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<title>Isogeny</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Isogeny</span></span>

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<p>In mathematics, particularly in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, an <b>isogeny</b> is a <a href="Morphism_of_algebraic_varieties" title="Morphism of algebraic varieties">morphism</a> of <a href="Algebraic_group" title="Algebraic group">algebraic groups</a> (also known as group varieties) that is <a href="Surjective_function" title="Surjective function">surjective</a> and has a finite <a href="Kernel_(algebra)" title="Kernel (algebra)">kernel</a>.

</p><p>If the <a href="Group_(mathematics)" title="Group (mathematics)">groups</a> are <a href="Abelian_variety" title="Abelian variety">abelian varieties</a>, then any morphism <span class="texhtml"><i>f</i> : <i>A</i> → <i>B</i></span> of the underlying algebraic varieties which is surjective with finite <a href="Fiber_(mathematics)" title="Fiber (mathematics)">fibres</a> is automatically an isogeny, provided that <span class="texhtml"><i>f</i>(1<sub><i>A</i></sub>) = 1<sub><i>B</i></sub></span>. Such an isogeny <span class="texhtml"><i>f</i></span> then provides a <a href="Group_homomorphism" title="Group homomorphism">group homomorphism</a> between the groups of <span class="texhtml"><i>k</i></span>-valued points of <span class="texhtml"><i>A</i></span> and <span class="texhtml"><i>B</i></span>, for any <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <span class="texhtml"><i>k</i></span> over which <span class="texhtml"><i>f</i></span> is defined.

</p><p>The terms "isogeny" and "isogenous" come from the Greek word ισογενη-ς, meaning "equal in kind or nature". The term "isogeny" was introduced by <a href="Andr%C3%A9_Weil" title="André Weil">Weil</a>; before this, the term "isomorphism" was somewhat confusingly used for what is now called an isogeny.

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<div class="mw-heading mw-heading2"><h2 id="Degree_of_isogeny">Degree of isogeny</h2></div>

<p>Let <span class="texhtml"><i>f</i> : <i>A</i> → <i>B</i></span> be isogeny between two algebraic groups.

This mapping induces a pullback mapping <span class="texhtml"><i>f*</i> : <i>K(B)</i> → <i>K(A)</i></span> between their <a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">rational function fields</a>. Since the mapping is nontrivial, it is a field embedding and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {im} f^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {im} f^{*}}</annotation>

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</math></span><img src="./0a1a1c4fc51f2e1a007cd4aac296a8b7f21d682b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.345ex; height:2.676ex;" alt="{\displaystyle \operatorname {im} f^{*}}" loading="lazy"></span> is a subfield of <span class="texhtml"><i>K(A)</i></span>. The <a href="Degree_of_a_field_extension" title="Degree of a field extension">degree</a> of the extension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(A)/\operatorname {im} f^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle K(A)/\operatorname {im} f^{*}}</annotation>

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</math></span><img src="./71536e835f17eac3e61cf64a8abcd081db52a391.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.512ex; height:2.843ex;" alt="{\displaystyle K(A)/\operatorname {im} f^{*}}" loading="lazy"></span> is called degree of isogeny:

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<annotation encoding="application/x-tex">{\displaystyle \deg f:=[K(A):\operatorname {im} f^{*}]}</annotation>

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</math></span><img src="./284de8dc997d317d2832bab0c41a6c6659168077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.092ex; height:2.843ex;" alt="{\displaystyle \deg f:=[K(A):\operatorname {im} f^{*}]}" loading="lazy"></span></dd></dl>

<p>Properties of degree:

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<annotation encoding="application/x-tex">{\displaystyle f:X\rightarrow Y}</annotation>

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</math></span><img src="./b215af1e965d0595a97ad2b21f7d0cbcf6281303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\rightarrow Y}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:Y\rightarrow Z}">

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<annotation encoding="application/x-tex">{\displaystyle g:Y\rightarrow Z}</annotation>

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</math></span><img src="./2a976067ecdfce0310e7d05b555ea816197f0410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.121ex; height:2.509ex;" alt="{\displaystyle g:Y\rightarrow Z}" loading="lazy"></span> are isogenies of algebraic groups, then: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(g\circ f)=\deg g\cdot \deg f}">

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<annotation encoding="application/x-tex">{\displaystyle \deg(g\circ f)=\deg g\cdot \deg f}</annotation>

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</math></span><img src="./3bb6d49f80ebaabb3bf440210be640c62b145a21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.807ex; height:2.843ex;" alt="{\displaystyle \deg(g\circ f)=\deg g\cdot \deg f}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle char\;K\nmid \deg f}</annotation>

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</math></span><img src="./e8feec2a2e2798c31cfa6d215ffd8284786ece19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.426ex; height:2.843ex;" alt="{\displaystyle char\;K\nmid \deg f}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg f=|\ker \;f|}">

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<annotation encoding="application/x-tex">{\displaystyle \deg f=|\ker \;f|}</annotation>

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</math></span><img src="./3bb3b727428698d7d8270f2a31efd5973cca5bb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.415ex; height:2.843ex;" alt="{\displaystyle \deg f=|\ker \;f|}" loading="lazy"></span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Case_of_abelian_varieties">Case of abelian varieties</h2></div>

<p>For <a href="Abelian_varieties" class="mw-redirect" title="Abelian varieties">abelian varieties</a>, such as <a href="Elliptic_curves" class="mw-redirect" title="Elliptic curves">elliptic curves</a>, this notion can also be formulated as follows:

</p><p>Let <i>E</i><sub>1</sub> and <i>E</i><sub>2</sub> be abelian varieties of the same dimension over a field <i>k</i>. An <b>isogeny</b> between <i>E</i><sub>1</sub> and <i>E</i><sub>2</sub> is a dense morphism <span class="texhtml"><i>f</i> : <i>E</i><sub>1</sub> → <i>E</i><sub>2</sub></span> of varieties that preserves basepoints (i.e. <i>f</i> maps the identity point on <i>E</i><sub>1</sub> to that on <i>E</i><sub>2</sub>).

</p><p>This is equivalent to the above notion, as every dense morphism between two abelian varieties of the same dimension is automatically surjective with finite fibres, and if it preserves identities then it is a homomorphism of groups.

</p><p>Two abelian varieties <i>E</i><sub>1</sub> and <i>E</i><sub>2</sub> are called <b>isogenous</b> if there is an isogeny <span class="texhtml"><i>E</i><sub>1</sub> → <i>E</i><sub>2</sub></span>. This can be shown to be an equivalence relation; in the case of elliptic curves, symmetry is due to the existence of the <a href="Dual_isogeny" class="mw-redirect" title="Dual isogeny">dual isogeny</a>. As above, every isogeny induces homomorphisms of the groups of the k-valued points of the abelian varieties.

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<ul><li><a href="Abelian_varieties_up_to_isogeny" class="mw-redirect" title="Abelian varieties up to isogeny">Abelian varieties up to isogeny</a></li>

<li><a href="Selmer_group" title="Selmer group">Selmer group</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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</style><cite id="CITEREFLang1983" class="citation book cs1"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (1983). <i>Abelian Varieties</i>. Springer Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-90875-7</bdi>.</cite></li>

<li><cite id="CITEREFMumford1974" class="citation book cs1"><a href="David_Mumford" title="David Mumford">Mumford, David</a> (1974). <i>Abelian Varieties</i>. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-560528-4</bdi>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-03-31" href="https://en.wikipedia.org/wiki/?title=Isogeny&oldid=1283268168">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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