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class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> </ul> <h3 id="structures">Structures</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h3 id="examples">Examples</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim = 1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, <a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim = 2</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></p> </li> </ul> </div></div> <h4 id="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="elliptic_cohomology">Elliptic cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a>, <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented</a><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <a class="existingWikiWord" href="/nlab/show/chromatic+level">chromatic level</a> 2</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supersingular+elliptic+curve">supersingular elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+elliptic+curve">derived elliptic curve</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+form">modular form</a>, <a class="existingWikiWord" href="/nlab/show/Jacobi+form">Jacobi form</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eisenstein+series">Eisenstein series</a>, <a class="existingWikiWord" href="/nlab/show/j-invariant">j-invariant</a>, <a class="existingWikiWord" href="/nlab/show/Weierstrass+sigma-function">Weierstrass sigma-function</a>, <a class="existingWikiWord" href="/nlab/show/Dedekind+eta+function">Dedekind eta function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+modular+form">topological modular form</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a></li> </ul> </li> </ul></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#DefinitionOverGeneralRing'>Over a general ring</a></li> <ul> <li><a href='#OverGeneralRingConceptualDefinition'>Conceptual definition</a></li> <li><a href='#OverGeneralRingAsSolutionsToEquations'>Coordinatized as solutions to cubic Weierstrass equations</a></li> <ul> <li><a href='#WeierstrassCubicDiscriminantAndJInvariant'>Weierstrass cubic, discriminant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>-invariant</a></li> <li><a href='#EllipticCurvesNodalCurvesCuspidalCurves'>Elliptic curves, Nodal curves, Cuspidal curves</a></li> <li><a href='#localization_at_2_and_3'>Localization at 2 and 3</a></li> </ul> </ul> <li><a href='#OverTheComplexNumbers'>Over the complex numbers</a></li> <ul> <li><a href='#in_terms_of_algebraic_geometry'>In terms of algebraic geometry</a></li> <li><a href='#InTermsOfCpomplexGeometry'>In terms of complex geometry</a></li> </ul> <li><a href='#OverTheRationalNumbers'>Over the rational numbers</a></li> <li><a href='#OverpAdics'>Over the <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>-adic integers 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>-adic numbers</a></li> </ul> <li><a href='#level_structures'>Level structures</a></li> <li><a href='#group_structure'>Group structure</a></li> <li><a href='#constructions'>Constructions</a></li> <ul> <li><a href='#formal_group_law'>Formal group law</a></li> <li><a href='#elliptic_cohomology_2'>Elliptic cohomology</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>Classically in <a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a>, an <em>elliptic curve</em> is a connected <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a> (a connected compact 1-dimensional <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>) of <a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a> 1, hence it is a <a class="existingWikiWord" href="/nlab/show/torus">torus</a> equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, or equivalently with <a class="existingWikiWord" href="/nlab/show/conformal+structure">conformal structure</a>.</p> <p>The curious term “elliptic” is a remnant from the 19th century, a back-formation which refers to <a class="existingWikiWord" href="/nlab/show/elliptic+functions">elliptic functions</a> (generalizing circular functions, i.e., the classical trigonometric functions) and their natural domains as Riemann surfaces.</p> <p>In more modern frameworks and in the generality of <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>, an elliptic curve over a <a class="existingWikiWord" href="/nlab/show/field">field</a> <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> or indeed over any <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> may be defined as a complete irreducible non-singular <a class="existingWikiWord" href="/nlab/show/algebraic+curve">algebraic curve</a> of <a class="existingWikiWord" href="/nlab/show/arithmetic+genus">arithmetic genus</a>-1 over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, or even as a certain type of algebraic <a class="existingWikiWord" href="/nlab/show/group+scheme">group scheme</a>.</p> <p>Elliptic curves have many remarkable properties, and their deeper arithmetic study is one of the most profound subjects in present-day mathematics.</p> <p>The <a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a> equipped with its canonical map to the <a class="existingWikiWord" href="/nlab/show/moduli+stack+of+formal+group+laws">moduli stack of formal group laws</a> plays a central role in <a class="existingWikiWord" href="/nlab/show/chromatic+homotopy+theory">chromatic homotopy theory</a> at <a class="existingWikiWord" href="/nlab/show/chromatic+level">chromatic level</a> 2, where it serves to parameterize <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology+theories">elliptic cohomology theories</a>.</p> <p>Elliptic curves over the complex numbers are also interpreted as those <a class="existingWikiWord" href="/nlab/show/worldsheets">worldsheets</a> in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> whose <a class="existingWikiWord" href="/nlab/show/correlators">correlators</a> are the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a>‘s <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a>, which is the <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a>. Via the <a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a> this connects to to the role of elliptic curves in <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology+theory">elliptic cohomology theory</a>.</p> <h2 id="definition">Definition</h2> <h3 id="DefinitionOverGeneralRing">Over a general ring</h3> <p>Elliptic curves over a general <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> (hence in <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>) are the well-behaved 1-dimensional <a class="existingWikiWord" href="/nlab/show/group+objects">group objects</a> parameterized over the <a class="existingWikiWord" href="/nlab/show/space">space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectrum</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>). (Notice the count of dimension: over the complex numbers a torus is complex 1-dimensional and in this sense one is looking at 1-dimensional group schemes here.) This we discuss below in</p> <ul> <li><a href="#OverGeneralRingConceptualDefinition">Conceptual definition</a></li> </ul> <p>More concretely there are various explicit and standard <a class="existingWikiWord" href="/nlab/show/coordinates">coordinazations</a> of elliptic curves as <a class="existingWikiWord" href="/nlab/show/affine+schemes">affine schemes</a>, hence as solution spaces to <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> <a class="existingWikiWord" href="/nlab/show/equations">equations</a>. This we discuss below in</p> <ul> <li><a href="#OverGeneralRingAsSolutionsToEquations">Coordinatized as solutions to cubic equations</a></li> </ul> <h4 id="OverGeneralRingConceptualDefinition">Conceptual definition</h4> <p> <div class='num_defn' id='DefinitionEllipticCurve'> <h6>Definition</h6> <p></p> <p>An <strong>elliptic curve</strong> over a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/group+scheme">group scheme</a> (a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/schemes">schemes</a>) 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>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math> that is a relative 1-dimensional, <a class="existingWikiWord" href="/nlab/show/smooth+scheme">smooth</a>, <a class="existingWikiWord" href="/nlab/show/proper+scheme">proper</a> curve over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>.</p> <p></p> </div> </p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This implies that an elliptic curve has <a class="existingWikiWord" href="/nlab/show/arithmetic+genus">arithmetic genus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> (by a direct argument concerning the <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>.)</p> </div> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>An elliptic curve over a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/positive+characteristic">positive characteristic</a> whose <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a> has <a class="existingWikiWord" href="/nlab/show/height+of+a+formal+group">height</a> equal to 2 is called a <em><a class="existingWikiWord" href="/nlab/show/supersingular+elliptic+curve">supersingular elliptic curve</a></em>. Otherwise the height equals 1 and the elliptic curve is called <em>ordinary</em>.</p> </div> <h4 id="OverGeneralRingAsSolutionsToEquations">Coordinatized as solutions to cubic Weierstrass equations</h4> <p>Elliptic curves are examples of solutions to <a class="existingWikiWord" href="/nlab/show/Diophantine+equations">Diophantine equations</a> of degree 3. We start by giving the equation valued over general rings, which is fairly complicated compared to the special case that it reduces to in the classical case over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>. The more elements in the ground ring are invertible, the more the equation may be simplified.</p> <p>(See <a href="#Silverman09">Silverman 09, III.1</a> for a textbook account and for instance (<a href="#QuickIntro">QuickIntro</a>) for a quick survey.)</p> <h5 id="WeierstrassCubicDiscriminantAndJInvariant">Weierstrass cubic, discriminant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>-invariant</h5> <div class="num_defn" id="GeneralWeierstrassCubic"> <h6 id="definition_3">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, then <a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski locally</a> 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>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cubic+curve">cubic curve</a> is a solution in the corresponding <a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> to an <a class="existingWikiWord" href="/nlab/show/equation">equation</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>y</mi> <mn>2</mn></msup><mo>+</mo><msub><mi>a</mi> <mn>1</mn></msub><mi>x</mi><mi>y</mi><mo>+</mo><msub><mi>a</mi> <mn>3</mn></msub><mi>y</mi><mo>=</mo><msup><mi>x</mi> <mn>3</mn></msup><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msub><mi>a</mi> <mn>4</mn></msub><mi>x</mi><mo>+</mo><msub><mi>a</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex"> y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 </annotation></semantics></math></div> <p>for <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">a_1, a_2, a_3, a_4, a_6</annotation></semantics></math>. This is called the <em><a class="existingWikiWord" href="/nlab/show/Weierstrass+equation">Weierstrass equation</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Much of the literature on elliptic curves considers def. <a class="maruku-ref" href="#GeneralWeierstrassCubic"></a> for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed field</a>, in which case there is no need to pass to a cover. But for the true global discussion necessary for the <a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a> one needs the full generality.</p> </div> <p>The <a class="existingWikiWord" href="/nlab/show/non-singular+algebraic+variety">non-singular</a> such solutions are the elliptic curves over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. Non-singularity is embodied in coordinates as follows.</p> <div class="num_defn" id="DiscriminantAndJInvariant"> <h6 id="definition_4">Definition</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>b</mi> <mn>2</mn></msub></mtd> <mtd><mo>≔</mo><msubsup><mi>a</mi> <mn>1</mn> <mn>2</mn></msubsup><mo>+</mo><mn>4</mn><msub><mi>a</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>b</mi> <mn>4</mn></msub></mtd> <mtd><mo>≔</mo><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>3</mn></msub><mo>+</mo><mn>2</mn><msub><mi>a</mi> <mn>4</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>b</mi> <mn>6</mn></msub></mtd> <mtd><mo>≔</mo><msubsup><mi>a</mi> <mn>3</mn> <mn>2</mn></msubsup><mo>+</mo><mn>4</mn><msub><mi>a</mi> <mn>6</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>b</mi> <mn>8</mn></msub></mtd> <mtd><mo>≔</mo><msubsup><mi>a</mi> <mn>1</mn> <mn>2</mn></msubsup><msub><mi>a</mi> <mn>6</mn></msub><mo>−</mo><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>3</mn></msub><msub><mi>a</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><msubsup><mi>a</mi> <mn>3</mn> <mn>2</mn></msubsup><mo>+</mo><mn>4</mn><msub><mi>a</mi> <mn>2</mn></msub><msub><mi>a</mi> <mn>6</mn></msub><mo>−</mo><msubsup><mi>a</mi> <mn>4</mn> <mn>2</mn></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} b_2 & \coloneqq a_1^2 + 4 a_2 \\ b_4 &\coloneqq a_1 a_3 + 2a_4 \\ b_6 & \coloneqq a_3^2 + 4 a_6 \\ b_8 & \coloneqq a_1^2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 + 4 a_2 a_6 - a_4^2 \end{aligned} </annotation></semantics></math></div> <p>and in terms of these</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>c</mi> <mn>4</mn></msub></mtd> <mtd><mo>≔</mo><msubsup><mi>b</mi> <mn>2</mn> <mn>2</mn></msubsup><mo>−</mo><mn>24</mn><msub><mi>b</mi> <mn>4</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>c</mi> <mn>6</mn></msub></mtd> <mtd><mo>≔</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mi>b</mi> <mn>2</mn> <mn>3</mn></msubsup><mo>+</mo><mn>36</mn><msub><mi>b</mi> <mn>2</mn></msub><msub><mi>b</mi> <mn>4</mn></msub><mo>−</mo><mn>216</mn><msub><mi>b</mi> <mn>6</mn></msub></mtd></mtr> <mtr><mtd><mi>Δ</mi></mtd> <mtd><mo>≔</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mi>b</mi> <mn>2</mn> <mn>2</mn></msubsup><msub><mi>b</mi> <mn>8</mn></msub><mo>−</mo><mn>8</mn><msubsup><mi>b</mi> <mn>4</mn> <mn>3</mn></msubsup><mo>−</mo><mn>27</mn><msubsup><mi>b</mi> <mn>6</mn> <mn>2</mn></msubsup><mo>+</mo><mn>9</mn><msub><mi>b</mi> <mn>2</mn></msub><msub><mi>b</mi> <mn>4</mn></msub><msub><mi>b</mi> <mn>6</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} c_4 & \coloneqq b_2^2 - 24 b_4 \\ c_6 & \coloneqq - b_2^3 + 36 b_2 b_4 - 216 b_6 \\ \Delta &\coloneqq - b_2^2 b_8 - 8 b_4^3 - 27 b_6^2 + 9 b_2 b_4 b_6 \end{aligned} \,. </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> is called the <strong><a class="existingWikiWord" href="/nlab/show/discriminant">discriminant</a></strong>.</p> <p>Finally let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>≔</mo><msubsup><mi>c</mi> <mn>4</mn> <mn>3</mn></msubsup><mo stretchy="false">/</mo><mi>Δ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> j \coloneqq c_4^3/\Delta \,, </annotation></semantics></math></div> <p>called the <strong><a class="existingWikiWord" href="/nlab/show/j-invariant">j-invariant</a></strong>.</p> </div> <div class="num_remark" id="RelationBetweenDiscriminantAndTheModularInvariants"> <h6 id="remark_3">Remark</h6> <p>In def. <a class="maruku-ref" href="#DiscriminantAndJInvariant"></a> the discriminant satisfies the relation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1728</mn><mi>Δ</mi><mo>=</mo><msubsup><mi>c</mi> <mn>4</mn> <mn>3</mn></msubsup><mo>−</mo><msubsup><mi>c</mi> <mn>6</mn> <mn>2</mn></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 1728 \Delta = c_4^3 - c_6^2 \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>Over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">R = \mathbb{C}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> the quantities <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">c_4</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">c_6</annotation></semantics></math> in def. <a class="maruku-ref" href="#DiscriminantAndJInvariant"></a> are proportional to the <a class="existingWikiWord" href="/nlab/show/modular+forms">modular forms</a> called the <a class="existingWikiWord" href="/nlab/show/Eisenstein+series">Eisenstein series</a> (see there) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">G_4</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">G_6</annotation></semantics></math>.</p> </div> <h5 id="EllipticCurvesNodalCurvesCuspidalCurves">Elliptic curves, Nodal curves, Cuspidal curves</h5> <p>The following is a definition if one takes the coordinate-description as fundamental. If one takes the more abstract characterization of def. <a class="maruku-ref" href="#EllipticCurve"></a> as fundamental then the following is a proposition.</p> <div class="num_defn"> <h6 id="definitionproposition">Definition/Proposition</h6> <p>A solution to the Weierstrass cubic, def. <a class="maruku-ref" href="#GeneralWeierstrassCubic"></a>, with modular invariants <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">c_4</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">c_6</annotation></semantics></math> and discriminant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> according to def. <a class="maruku-ref" href="#DiscriminantAndJInvariant"></a> is</p> <ol> <li> <p>an <em>elliptic curve</em> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta \neq 0</annotation></semantics></math>;</p> </li> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/nodal+curve">nodal curve</a></em> or <em><a class="existingWikiWord" href="/nlab/show/cubic+curve">cubic curve</a> with nodal <a class="existingWikiWord" href="/nlab/show/singular+point+of+an+algebraic+variety">singularity</a></em> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta = 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>4</mn></msub><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c_4 \neq 0</annotation></semantics></math>;</p> </li> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/cuspidal+curve">cuspidal curve</a></em> or <em><a class="existingWikiWord" href="/nlab/show/cubic+curve">cubic curve</a> with <a class="existingWikiWord" href="/nlab/show/cusp">cusp</a> <a class="existingWikiWord" href="/nlab/show/singular+point+of+an+algebraic+variety">singularity</a></em> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta = 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>4</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c_4 = 0</annotation></semantics></math> (which by remark <a class="maruku-ref" href="#RelationBetweenDiscriminantAndTheModularInvariants"></a> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>4</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c_4 = 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>6</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c_6 = 0</annotation></semantics></math>)</p> </li> </ol> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>Adding the nodal curve to the <a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a> yields its <a class="existingWikiWord" href="/nlab/show/compactification">compactification</a>, and the <a class="existingWikiWord" href="/nlab/show/formal+neighbourhood">formal neighbourhood</a> of the nodal curve in that compactification is known as the <em><a class="existingWikiWord" href="/nlab/show/Tate+curve">Tate curve</a></em>.</p> </div> <h5 id="localization_at_2_and_3">Localization at 2 and 3</h5> <div class="num_prop" id="WeierstrassEquationLocalAt2And3"> <h6 id="proposition">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math> is invertible in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> (is a <a class="existingWikiWord" href="/nlab/show/unit">unit</a> ), and hence generally over the <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">localization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R[\frac{1}{2}]</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> at 2, the general <a class="existingWikiWord" href="/nlab/show/Weierstrass+equation">Weierstrass equation</a>, def. <a class="maruku-ref" href="#GeneralWeierstrassCubic"></a>, is equivalent, to the equation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mi>x</mi> <mn>3</mn></msup><mo>+</mo><msub><mi>b</mi> <mn>2</mn></msub><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><mn>2</mn><msub><mi>b</mi> <mn>4</mn></msub><mi>x</mi><mo>+</mo><msub><mi>b</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex"> y^2 = 4 x^3 + b_2 x^2 + 2 b_4 x + b_6 </annotation></semantics></math></div> <p>with the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> identified as in def. <a class="maruku-ref" href="#DiscriminantAndJInvariant"></a>.</p> <p>If moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math> is also invertible in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, hence generally over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">[</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R[\frac{1}{2}, \frac{1}{3}]</annotation></semantics></math> then this equation is equivalent to just</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>x</mi> <mn>3</mn></msup><mo>−</mo><mn>27</mn><msub><mi>c</mi> <mn>4</mn></msub><mi>x</mi><mo>−</mo><mn>54</mn><msub><mi>c</mi> <mn>6</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> y^2 = x^3 - 27 c_4 x - 54 c_6 \,. </annotation></semantics></math></div></div> <h3 id="OverTheComplexNumbers">Over the complex numbers</h3> <h4 id="in_terms_of_algebraic_geometry">In terms of algebraic geometry</h4> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>If the <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">R= \mathbb{C}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, then complex tori are indeed the solutions to the <a class="existingWikiWord" href="/nlab/show/Weierstrass+equation">Weierstrass equation</a> as in prop. <a class="maruku-ref" href="#WeierstrassEquationLocalAt2And3"></a>, parameterized by a <a class="existingWikiWord" href="/nlab/show/torus">torus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><mi>ℂ</mi><mo stretchy="false">/</mo><mi>Λ</mi></mrow><annotation encoding="application/x-tex">z \in \mathbb{C}/\Lambda</annotation></semantics></math> (as discussed in the section <a href="#InTermsOfCpomplexGeometry">in terms of complex geometry</a>) via the <a class="existingWikiWord" href="/nlab/show/Weierstrass+elliptic+function">Weierstrass elliptic function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>℘</mi></mrow><annotation encoding="application/x-tex">\wp</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mi>℘</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo><mi>y</mi><mo>=</mo><mi>℘</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x = \wp(z), y = \wp'(z), )</annotation></semantics></math> in the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>℘</mi><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>℘</mi><mo stretchy="false">(</mo><mi>z</mi><msup><mo stretchy="false">)</mo> <mn>3</mn></msup><mo>−</mo><msub><mi>g</mi> <mn>2</mn></msub><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>g</mi> <mn>3</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \wp'(z)^2 + 4 \wp(z)^3 - g_2 p(z) - g_3 \,. </annotation></semantics></math></div></div> <p>See e.g. (<a href="#Hain08">Hain 08, section 5</a>) on how complex elliptic curves are expressed in this algebraic geometric fashion.</p> <h4 id="InTermsOfCpomplexGeometry">In terms of complex geometry</h4> <div class="num_prop" id="CharacterizationOverC"> <h6 id="proposition_2">Proposition</h6> <p>An <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a>, def. <a class="maruku-ref" href="#EllipticCurve"></a>, over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> is equivalently</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</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> of <a class="existingWikiWord" href="/nlab/show/genus">genus</a> 1 with a base point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \in X</annotation></semantics></math> (e.g.<a href="#Hain08">Hain 08, def. 1.1</a>)</p> </li> <li> <p>a quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">/</mo><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}/\Lambda</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/lattice">lattice</a> in <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>;</p> </li> <li> <p>a compact complex <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> of dimension 1.</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/smooth+scheme">smooth</a> <a class="existingWikiWord" href="/nlab/show/algebraic+curve">algebraic curve</a> of degree 3 in <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">\mathcal{P}</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>From the second definition it follows that to study the <a class="existingWikiWord" href="/nlab/show/moduli+space+of+elliptic+curves">moduli space of elliptic curves</a> it suffices to study the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of <a class="existingWikiWord" href="/nlab/show/lattices">lattices</a> in <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>.</p> </div> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/framed+elliptic+curve">framed elliptic curve</a></strong> is an elliptic curve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,P)</annotation></semantics></math> in the sense of the first item in prop. <a class="maruku-ref" href="#CharacterizationOverC"></a>, together with an <a class="existingWikiWord" href="/nlab/show/ordering">ordered</a> <a class="existingWikiWord" href="/nlab/show/basis">basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_1(X, \mathbb{Z})</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(a \cdot b) = 1 </annotation></semantics></math></p> <p>For <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> a natural number, a <strong><a class="existingWikiWord" href="/nlab/show/level+n-structure+on+an+elliptic+curve">level n-structure on an elliptic curve</a></strong> over the complex numbers is similar data but with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> only in the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/n\mathbb{Z}</annotation></semantics></math>.</p> <p>A <strong>framed lattice</strong> in <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> is a lattice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> together with an ordered basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>λ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>λ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\lambda_1, \lambda_2)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Im</mi><mo stretchy="false">(</mo><msub><mi>λ</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><msub><mi>λ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Im(\lambda_2/\lambda_1) \gt 0</annotation></semantics></math>.</p> </div> <p>Hence a framed elliptic curve is the quotient of the complex plane by a lattice <em>together</em> with the information on how this quotient was obtained. This is useful for describing the <a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a> over the complex numbers.</p> <h3 id="OverTheRationalNumbers">Over the rational numbers</h3> <p>Over the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>: <a href="http://www.sagemath.org/doc/reference/plane_curves/sage/schemes/elliptic_curves/ell_rational_field.html">Sagemath: Elliptic curves over the rational numbers</a></p> <h3 id="OverpAdics">Over the <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>-adic integers 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>-adic numbers</h3> <p>Over the <a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a>, see (<a href="#Conrad07">Conrad 07</a>).</p> <p>Over the <a class="existingWikiWord" href="/nlab/show/p-adic+numbers">p-adic numbers</a>, see (<a href="#Winter11">Winter 11</a>).</p> <h2 id="level_structures">Level structures</h2> <p>In the case over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> an elliptic curve <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> is equivalently the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of the <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a> by a framed <a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>. If here one remembers the structure given by that framed lattice, this means equivalently to remember an ordered basis</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (a_1, a_2)\in H_1(\Sigma, \mathbb{Z}) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> group 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> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>.</p> <p>If here one replaces the integers by a <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/n\mathbb{Z}</annotation></semantics></math> then one obtains what is called a <em><a class="existingWikiWord" href="/nlab/show/level+structure+on+an+elliptic+curve">level-n structure on an elliptic curve</a></em>. Level-<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> structures on elliptic curves may also be defined over general rings.</p> <p>These structures are useful in that the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <a class="existingWikiWord" href="/nlab/show/elliptic+curves+with+level-n+structure">elliptic curves with level-n structure</a> (a <em><a class="existingWikiWord" href="/nlab/show/modular+curve">modular curve</a></em> in the case over the complex numbers) provides a finite <a class="existingWikiWord" href="/nlab/show/covering">covering</a> of the full <a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a>.</p> <h2 id="group_structure">Group structure</h2> <p>A vital reason for the importance of elliptic curves in number theory, arithmetic geometry, and elsewhere is their group structure. This group structure is present by fiat in Definition <a class="maruku-ref" href="#DefinitionEllipticCurve"></a>, and can be constructed given a Weierstrass equation.</p> <p>The <a class="existingWikiWord" href="/nlab/show/torsion+element">torsion elements</a> of this group structure when the elliptic curve is defined over a <a class="existingWikiWord" href="/nlab/show/number+field">number field</a> have good Galois-theoretic properties. See <a class="existingWikiWord" href="/nlab/show/torsion+points+of+an+elliptic+curve">torsion points of an elliptic curve</a> for more.</p> <h2 id="constructions">Constructions</h2> <h3 id="formal_group_law">Formal group law</h3> <p>Given an <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>Spec</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">E \to Spec R</annotation></semantics></math>, we get a <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat E</annotation></semantics></math> by completing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> along its identity <a class="existingWikiWord" href="/nlab/show/section">section</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_0</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>E</mi><mo>→</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>σ</mi> <mn>0</mn></msub></mrow></mover><mi>E</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> E \to Spec(R) \stackrel{\sigma_0}{\to} E \,, </annotation></semantics></math></div> <p>we get a <a class="existingWikiWord" href="/nlab/show/ringed+space">ringed space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo>,</mo><msub><mover><mi>O</mi><mo stretchy="false">^</mo></mover> <mrow><mi>E</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\hat E, \hat O_{E,0})</annotation></semantics></math></p> <div class="num_example"> <h6 id="example">Example</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/field">field</a> <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>, then the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>O</mi><mo stretchy="false">^</mo></mover> <mrow><mi>E</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>≃</mo><mi>k</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\hat O_{E,0} \simeq k[ [z] ]</annotation></semantics></math></p> <p>then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>O</mi><mo stretchy="false">^</mo></mover> <mrow><mi>E</mi><mo>×</mo><mi>E</mi><mo>,</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><mo>≃</mo><msub><mover><mi>O</mi><mo stretchy="false">^</mo></mover> <mrow><mi>E</mi><mo>,</mo><mn>0</mn></mrow></msub><msub><mover><mo>⊗</mo><mo stretchy="false">^</mo></mover> <mi>k</mi></msub><msub><mover><mi>O</mi><mo stretchy="false">^</mo></mover> <mrow><mi>E</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>≃</mo><mi>k</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \hat O_{E \times E, (0,0)} \simeq \hat O_{E,0} \hat \otimes_k \hat O_{E,0} \simeq k[[x,y]] </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_2">Example</h6> <p><strong>(Jacobi quartics)</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>δ</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><mi>ϵ</mi><msup><mi>x</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex"> y^2 = 1- 2 \delta x^2 + \epsilon x^4 </annotation></semantics></math></div> <p>defines <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>Y</mi> <mi>Z</mi></msub><mo>,</mo><mi>ϵ</mi><mo>,</mo><mi>δ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R = \mathbb{Z}[Y_Z,\epsilon, \delta]</annotation></semantics></math>.</p> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a> is <strong>Euler’s formal group law</strong></p> <div class="maruku-equation" id="eq:EulerFormalGroupLaw"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mfrac><mrow><mi>x</mi><msqrt><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>δ</mi><msup><mi>y</mi> <mn>2</mn></msup><mo>+</mo><mi>ϵ</mi><msup><mi>y</mi> <mn>4</mn></msup></mrow></msqrt><mo>+</mo><mi>y</mi><msqrt><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>δ</mi><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><mi>ϵ</mi><msup><mi>x</mi> <mn>4</mn></msup></mrow></msqrt></mrow><mrow><mn>1</mn><mo>−</mo><mi>ϵ</mi><msup><mi>x</mi> <mn>2</mn></msup><msup><mi>y</mi> <mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> f(x,y) \;=\; \frac{ x{\sqrt{1- 2 \delta y^2 + \epsilon y^4}} + y{\sqrt{1- 2 \delta x^2 + \epsilon x^4}} } {1- \epsilon x^2 y^2} </annotation></semantics></math></div> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>≔</mo><mi>ϵ</mi><mo stretchy="false">(</mo><msup><mi>δ</mi> <mn>2</mn></msup><mo>−</mo><mi>ϵ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta \coloneqq \epsilon(\delta^2 - \epsilon)^2 \neq 0</annotation></semantics></math> then this is a non-trivial elliptic curve.</p> </div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta = 0</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>G</mi> <mi>m</mi></msub><mo>,</mo><msub><mi>G</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">f(x,y) \simeq G_m, G_a</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/additive+group">additive</a> or <a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative</a> formal group law corresponding to <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> and <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> <a class="existingWikiWord" href="/nlab/show/KU">KU</a>, respectively).</p> <h3 id="elliptic_cohomology_2">Elliptic cohomology</h3> <p>Elliptic curves, via their <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a>s, give the name to <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a> theories.</p> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A+Survey+of+Elliptic+Cohomology+-+formal+groups+and+cohomology">A Survey of Elliptic Cohomology - formal groups and cohomology</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tate+curve">Tate curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Szpiro%27s+conjecture">Szpiro's conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cubic+curve">cubic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+form">modular form</a>, <a class="existingWikiWord" href="/nlab/show/Jacobi+form">Jacobi form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+modular+form">topological modular form</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modularity+theorem">modularity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> </ul> <h2 id="references">References</h2> <p>Classical accounts of the general case include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Rapoport">Michael Rapoport</a>, <em>Les Schémas de Modules de Courbes Elliptiques</em> In: Deligne P., Kuijk W. (eds) <em>Modular Functions of One Variable II</em>, Lecture Notes in Mathematics, vol 349. Springer (1973) (<a href="https://doi.org/10.1007/978-3-540-37855-6_4">doi:10.1007/978-3-540-37855-6_4</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <em>Courbes Elliptiques: Formulaire (d’apres J. Tate)</em> (<a href="http://modular.math.washington.edu/Tables/antwerp/deligne/">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicholas+Katz">Nicholas Katz</a>, <a class="existingWikiWord" href="/nlab/show/Barry+Mazur">Barry Mazur</a>, <em>Arithmetic moduli of elliptic curves, Annals of Mathematics Studies</em>, vol. 108, Princeton University Press, 1985 (<a href="https://www.jstor.org/stable/j.ctt1b9s05p">jstor:j.ctt1b9s05p</a>)</p> </li> </ul> <p>Introductory lecture notes for elliptic curves over the complex numbers include</p> <ul> <li id="Hain08"> <p>Richard Hain, <em>Lectures on Moduli Spaces of Elliptic Curves</em> (<a href="http://arxiv.org/abs/0812.1803">arXiv:0812.1803</a>)</p> </li> <li id="Neeman07"> <p><a class="existingWikiWord" href="/nlab/show/Amnon+Neeman">Amnon Neeman</a>, section 1.2 of <em>Algebraic and analytic geometry</em>, London Math. Soc. Lec. Note Series <strong>345</strong>, 2007 (<a href="http://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/algebraic-and-analytic-geometry">publisher</a>)</p> </li> </ul> <p>and for the general case:</p> <ul> <li id="QuickIntro"> <p><em>A quick introduction to elliptic curves</em> (<a href="http://people.reed.edu/~jerry/311/ecintro.pdf">pdf</a>)</p> </li> <li> <p>R. Sujatha, <em>Elliptic Curves & Number Theory</em> (<a class="existingWikiWord" href="/nlab/files/SujahtaElliptic.pdf" title="pdf">pdf</a>)</p> </li> <li id="Sutherland21"> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Sutherland">Andrew Sutherland</a>, <em>Elliptic curves</em>, 2021 (<a href="https://ocw.mit.edu/courses/18-783-elliptic-curves-spring-2021/">web</a>)</p> </li> </ul> <p>See also</p> <ul> <li id="AndoHopkinsStrickland01"> <p><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, appendix B of: <em>Elliptic spectra, the Witten genus, and the theorem of the cube</em>, Inventiones Mathematicae, 146:595–687, 2001, (<a href="http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf">pdf</a>, <a href="https://doi.org/10.1007/s002220100175">doi:10.1007/s002220100175</a>)</p> <p>(review of general elliptic curves in the context of <a class="existingWikiWord" href="/nlab/show/elliptic+genera">elliptic genera</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+spectra">elliptic spectra</a> and the <a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a>)</p> </li> </ul> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a> of elliptic curves:</p> <ul> <li> <p>Antonia W. Bluher, <em>Formal groups, elliptic curves, and some theorems of Couveignes</em>, in: J.P. Buhler (eds.) <em>Algorithmic Number Theory</em> ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer 1998 (<a href="https://arxiv.org/abs/math/9708215">arXiv:math/9708215</a>, <a href="https://doi.org/10.1007/BFb0054887">doi:10.1007/BFb0054887</a>)</p> </li> <li id="Friedl17"> <p>Stefan Friedl, <em>An elementary proof of the group law for elliptic curves</em> (<a href="https://arxiv.org/abs/1710.00214">arXiv:1710.00214</a>)</p> </li> </ul> <p>For more along these lines see also at <em><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></em>.</p> <p>In the context of <a class="existingWikiWord" href="/nlab/show/elliptic+fibrations">elliptic fibrations</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Rick+Miranda">Rick Miranda</a>, <em>The basic theory of elliptic surfaces</em>, lecture notes 1988 (<a href="http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf">pdf</a>)</li> </ul> <p>A general textbook account is</p> <ul> <li id="Silverman09"><a class="existingWikiWord" href="/nlab/show/Joseph+Silverman">Joseph Silverman</a>, <em>The arithmetic of elliptic curves</em>, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094 (2010i:11005)</li> </ul> <p>Discussion over the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> includes</p> <ul> <li>Sagemath <em><a href="http://www.sagemath.org/doc/reference/plane_curves/sage/schemes/elliptic_curves/ell_rational_field.html">Elliptic curves over the rational numbers</a></em></li> </ul> <p>Discussion of elliptic curves over the <a class="existingWikiWord" href="/nlab/show/p-adic+numbers">p-adic numbers</a> includes</p> <ul> <li id="Conrad07"> <p><a class="existingWikiWord" href="/nlab/show/Brian+Conrad">Brian Conrad</a>, <em>Arithmetic moduli of generalized elliptic curves</em>, J. Inst. Math. Jussieu 6 (2007), no. 2, 209-278. (<a href="http://math.stanford.edu/~conrad/papers/kmpaper.pdf">pdf</a>)</p> </li> <li id="Winter11"> <p>Rosa Winter, <em>Elliptic curves over <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></em>, 2011 (<a href="http://www.math.leidenuniv.nl/scripties/BachWinter.pdf">pdf</a>)</p> </li> </ul> <p>See also</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/A+Survey+of+Elliptic+Cohomology+-+elliptic+curves">A Survey of Elliptic Cohomology - elliptic curves</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 27, 2023 at 11:50:40. 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