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<title>Automorphic L-function</title>

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<span id="openzim-page-title" class="mw-page-title-main">Automorphic <i>L</i>-function</span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <b>automorphic <i>L</i>-function</b> is a function <i>L</i>(<i>s</i>,π,<i>r</i>) of a complex variable <i>s</i>, associated to an <a href="Automorphic_representation" class="mw-redirect" title="Automorphic representation">automorphic representation</a> π of a <a href="Reductive_group" title="Reductive group">reductive group</a> <i>G</i> over a <a href="Global_field" title="Global field">global field</a> and a finite-dimensional complex representation <i>r</i> of the <a href="Langlands_dual_group" title="Langlands dual group">Langlands dual group</a> <sup><i>L</i></sup><i>G</i> of <i>G</i>, generalizing the <a href="Dirichlet_L-series" class="mw-redirect" title="Dirichlet L-series">Dirichlet L-series</a> of a <a href="Dirichlet_character" title="Dirichlet character">Dirichlet character</a> and the <a href="Mellin_transform" title="Mellin transform">Mellin transform</a> of a <a href="Modular_form" title="Modular form">modular form</a>. They were introduced by <a href="Robert_Langlands" title="Robert Langlands">Langlands</a> (<a href="#CITEREFLanglands1967">1967</a>, <a href="#CITEREFLanglands1970">1970</a>, <a href="#CITEREFLanglands1971">1971</a>).

</p><p><a href="#CITEREFBorel1979">Borel (1979)</a> and <a href="#CITEREFArthurGelbart1991">Arthur & Gelbart (1991)</a> gave surveys of automorphic L-functions.

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

<p>Automorphic <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 L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span>-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases).

</p><p>The L-function <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 L(s,\pi ,r)}">

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<annotation encoding="application/x-tex">{\displaystyle L(s,\pi ,r)}</annotation>

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</math></span><img src="./831ca56d99245686bc1667c45fde73e365562592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.931ex; height:2.843ex;" alt="{\displaystyle L(s,\pi ,r)}" loading="lazy"></span> should be a product over the places <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 v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> of <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 F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> of local <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 L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</p><p><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 L(s,\pi ,r)=\prod _{v}L(s,\pi _{v},r_{v})}">

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<annotation encoding="application/x-tex">{\displaystyle L(s,\pi ,r)=\prod _{v}L(s,\pi _{v},r_{v})}</annotation>

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</math></span><img src="./08f3f23545c37c7d49635f3e3d7b1cd190bf3df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.369ex; height:5.509ex;" alt="{\displaystyle L(s,\pi ,r)=\prod _{v}L(s,\pi _{v},r_{v})}" loading="lazy"></span>

</p><p>Here the <a href="Automorphic_representation" class="mw-redirect" title="Automorphic representation">automorphic representation</a> <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 \pi =\otimes \pi _{v}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi =\otimes \pi _{v}}</annotation>

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</math></span><img src="./98cea6a78eeb514e847a1d4aa9b731eee8ecaf09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.593ex; height:2.343ex;" alt="{\displaystyle \pi =\otimes \pi _{v}}" loading="lazy"></span> is a tensor product of the representations <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 \pi _{v}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{v}}</annotation>

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</math></span><img src="./f21165a2b8dd291d122fe5b30a9625e978f671e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.355ex; height:2.009ex;" alt="{\displaystyle \pi _{v}}" loading="lazy"></span> of local groups.

</p><p>The L-function is expected to have an analytic continuation as a meromorphic function of all complex <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 s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span>, and satisfy a functional equation

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<annotation encoding="application/x-tex">{\displaystyle L(s,\pi ,r)=\epsilon (s,\pi ,r)L(1-s,\pi ,r^{\lor })}</annotation>

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</math></span><img src="./92ffac1cbfe712561c19ca0a09cfd0ad18764927.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.584ex; height:3.009ex;" alt="{\displaystyle L(s,\pi ,r)=\epsilon (s,\pi ,r)L(1-s,\pi ,r^{\lor })}" loading="lazy"></span>

</p><p>where the factor <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 \epsilon (s,\pi ,r)}">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon (s,\pi ,r)}</annotation>

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</math></span><img src="./0a20ec674bf0c077329ae644671fae48bf872a20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.292ex; height:2.843ex;" alt="{\displaystyle \epsilon (s,\pi ,r)}" loading="lazy"></span> is a product of "local constants"

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<annotation encoding="application/x-tex">{\displaystyle \epsilon (s,\pi ,r)=\prod _{v}\epsilon (s,\pi _{v},r_{v},\psi _{v})}</annotation>

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</math></span><img src="./b28eaa4a687117837b68f000a44b4c5c501a457e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.669ex; height:5.509ex;" alt="{\displaystyle \epsilon (s,\pi ,r)=\prod _{v}\epsilon (s,\pi _{v},r_{v},\psi _{v})}" loading="lazy"></span>

</p><p>almost all of which are 1.

</p>

<div class="mw-heading mw-heading2"><h2 id="General_linear_groups">General linear groups</h2></div>

<p><a href="#CITEREFGodementJacquet1972">Godement & Jacquet (1972)</a> constructed the automorphic L-functions for general linear groups with <i>r</i> the standard representation (so-called <a href="Standard_L-function" title="Standard L-function">standard L-functions</a>) and verified analytic continuation and the functional equation, by using a generalization of the method in <a href="Tate's_thesis" title="Tate's thesis">Tate's thesis</a>. Ubiquitous in the Langlands Program are <a href="Rankin%E2%80%93Selberg_method" title="Rankin–Selberg method">Rankin-Selberg</a> products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the <a href="Langlands%E2%80%93Shahidi_method" title="Langlands–Shahidi method">Langlands–Shahidi method</a>.

</p><p>In general, the <a href="Langlands_functoriality" class="mw-redirect" title="Langlands functoriality">Langlands functoriality</a> conjectures imply that automorphic L-functions of a connected <a href="Reductive_group" title="Reductive group">reductive group</a> are equal to products of automorphic L-functions of general linear groups. A proof of Langlands functoriality would also lead towards a thorough understanding of the analytic properties of automorphic L-functions.

</p>

<ul><li><a href="Grand_Riemann_hypothesis" title="Grand Riemann hypothesis">Grand Riemann hypothesis</a></li></ul>

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

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</style><cite id="CITEREFArthurGelbart1991" class="citation cs2">Arthur, James; <a href="Stephen_Gelbart" title="Stephen Gelbart">Gelbart, Stephen</a> (1991), "Lectures on automorphic L-functions", in Coates, John; Taylor, M. J. (eds.), <a rel="nofollow" class="external text" href="http://www.claymath.org/library/cw/arthur/pdf/automorphic-L.pdf"><i>L-functions and arithmetic (Durham, 1989)</i></a> <span class="cs1-format">(PDF)</span>, London Math. Soc. Lecture Note Ser., vol. 153, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, pp. <span class="nowrap">1–</span>59, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511526053.003">10.1017/CBO9780511526053.003</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-38619-7</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1110389">1110389</a></cite></li>

<li><cite id="CITEREFBorel1979" class="citation cs2"><a href="Armand_Borel" title="Armand Borel">Borel, Armand</a> (1979), "Automorphic L-functions", in <a href="Armand_Borel" title="Armand Borel">Borel, Armand</a>; Casselman, W. (eds.), <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.ams.org/publications/online-books/pspum332-index"><i>Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2</i></a></span>, vol. XXXIII, Providence, R.I.: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, pp. <span class="nowrap">27–</span>61, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fpspum%2F033.2%2F546608">10.1090/pspum/033.2/546608</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-1437-6</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0546608">0546608</a></cite></li>

<li><cite id="CITEREFCogdellKimMurty2004" class="citation cs2">Cogdell, James W.; Kim, Henry H.; Murty, Maruti Ram (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jb3ZCp0-MQsC"><i>Lectures on automorphic L-functions</i></a>, Fields Institute Monographs, vol. 20, Providence, R.I.: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-3516-6</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2071722">2071722</a></cite></li>

<li><cite id="CITEREFGelbartPiatetski-ShapiroRallis1987" class="citation cs2"><a href="Stephen_Gelbart" title="Stephen Gelbart">Gelbart, Stephen</a>; Piatetski-Shapiro, Ilya; Rallis, Stephen (1987), <i>Explicit Constructions of Automorphic L-Functions</i>, Lecture Notes in Mathematics, vol. 1254, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0078125">10.1007/BFb0078125</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-17848-4</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0892097">0892097</a></cite></li>

<li><cite id="CITEREFGodementJacquet1972" class="citation cs2"><a href="Roger_Godement" title="Roger Godement">Godement, Roger</a>; Jacquet, Hervé (1972), <i>Zeta Functions of Simple Algebras</i>, Lecture Notes in Mathematics, vol. 260, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0070263">10.1007/BFb0070263</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-05797-0</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0342495">0342495</a></cite></li>

<li><cite id="CITEREFJacquetPiatetski-ShapiroShalika1983" class="citation cs2">Jacquet, H.; Piatetski-Shapiro, I. I.; Shalika, J. A. (1983), "Rankin-Selberg Convolutions", <i>Amer. J. Math.</i>, <b>105</b> (2): <span class="nowrap">367–</span>464, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2374264">10.2307/2374264</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2374264">2374264</a></cite></li>

<li><cite id="CITEREFLanglands1967" class="citation cs2">Langlands, Robert (1967), <a rel="nofollow" class="external text" href="http://publications.ias.edu/rpl/section/21"><i>Letter to Prof. Weil</i></a></cite></li>

<li><cite id="CITEREFLanglands1970" class="citation cs2">Langlands, R. P. (1970), "Problems in the theory of automorphic forms", <a rel="nofollow" class="external text" href="http://publications.ias.edu/rpl/section/21"><i>Lectures in modern analysis and applications, III</i></a>, Lecture Notes in Math, vol. 170, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, pp. <span class="nowrap">18–</span>61, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0079065">10.1007/BFb0079065</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-05284-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0302614">0302614</a></cite></li>

<li><cite id="CITEREFLanglands1971" class="citation cs2">Langlands, Robert P. (1971) [1967], <a rel="nofollow" class="external text" href="http://publications.ias.edu/rpl/paper/37"><i>Euler products</i></a>, Yale University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-300-01395-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0419366">0419366</a></cite></li>

<li><cite id="CITEREFShahidi1981" class="citation cs2">Shahidi, F. (1981), "On certain "L"-functions", <i>Amer. J. Math.</i>, <b>103</b> (2): <span class="nowrap">297–</span>355, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2374219">10.2307/2374219</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2374219">2374219</a></cite></li></ul>

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</style><div id="L-functions_in_number_theory183" style="font-size:114%;margin:0 4em"><a href="L-function" title="L-function"><i>L</i>-functions</a> in <a href="Number_theory" title="Number theory">number theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analytic examples</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a></li>

<li><a href="Dirichlet_L-function" title="Dirichlet L-function">Dirichlet <i>L</i>-functions</a></li>

<li><a href="L-function_with_Gr%C3%B6ssencharakter" class="mw-redirect" title="L-function with Grössencharakter"><i>L</i>-functions of Hecke characters</a></li>

<li><a href="Selberg_class" title="Selberg class">Selberg class</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic examples</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dedekind_zeta_function" title="Dedekind zeta function">Dedekind zeta functions</a></li>

<li><a href="Artin_L-function" title="Artin L-function">Artin <i>L</i>-functions</a></li>

<li><a href="Hasse%E2%80%93Weil_zeta_function" title="Hasse–Weil zeta function">Hasse–Weil <i>L</i>-functions</a></li>

<li><a href="Motivic_L-function" title="Motivic L-function">Motivic <i>L</i>-functions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Class_number_formula" title="Class number formula">Analytic class number formula</a></li>

<li><a href="Riemann%E2%80%93von_Mangoldt_formula" title="Riemann–von Mangoldt formula">Riemann–von Mangoldt formula</a></li>

<li><a href="Weil_conjectures" title="Weil conjectures">Weil conjectures</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analytic conjectures</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li>

<li><a href="Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">Generalized Riemann hypothesis</a></li>

<li><a href="Lindel%C3%B6f_hypothesis" title="Lindelöf hypothesis">Lindelöf hypothesis</a></li>

<li><a href="Ramanujan%E2%80%93Petersson_conjecture" title="Ramanujan–Petersson conjecture">Ramanujan–Petersson conjecture</a></li>

<li><a href="Artin_conjecture_(L-functions)" class="mw-redirect" title="Artin conjecture (L-functions)">Artin conjecture</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic conjectures</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Birch_and_Swinnerton-Dyer_conjecture" title="Birch and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</a></li>

<li><a href="Special_values_of_L-functions" title="Special values of L-functions">Deligne's conjecture</a></li>

<li><a href="Beilinson_conjectures" class="mw-redirect" title="Beilinson conjectures">Beilinson conjectures</a></li>

<li><a href="Bloch%E2%80%93Kato_conjecture_(L-functions)" class="mw-redirect" title="Bloch–Kato conjecture (L-functions)">Bloch–Kato conjecture</a></li>

<li><a href="Langlands_program" title="Langlands program">Langlands conjecture</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="P-adic_L-function" title="P-adic L-function"><i>p</i>-adic <i>L</i>-functions</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Main_conjecture_of_Iwasawa_theory" title="Main conjecture of Iwasawa theory">Main conjecture of Iwasawa theory</a></li>

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

<li><a href="Euler_system" title="Euler system">Euler system</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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