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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="arithmetic_geometry">Arithmetic geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/number+theory">number theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/arithmetic+topology">arithmetic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+arithmetic+geometry">higher arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+arithmetic+geometry">E-∞ arithmetic geometry</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/number">number</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>, <a class="existingWikiWord" href="/nlab/show/integer+number">integer number</a>, <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a>, <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, <a class="existingWikiWord" href="/nlab/show/irrational+number">irrational number</a>, <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a>, <a class="existingWikiWord" href="/nlab/show/quaternion">quaternion</a>, <a class="existingWikiWord" href="/nlab/show/octonion">octonion</a>, <a class="existingWikiWord" href="/nlab/show/adic+number">adic number</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+number">cardinal number</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+number">ordinal number</a>, <a class="existingWikiWord" href="/nlab/show/surreal+number">surreal number</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+arithmetic">Peano arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/second-order+arithmetic">second-order arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transfinite+arithmetic">transfinite arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+arithmetic">cardinal arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+arithmetic">ordinal arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+field">prime field</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+integer">p-adic integer</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+rational+number">p-adic rational number</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+complex+number">p-adic complex number</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></strong>, <a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+scheme">arithmetic scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+curve">arithmetic curve</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+genus">arithmetic genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chern-Simons+theory">arithmetic Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chow+group">arithmetic Chow group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil-%C3%A9tale+topology+for+arithmetic+schemes">Weil-étale topology for arithmetic schemes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absolute+cohomology">absolute cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+conjecture+on+Tamagawa+numbers">Weil conjecture on Tamagawa numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borger%27s+absolute+geometry">Borger's absolute geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Iwasawa-Tate+theory">Iwasawa-Tate theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic jet space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adelic+integration">adelic integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shtuka">shtuka</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenioid">Frobenioid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Arakelov+geometry">Arakelov geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Riemann-Roch+theorem">arithmetic Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+algebraic+K-theory">differential algebraic K-theory</a></p> </li> </ul> </div></div> <h4 id="analytic_geometry">Analytic geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/analytic+geometry">analytic geometry</a> (<a class="existingWikiWord" href="/nlab/show/complex+analytic+geometry">complex</a>, <a class="existingWikiWord" href="/nlab/show/rigid+analytic+geometry">rigid</a>, <a class="existingWikiWord" href="/nlab/show/global+analytic+geometry">global</a>)</strong></p> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>+<a class="existingWikiWord" href="/nlab/show/analysis">analysis</a>/<a class="existingWikiWord" href="/nlab/show/analytic+number+theory">analytic number theory</a></p> <h2 id="basic_concepts">Basic concepts</h2> <p><a class="existingWikiWord" href="/nlab/show/analytic+function">analytic function</a></p> <p><a class="existingWikiWord" href="/nlab/show/analytic+space">analytic space</a>, <a class="existingWikiWord" href="/nlab/show/analytic+variety">analytic variety</a>, <a class="existingWikiWord" href="/nlab/show/Berkovich+space">Berkovich space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/polydisc">polydisc</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affinoid+algebra">affinoid algebra</a>, <a class="existingWikiWord" href="/nlab/show/analytic+spectrum">analytic spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+%E2%88%9E-groupoid">analytic ∞-groupoid</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/analytification">analytification</a></p> <h2 id="theorems">Theorems</h2> <p><a class="existingWikiWord" href="/nlab/show/GAGA">GAGA</a></p> </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='#Formalizations'>Formalizations</a></li> <li><a href='#Overview'>Overview</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>There is a noticeable <a class="existingWikiWord" href="/nlab/show/analogy">analogy</a> between phenomena (<a class="existingWikiWord" href="/nlab/show/theorems">theorems</a>) in the theory of <a class="existingWikiWord" href="/nlab/show/number+fields">number fields</a> and those in the theory of <a class="existingWikiWord" href="/nlab/show/function+fields">function fields</a> over <a class="existingWikiWord" href="/nlab/show/finite+fields">finite fields</a> (<a href="#Weil39">Weil 39</a>, <a href="#Iwasaw69">Iwasawa 69</a>, <a href="#MazurWiles83">Mazur-Wiles 83</a>), hence between the theories of the two kinds of <em><a class="existingWikiWord" href="/nlab/show/global+fields">global fields</a></em>. When regarding <a class="existingWikiWord" href="/nlab/show/number+theory">number theory</a> dually as <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>, then one may see that this analogy extends further to include <a class="existingWikiWord" href="/nlab/show/complex+analytic+geometry">complex analytic geometry</a>, the theory of <a class="existingWikiWord" href="/nlab/show/complex+curves">complex curves</a> (e.g. <a href="#Frenkel05">Frenkel 05</a>).</p> <p>At a very basic level the analogy may be plausible from the fact that both the <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> as well as the <a class="existingWikiWord" href="/nlab/show/polynomial+rings">polynomial rings</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><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q[x]</annotation></semantics></math> (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>) are <a class="existingWikiWord" href="/nlab/show/principal+ideal+domains">principal ideal domains</a> with <a class="existingWikiWord" href="/nlab/show/finite+group">finite</a> <a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, all <a class="existingWikiWord" href="/nlab/show/quotients">quotients</a> being finite rings and with infinitely many <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a>, which already implies that a lot of <a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a> over these rings is similar. Since <a class="existingWikiWord" href="/nlab/show/number+fields">number fields</a> are the <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/field+extensions">field extensions</a> of the <a class="existingWikiWord" href="/nlab/show/field+of+fractions">field of fractions</a> of <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>, namely the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</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">\mathbb{Q}</annotation></semantics></math>, and since <a class="existingWikiWord" href="/nlab/show/function+fields">function fields</a> are just the finite-dimensional field extensions of the fields of fractions <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>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q(x)</annotation></semantics></math> of <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>x</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q[x]</annotation></semantics></math>, this similarity plausibly extends to these extensions.</p> <p>The <a class="existingWikiWord" href="/nlab/show/entire+function">entire</a> <a class="existingWikiWord" href="/nlab/show/holomorphic+functions">holomorphic functions</a> on the <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a> are, while not quite an principal ideal domain, still a <a class="existingWikiWord" href="/nlab/show/B%C3%A9zout+domain">Bézout domain</a>, but in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a> the <a class="existingWikiWord" href="/nlab/show/integers">integers</a> and the <a class="existingWikiWord" href="/nlab/show/polynomial+rings">polynomial rings</a> over <a class="existingWikiWord" href="/nlab/show/finite+fields">finite fields</a> are only Bézout domains as well.</p> <p>But the analogy ranges much deeper than this similarity alone might suggest. For instance (<a href="#Weil39">Weil 39</a>) defined an invariant of a <a class="existingWikiWord" href="/nlab/show/number+field">number field</a> – the <em><a class="existingWikiWord" href="/nlab/show/genus+of+a+number+field">genus of a number field</a></em>– which is analogous to the <a class="existingWikiWord" href="/nlab/show/genus+of+a+curve">genus</a> of the <a class="existingWikiWord" href="/nlab/show/algebraic+curve">algebraic curve</a> on which a given <a class="existingWikiWord" href="/nlab/show/function+field">function field</a> is the <a class="existingWikiWord" href="/nlab/show/rational+functions">rational functions</a>. This is such as to make the statement of the <a class="existingWikiWord" href="/nlab/show/Riemann-Roch+theorem">Riemann-Roch theorem</a> for <a class="existingWikiWord" href="/nlab/show/algebraic+curves">algebraic curves</a> extend to <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a> (<a href="#Neukirch92">Neukirch 92, chapter II, prop.(3.6)</a>).</p> <p>Another notable part of the analogy is the fact that there are natural analogs of the <a class="existingWikiWord" href="/nlab/show/Riemann+zeta+function">Riemann zeta function</a> in all three columns of the analogy. This aspect has found attention notably through the lens of regarding <a class="existingWikiWord" href="/nlab/show/number+fields">number fields</a> as <a class="existingWikiWord" href="/nlab/show/rational+functions">rational functions</a> on “<a class="existingWikiWord" href="/nlab/show/arithmetic+curves">arithmetic curves</a> over the would-be <a class="existingWikiWord" href="/nlab/show/field+with+one+element">field with one element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_1</annotation></semantics></math>”.</p> <p>The analogy between <a class="existingWikiWord" href="/nlab/show/p-adic+numbers">p-adic numbers</a> and <a class="existingWikiWord" href="/nlab/show/Laurent+series">Laurent series</a> 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> is strengthened by (<a href="#FontaineWinterberger79">Fontaine-Winterberger 79</a>), which shows that the absolute <a class="existingWikiWord" href="/nlab/show/Galois+groups">Galois groups</a> of the <a class="existingWikiWord" href="/nlab/show/perfect+field">perfection</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_p((t))</annotation></semantics></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℚ</mi> <mi>p</mi></msub><mo stretchy="false">[</mo><msup><mi>p</mi> <mfrac><mn>1</mn><mrow><msup><mi>p</mi> <mn>∞</mn></msup></mrow></mfrac></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Q}_p[p^{\frac{1}{p^\infty}}]</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a>. For more review of this see also (<a href="#Hartl06">Hartl 06</a>). (The generalization of this to higher dimensions is the topic of <a class="existingWikiWord" href="/nlab/show/perfectoid+spaces">perfectoid spaces</a>.)</p> <p>It is also the function field analogy which induces the conjecture of the <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+correspondence">geometric Langlands correspondence</a> by analogy from the number-theoretic <a class="existingWikiWord" href="/nlab/show/Langlands+correspondence">Langlands correspondence</a>. Here one finds that the <a class="existingWikiWord" href="/nlab/show/moduli+stack+of+bundles">moduli stack of bundles</a> over a <a class="existingWikiWord" href="/nlab/show/complex+curve">complex curve</a> is analogous in absolute <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a> to the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> of the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> with coefficients in the <a class="existingWikiWord" href="/nlab/show/ring+of+adeles">ring of adeles</a> of a number field, on which <a class="existingWikiWord" href="/nlab/show/unramified">unramified</a> <a class="existingWikiWord" href="/nlab/show/automorphic+representations">automorphic representations</a> are functions. Under this analogy the <a class="existingWikiWord" href="/nlab/show/Weil+conjecture+on+Tamagawa+numbers">Weil conjecture on Tamagawa numbers</a> may be regarded as giving the <a class="existingWikiWord" href="/nlab/show/groupoid+cardinality">groupoid cardinality</a> of the <a class="existingWikiWord" href="/nlab/show/moduli+stack+of+bundles">moduli stack of bundles</a> in <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>.</p> <p>In summary then the analogy says that the theories of <a class="existingWikiWord" href="/nlab/show/number+fields">number fields</a> and of <a class="existingWikiWord" href="/nlab/show/function+fields">function fields</a> both look much like a <a class="existingWikiWord" href="/nlab/show/global+analytic+geometry">global analytic geometry</a>-version of the theory of <a class="existingWikiWord" href="/nlab/show/complex+curves">complex curves</a>.</p> <h2 id="Formalizations">Formalizations</h2> <p>To date the function field analogy remains just that, an <a class="existingWikiWord" href="/nlab/show/analogy">analogy</a>, though various research programs may be thought of as trying to provide a context in which the analogy would become a consequence of a systematic theory (see e.g. the introduction of <a href="#vdGeer05">v.d. Geer et al 05</a>). This includes</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Arakelov+geometry">Arakelov geometry</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+analytic+geometry">global analytic geometry</a>.</p> </li> <li> <p>geometry “over <a class="existingWikiWord" href="/nlab/show/F1">F1</a>”.</p> </li> </ul> <p>Regarding the last point, in particular <a class="existingWikiWord" href="/nlab/show/Borger%27s+absolute+geometry">Borger's absolute geometry</a> (<a href="#Borger09">Borger 09</a>) makes precise the analogy between <a class="existingWikiWord" href="/nlab/show/Spec%28Z%29">Spec(Z)</a> and the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</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>z</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[z]</annotation></semantics></math>/<a class="existingWikiWord" href="/nlab/show/entire+holomorphic+function">entire holomorphic function</a>-ring <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> by interpreting the analog of the canonical <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> <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> on the latter two as the <a class="existingWikiWord" href="/nlab/show/Fermat+quotient">Fermat quotient</a> operation, and more generally by interpreting the lift of this to arithmetic spaces over <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> as lifts of <a class="existingWikiWord" href="/nlab/show/Frobenius+homomorphisms">Frobenius homomorphisms</a> as given by <a class="existingWikiWord" href="/nlab/show/Lambda-ring">Lambda-ring</a> structures. See at <em><a href="Borger%27s+absolute+geometry#Motivation">Borger’s absolute geometry – Motivation</a></em> for more on this.</p> <p>In this context the analogy between geometry over <a class="existingWikiWord" href="/nlab/show/number+fields">number fields</a> and over <a class="existingWikiWord" href="/nlab/show/function+fields">function fields</a> is made precise by showing (<a href="#Borger09">Borger 09, section 7</a>) that for any smooth connected curve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">/</mo><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow><annotation encoding="application/x-tex">S/\mathbb{F}_q</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/finite+field">finite field</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> the standard <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> of (“big”) <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">/</mo><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spec(S/\mathbb{F}_q)\longrightarrow Spec(\mathbb{F}_q) </annotation></semantics></math></div> <p>factors through an alternative base topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Spec</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde Spec(\mathbb{F}_q)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">/</mo><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>Spec</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spec(S/\mathbb{F}_q)\longrightarrow \widetilde Spec(\mathbb{F}_q) \longrightarrow Spec(\mathbb{F}_q) </annotation></semantics></math></div> <p>which, while different from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(\mathbb{F}_q)</annotation></semantics></math> is “close” to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(\mathbb{F}_q)</annotation></semantics></math> in some precise sense, but which has the advantage that its construction does exist for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">q = 1</annotation></semantics></math> in that there is directly analogous</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>Spec</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Spec(\mathbb{Z}) \longrightarrow \widetilde Spec(\mathbb{F}_1) \,, </annotation></semantics></math></div> <p>where the notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Spec</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde Spec(\mathbb{F}_1)</annotation></semantics></math> here stands for Borger’s the topos over <a class="existingWikiWord" href="/nlab/show/Lambda-rings">Lambda-rings</a>, see at <em><a class="existingWikiWord" href="/nlab/show/Borger%27s+absolute+geometry">Borger's absolute geometry</a></em> for the actual details.</p> <h2 id="Overview">Overview</h2> <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> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/analogies">analogies</a> in the <a class="existingWikiWord" href="/nlab/show/Langlands+program">Langlands program</a>:</strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic</a> <a class="existingWikiWord" href="/nlab/show/Langlands+correspondence">Langlands correspondence</a></th><th><a class="existingWikiWord" href="/nlab/show/geometric+Langlands+correspondence">geometric Langlands correspondence</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring+of+integers">ring of integers</a> of <a class="existingWikiWord" href="/nlab/show/global+field">global field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</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;"><a class="existingWikiWord" href="/nlab/show/Galois+group">Galois group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of <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;"><a class="existingWikiWord" href="/nlab/show/Galois+representation">Galois representation</a></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;"><a class="existingWikiWord" href="/nlab/show/idele+class+group">idele class group</a> mod <a class="existingWikiWord" href="/nlab/show/integral+adeles">integral adeles</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/moduli+stack+of+line+bundles">moduli stack of line bundles</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;">nonabelian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math> “</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/moduli+stack+of+vector+bundles">moduli stack of vector bundles</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;"><a class="existingWikiWord" href="/nlab/show/automorphic+representation">automorphic representation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hitchin+connection">Hitchin connection</a> <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a> on bundle of <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> over the moduli stack</td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/MKR+analogy">MKR analogy</a> in <a class="existingWikiWord" href="/nlab/show/arithmetic+topology">arithmetic topology</a></li> </ul> <h2 id="references">References</h2> <p>Original articles includes</p> <ul> <li id="Weil39"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Weil">André Weil</a>, <em>Sur l’analogie entre les corps de nombres algébrique et les corps de fonctions algébrique</em>, Revue Scient. 77, 104-106, 1939</p> </li> <li id="Iwasaw69"> <p><a class="existingWikiWord" href="/nlab/show/Kenkichi+Iwasawa">Kenkichi Iwasawa</a>, <em>Analogies between number fields and function fields</em>, in <em>Some Recent Advances in the Basic Sciences</em>, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966), Belfer Graduate School of Science, Yeshiva Univ., New York, pp. 203–208, <a href="http://www.ams.org/mathscinet-getitem?mr=0255510">MR 0255510</a></p> <p>for more on this see: Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Main_conjecture_of_Iwasawa_theory">Main conjecture of Iwasawa theory</a></em></p> </li> <li id="FontaineWinterberger79"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marc+Fontaine">Jean-Marc Fontaine</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Wintenberger">Jean-Pierre Wintenberger</a>, <em>Extensions algébrique et corps des normes des extensions APF des corps locaux</em>, C. R. Acad. Sci. Paris Sér. A–B 288(8) (1979), A441–A444</p> </li> <li id="MazurWiles83"> <p><a class="existingWikiWord" href="/nlab/show/Barry+Mazur">Barry Mazur</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Wiles">Andrew Wiles</a>, <em>Analogies between function fields and number fields</em>, American Journal of Mathematics Vol. 105, No. 2 (Apr., 1983), pp. 507-521 (<a href="http://www.jstor.org/stable/2374266">JStor</a>)</p> </li> </ul> <p>Textbook accounts include</p> <ul> <li id="Neukirch92"> <p><a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Neukirch">Jürgen Neukirch</a>, <em>Algebraische Zahlentheorie</em> (1992), English translation <em>Algebraic Number Theory</em>, Grundlehren der Mathematischen Wissenschaften 322, 1999 (<a href="http://www.plouffe.fr/simon/math/Algebraic%20Number%20Theory.pdf">pdf</a>)</p> </li> <li id="Rosen02"> <p>Michael Rosen, <em>Number theory in function fields</em>, Graduate texts in mathematics, 2002</p> </li> </ul> <p>Tables showing the parallels between number fields and function fields are in</p> <ul> <li id="Goss92"> <p><a class="existingWikiWord" href="/nlab/show/David+Goss">David Goss</a>, <em>Dictionary</em>, in David Goss, David R. Hayes, Michael Rosen (eds.) <em>The Arithmetic of Function Fields</em>, Ohio State Univ. Math. Res. Inst. Publ., 2, de Gruyter, Berlin, 1992, pp. 475-482,</p> </li> <li id="Poonen06"> <p><a class="existingWikiWord" href="/nlab/show/Bjorn+Poonen">Bjorn Poonen</a>, section 2.6 of <em>Lectures on rational points on curves</em>, 2006 (<a href="http://math.mit.edu/~poonen/papers/curves.pdf">pdf</a>)</p> </li> <li id="Hartl06"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Hartl">Urs Hartl</a>, <em>A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic</em> (<a href="http://arxiv.org/abs/math/0607182">arXiv:math/0607182</a>)</p> </li> </ul> <p>See also</p> <ul> <li>M. Blickle,<a class="existingWikiWord" href="/nlab/show/H%C3%A9l%C3%A8ne+Esnault">Hélène Esnault</a>, K. Rülling, <em>Characteristic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> and <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> analogies, and somemotivic cohomology</em> (<a class="existingWikiWord" href="/nlab/files/EsnaultAnalogy.pdf" title="pdf">pdf</a>)</li> </ul> <p>A collection of more recent developments is in</p> <ul> <li id="vdGeer05">van der Geer et al (eds.) <em>Number Fields and Function Fields – Two Parallel Worlds</em>, Birkhäuser 2005 (<a href="http://www.springer.com/birkhauser/mathematics/book/978-0-8176-4397-3">publisher page</a>)</li> </ul> <p>Discussion including also the complex-analytic side includes</p> <ul> <li id="Frenkel05"><a class="existingWikiWord" href="/nlab/show/Edward+Frenkel">Edward Frenkel</a>, section 2 of <em>Lectures on the Langlands Program and Conformal Field Theory</em> (<a href="http://arxiv.org/abs/hep-th/0512172">arXiv:hep-th/0512172</a>).</li> </ul> <p>and a comparison of the number theory to that of <a class="existingWikiWord" href="/nlab/show/foliations">foliations</a> is in</p> <ul> <li id="Deninger07"><a class="existingWikiWord" href="/nlab/show/Christopher+Deninger">Christopher Deninger</a>, <em>Analogies between analysis on foliated spaces and arithmetic geometry</em> (<a href="http://arxiv.org/abs/0709.2801">arXiv:0709.2801</a>)</li> </ul> <p>An actual formalization of the analogy between geometry over number fields and function fields is in</p> <ul> <li id="Borger09"><a class="existingWikiWord" href="/nlab/show/James+Borger">James Borger</a>, section 7 of <em>Lambda-rings and the field with one element</em> (<a href="http://arxiv.org/abs/0906.3146">arXiv/0906.3146</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 3, 2024 at 01:54:43. 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