">ring of adeles</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>∈</mo><mi>Σ</mi></mrow> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/restricted+product">restricted product</a> of <a class="existingWikiWord" href="/nlab/show/function+rings">function rings</a> on all punctured <a class="existingWikiWord" href="/nlab/show/formal+disks">formal disks</a> around all points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>∈</mo><mi>Σ</mi></mrow></msub><mi>ℂ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>z</mi> <mi>x</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] </annotation></semantics></math> (function ring on all <a class="existingWikiWord" href="/nlab/show/formal+disks">formal disks</a> around all points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕀</mi> <mi>K</mi></msub><mo>=</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔸</mi> <mi>K</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{I}_K = GL_1(\mathbb{A}_K)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/group+of+ideles">group of ideles</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>∈</mo><mi>Σ</mi></mrow> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Galois+group">Galois group</a></td><td style="text-align: left;">“</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(\Sigma)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Galois+representation">Galois representation</a></td><td style="text-align: left;">“</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flat+connection">flat connection</a> (“<a class="existingWikiWord" href="/nlab/show/local+system">local system</a>”) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/class+field+theory">class field theory</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/class+field+theory">class field theory</a></td><td style="text-align: left;">“</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+class+field+theory">geometric class field theory</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hilbert+reciprocity+law">Hilbert reciprocity law</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Artin+reciprocity+law">Artin reciprocity law</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Weil+reciprocity+law">Weil reciprocity law</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>\</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔸</mi> <mi>K</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(K)\backslash GL_1(\mathbb{A}_K)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/idele+class+group">idele class group</a>)</td><td style="text-align: left;">“</td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>\</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔸</mi> <mi>K</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})</annotation></semantics></math></td><td style="text-align: left;">“</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bun</mi> <mrow><msub><mi>GL</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bun_{GL_1}(\Sigma)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/moduli+stack+of+bundles">moduli stack of line bundles</a>, by <a class="existingWikiWord" href="/nlab/show/Weil+uniformization+theorem">Weil uniformization theorem</a>)</td></tr> <tr><td style="text-align: left;"><strong>non-abelian class field theory and automorphy</strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">number field <a class="existingWikiWord" href="/nlab/show/Langlands+correspondence">Langlands correspondence</a></td><td style="text-align: left;">function field <a class="existingWikiWord" href="/nlab/show/Langlands+correspondence">Langlands correspondence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+Langlands+correspondence">geometric Langlands correspondence</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>\</mo><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>𝔸</mi> <mi>K</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/constant+sheaves">constant sheaves</a> on this <a class="existingWikiWord" href="/nlab/show/stack">stack</a> form <a class="existingWikiWord" href="/nlab/show/unramified">unramified</a> <a class="existingWikiWord" href="/nlab/show/automorphic+representations">automorphic representations</a>)</td><td style="text-align: left;">“</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bun</mi> <mrow><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bun_{GL_n(\mathbb{C})}(\Sigma)</annotation></semantics></math> (<a href="moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence">moduli stack of bundles on the curve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, by <a class="existingWikiWord" href="/nlab/show/Weil+uniformization+theorem">Weil uniformization theorem</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a href="Weil+conjecture+on+Tamagawa+numbers#NumberFieldCase">Tamagawa-Weil for number fields</a></td><td style="text-align: left;"><a href="Weil+conjecture+on+Tamagawa+numbers#FunctionFieldCase">Tamagawa-Weil for function fields</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/theta+functions">theta functions</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hecke+theta+function">Hecke theta function</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/functional+determinant">functional determinant</a> <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">line bundle</a> of <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a>/chiral <a class="existingWikiWord" href="/nlab/show/Laplace+operator">Laplace operator</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/zeta+functions">zeta functions</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dedekind+zeta+function">Dedekind zeta function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Weil+zeta+function">Weil zeta function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/zeta+function+of+a+Riemann+surface">zeta function of a Riemann surface</a>/<a class="existingWikiWord" href="/nlab/show/zeta+function+of+an+elliptic+differential+operator">of the Laplace operator</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/higher+dimensional+arithmetic+geometry">higher dimensional spaces</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/zeta+functions">zeta functions</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hasse-Weil+zeta+function">Hasse-Weil zeta function</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/genus+of+a+curve">genus of a curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+genus">arithmetic genus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/curve">curve</a>, <a class="existingWikiWord" href="/nlab/show/singular+point+of+a+curve">singular point of a curve</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Genus_%28mathematics%29">Genus</a></em></li> </ul> <p>Various approaches to the classification of surfaces are described in introductory texts such as</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/N.+D.+Gilbert">N. D. Gilbert</a> and <a class="existingWikiWord" href="/nlab/show/T.+Porter">T. Porter</a>, Knots and Surfaces, Oxford U.P., 1994.</p> </li> <li> <p><em><a href="http://lamington.wordpress.com/2009/09/23/how-to-see-the-genus/">How to see the genus</a></em></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 19, 2021 at 15:23:44. See the <a href="/nlab/history/genus+of+a+surface" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/genus+of+a+surface" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/5844/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/genus+of+a+surface/6" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/genus+of+a+surface" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/genus+of+a+surface" accesskey="S" class="navlink" id="history" rel="nofollow">History (6 revisions)</a> <a href="/nlab/show/genus+of+a+surface/cite" style="color: black">Cite</a> <a href="/nlab/print/genus+of+a+surface" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/genus+of+a+surface" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>