Main conjecture of Iwasawa theory

<h2 id="motivation">Motivation</h2>
Iwasawa (1969a) was partly motivated by an analogy with <a href="/facts/Weil_conjectures/JPSk0G5j">Weil's description</a> of the zeta function of an <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> over a <a href="/facts/Finite_field/UkMGJo3m">finite field</a> in terms of eigenvalues of the <a href="/facts/Frobenius_endomorphism/b9hkmc97">Frobenius endomorphism</a> on its <a href="/facts/Jacobian_variety/HFedCxIn">Jacobian variety</a>. In this analogy,

<ul><li>The action of the Frobenius corresponds to the action of the group Γ.</li>
<li>The Jacobian of a curve corresponds to a module X over Γ defined in terms of ideal class groups.</li>
<li>The zeta function of a curve over a finite field corresponds to a p-adic L-function.</li>
<li>Weil's theorem relating the eigenvalues of Frobenius to the zeros of the zeta function of the curve corresponds to Iwasawa's main conjecture relating the action of the <a href="/facts/Iwasawa_algebra/bw69GcN9">Iwasawa algebra</a> on X to zeros of the p-adic zeta function.</li></ul>
<h2 id="history">History</h2>
The main conjecture of Iwasawa theory was formulated as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by Mazur & Wiles (1984) for Q, and for all <a href="/facts/Totally_real_number_field/4rulk9kd">totally real number fields</a> by Wiles (1990). These proofs were modeled upon <a href="/facts/Ken_Ribet/6hU9bzw8">Ken Ribet</a>'s proof of the converse to Herbrand's theorem (the <a href="/facts/Herbrand%E2%80%93Ribet_theorem/nl1vLJeG">Herbrand–Ribet theorem</a>).
<a href="/facts/Karl_Rubin/3xhVe6yE">Karl Rubin</a> found a more elementary proof of the Mazur–Wiles theorem by using <a href="/facts/Thaine%27s_method/LDg5HXbJ">Thaine's method</a> and Kolyvagin's <a href="/facts/Euler_system/PSzqCxp5">Euler systems</a>, described in Lang (1990) and Washington (1997), and later proved other generalizations of the main conjecture for imaginary <a href="/facts/Quadratic_field/dUnSDN4e">quadratic fields</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
In 2014, <a href="/facts/Christopher_Skinner/CFBcb0jn">Christopher Skinner</a> and <a href="/facts/Eric_Urban/F2Wzynxx">Eric Urban</a> proved several cases of the main conjectures for a large class of <a href="/facts/Modular_form/PcnbYR1f">modular forms</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> As a consequence, for a <a href="/facts/Modular_elliptic_curve/7FVv9htb">modular elliptic curve</a> over the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>, they prove that the vanishing of the <a href="/facts/Hasse%E2%80%93Weil_L-function/1THWBhB2">Hasse–Weil L-function</a> L(E, s) of E at s = 1 implies that the p-adic <a href="/facts/Selmer_group/jnT5RIqE">Selmer group</a> of E is infinite. Combined with theorems of <a href="/facts/Benedict_Gross/XS6uAajW">Gross</a>-<a href="/facts/Don_Zagier/8sRgRUUj">Zagier</a> and <a href="/facts/Victor_Kolyvagin/7H6wd82H">Kolyvagin</a>, this gave a conditional proof (on the <a href="/facts/Tate%E2%80%93Shafarevich_conjecture/yCDadEcM">Tate–Shafarevich conjecture</a>) of the conjecture that E has infinitely many rational points if and only if L(E, 1) = 0, a (weak) form of the <a href="/facts/Birch%E2%80%93Swinnerton-Dyer_conjecture/Issh7Yfr">Birch–Swinnerton-Dyer conjecture</a>. These results were used by <a href="/facts/Manjul_Bhargava/DPSrSaXE">Manjul Bhargava</a>, Skinner, and <a href="/facts/Wei_Zhang_(mathematician)/vpkddvyJ">Wei Zhang</a> to prove that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="statement">Statement</h2>
<ul><li>p is a prime number.</li>
<li>Fn is the field Q(ζ) where ζ is a root of unity of order pn+1.</li>
<li>Γ is the largest subgroup of the <a href="/facts/Absolute_Galois_group/fvcDwzWq">absolute Galois group</a> of F∞ isomorphic to the <a href="/facts/P-adic_integers/SDgDbkLF">p-adic integers</a>.</li>
<li>γ is a topological generator of Γ.</li>
<li>Ln is the p-<a href="/facts/Hilbert_class_field/UbGUNu89">Hilbert class field</a> of Fn.</li>
<li>Hn is the Galois group Gal(Ln/Fn), isomorphic to the subgroup of elements of the ideal class group of Fn whose order is a power of p.</li>
<li>H∞ is the <a href="/facts/Inverse_limit/f6QecRUu">inverse limit</a> of the Galois groups Hn.</li>
<li>V is the vector space H∞⊗ZpQp.</li>
<li>ω is the <a href="/facts/Teichm%C3%BCller_character/MtcdHbTo">Teichmüller character</a>.</li>
<li>Vi is the ωi eigenspace of V.</li>
<li>hp(ωi,T) is the characteristic polynomial of γ acting on the vector space Vi.</li>
<li>Lp is the <a href="/facts/P-adic_L_function/9LE4oSTW">p-adic L function</a> with Lp(ωi,1–k) = –Bk(ωi–k)/k, where B is a <a href="/facts/Generalized_Bernoulli_number/RH0mblhs">generalized Bernoulli number</a>.</li>
<li>u is the unique p-adic number satisfying γ(ζ) = ζu for all p-power roots of unity ζ.</li>
<li>Gp is the <a href="/facts/Power_series/vYh6sTps">power series</a> with Gp(ωi,us–1) = Lp(ωi,s).</li></ul>
The main conjecture of Iwasawa theory proved by Mazur and Wiles states that if i is an odd integer not congruent to 1 mod p–1 then the ideals of 
 
 
 
 
 
 Z
 
 
 p
 
 
 [
 [
 T
 ]
 ]
 
 
 {\displaystyle \mathbf {Z} _{p}[[T]]}
 
 generated by hp(ωi,T) and Gp(ω1–i,T) are equal.

<h2 id="notes">Notes</h2>

<h2 id="sources">Sources</h2>

<ul><li>Baker, Matt (2014-03-10), <a href="https://mattbaker.blog/2014/03/10/the-bsd-conjecture-is-true-for-most-elliptic-curves/">"The BSD conjecture is true for most elliptic curves"</a>, Matt Baker's Math Blog, retrieved 2019-02-24</li>
<li>Bhargava, Manjul; Skinner, Christopher; Zhang, Wei (2014-07-07), "A majority of elliptic curves over $\mathbb Q$ satisfy the Birch and Swinnerton-Dyer conjecture", <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1407.1826">1407.1826</a> [<a href="https://arxiv.org/archive/math.NT">math.NT</a>]</li>
<li><a href="/facts/John_Coates_(mathematician)/P7XvMrHY">Coates, John</a>; <a href="/facts/Sujatha_Ramdorai/70rf2Tqu">Sujatha, R.</a> (2006), Cyclotomic Fields and Zeta Values, Springer Monographs in Mathematics, <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-33068-4, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1100.11002">1100.11002</a></li>
<li>Iwasawa, Kenkichi (1964), "On some modules in the theory of cyclotomic fields", Journal of the Mathematical Society of Japan, 16: 42–82, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2969%2Fjmsj%2F01610042">10.2969/jmsj/01610042</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5645">0025-5645</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0215811">0215811</a></li>
<li>Iwasawa, Kenkichi (1969a), "Analogies between number fields and function fields", Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966), Belfer Graduate School of Science, Yeshiva Univ., New York, pp. 203–208, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0255510">0255510</a></li>
<li>Iwasawa, Kenkichi (1969b), "On p-adic L-functions", <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>, Second Series, 89 (1): 198–205, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1970817">10.2307/1970817</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1970817">1970817</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0269627">0269627</a></li>
<li><a href="/facts/Mahesh_Kakde/7pdzfrKa">Kakde, Mahesh</a> (2013), "The main conjecture of Iwasawa theory for totally real fields", <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>, 193 (3): 539–626, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1008.0142">1008.0142</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2013InMat.193..539K">2013InMat.193..539K</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00222-012-0436-x">10.1007/s00222-012-0436-x</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3091976">3091976</a></li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1990), <a href="https://books.google.com/books?isbn=0-387-96671-4">Cyclotomic fields I and II</a>, <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, vol. 121, With an appendix by <a href="/facts/Karl_Rubin/3xhVe6yE">Karl Rubin</a> (Combined 2nd ed.), Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-96671-7, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0704.11038">0704.11038</a></li>
<li><a href="/facts/Yuri_I._Manin/gDlvYGCq">Manin, Yu I.</a>; Panchishkin, A. A. (2007), Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, vol. 49 (Second ed.), <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-20364-3, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0938-0396">0938-0396</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1079.11002">1079.11002</a></li>
<li><a href="/facts/Barry_Mazur/4tGbAnCX">Mazur, Barry</a>; <a href="/facts/Andrew_Wiles/bdc3k3pQ">Wiles, Andrew</a> (1984), "Class fields of abelian extensions of Q", <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>, 76 (2): 179–330, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1984InMat..76..179M">1984InMat..76..179M</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01388599">10.1007/BF01388599</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0742853">0742853</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122576427">122576427</a></li>
<li>Skinner, Christopher; Urban, Eric (2014), "The Iwasawa main conjectures for GL2", <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>, 195 (1): 1–277, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2014InMat.195....1S">2014InMat.195....1S</a>, <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.363.2008">10.1.1.363.2008</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00222-013-0448-1">10.1007/s00222-013-0448-1</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3148103">3148103</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120848645">120848645</a></li>
<li><a href="/facts/Lawrence_C._Washington/SHyW0o8j">Washington, Lawrence C.</a> (1997), <a href="https://books.google.com/books?isbn=0-387-94762-0">Introduction to cyclotomic fields</a>, Graduate Texts in Mathematics, vol. 83 (2nd ed.), Berlin, New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94762-4</li>
<li><a href="/facts/Andrew_Wiles/bdc3k3pQ">Wiles, Andrew</a> (1990), "The Iwasawa conjecture for totally real fields", <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>, Second Series, 131 (3): 493–540, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1971468">10.2307/1971468</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1971468">1971468</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1053488">1053488</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Wiles 1990, Kakde 2013 - Wiles, Andrew (1990), "The Iwasawa conjecture for totally real fields", Annals of Mathematics, Second Series, 131 (3): 493–540, doi:10.2307/1971468, ISSN 0003-486X, JSTOR 1971468, MR 1053488 <a href="https://doi.org/10.2307%2F1971468" target="_blank">https://doi.org/10.2307%2F1971468</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Manin & Panchishkin 2007, p. 246. - Manin, Yu I.; Panchishkin, A. A. (2007), Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, vol. 49 (Second ed.), ISBN 978-3-540-20364-3, ISSN 0938-0396, Zbl 1079.11002 <a href="https://search.worldcat.org/issn/0938-0396" target="_blank">https://search.worldcat.org/issn/0938-0396</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Skinner & Urban 2014, pp. 1–277. - Skinner, Christopher; Urban, Eric (2014), "The Iwasawa main conjectures for GL2", Inventiones Mathematicae, 195 (1): 1–277, Bibcode:2014InMat.195....1S, CiteSeerX 10.1.1.363.2008, doi:10.1007/s00222-013-0448-1, MR 3148103, S2CID 120848645 <a href="https://ui.adsabs.harvard.edu/abs/2014InMat.195....1S" target="_blank">https://ui.adsabs.harvard.edu/abs/2014InMat.195....1S</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Bhargava, Skinner & Zhang 2014. - Bhargava, Manjul; Skinner, Christopher; Zhang, Wei (2014-07-07), "A majority of elliptic curves over $\mathbb Q$ satisfy the Birch and Swinnerton-Dyer conjecture", arXiv:1407.1826 [math.NT] <a href="https://arxiv.org/abs/1407.1826" target="_blank">https://arxiv.org/abs/1407.1826</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Baker 2014. - Baker, Matt (2014-03-10), "The BSD conjecture is true for most elliptic curves", Matt Baker's Math Blog, retrieved 2019-02-24 <a href="https://mattbaker.blog/2014/03/10/the-bsd-conjecture-is-true-for-most-elliptic-curves/" target="_blank">https://mattbaker.blog/2014/03/10/the-bsd-conjecture-is-true-for-most-elliptic-curves/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

Main conjecture of Iwasawa theory open-in-new

Main conjecture of Iwasawa theory