Statements in group theory which imply deep results in number theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T13:33:01Z http://mathoverflow.net/feeds/question/7712 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7712/statements-in-group-theory-which-imply-deep-results-in-number-theory Statements in group theory which imply deep results in number theory Simon 2009-12-03T20:26:26Z 2010-10-28T14:13:16Z <p>Can we name some examples of theorems in group theory which imply (in a relatively straight-forward way) interesting theorems or phenomena in number theory?</p> <p>Here are two examples I thought of:</p> <p>The existence of Golod-Shafarevich towers of Hilbert class fields follows from an inequality on the dimensions of the first two cohomology groups of the ground field.</p> <p>Iwasawa's theorem on the size of the $p$ part of the class groups in <code>$\mathbb{Z}_p$</code>-extensions follows from studying the structure of <code>$\mathbb{Z}_p[\![T]\!]$</code>-modules.</p> <p>Can you name some others?</p> http://mathoverflow.net/questions/7712/statements-in-group-theory-which-imply-deep-results-in-number-theory/7713#7713 Answer by Ilya Nikokoshev for Statements in group theory which imply deep results in number theory Ilya Nikokoshev 2009-12-03T20:34:02Z 2009-12-03T20:40:21Z <p>I'm sure you omitted this just because it's too classic: <a href="http://en.wikipedia.org/wiki/Solvable%5Fgroup" rel="nofollow">big part of group theory</a> was invented to prove that <em>most roots of unity cannot be constructed by quadratic (or cyclic) extensions</em>. </p> <p>It's still the best, most direct connection between [nt.number-theory] and [gr.group-theory] I know of.</p> <p>For a more "advanced" version of this, do <a href="http://en.wikipedia.org/wiki/Hilbert%27s%5FTheorem%5F90" rel="nofollow">computations of group cohomology</a> count? </p> http://mathoverflow.net/questions/7712/statements-in-group-theory-which-imply-deep-results-in-number-theory/7723#7723 Answer by Rebecca Bellovin for Statements in group theory which imply deep results in number theory Rebecca Bellovin 2009-12-03T22:49:01Z 2009-12-03T22:49:01Z <p>The fact (from class field theory) that ideals become principal in the Hilbert class field follows from the fact that the Verlagerung $V:G^{\text{ab}}\rightarrow H^{\text{ab}}$ is zero if $G$ is any finite group and $H$ is its commutator subgroup.</p> http://mathoverflow.net/questions/7712/statements-in-group-theory-which-imply-deep-results-in-number-theory/7725#7725 Answer by Guillermo Mantilla for Statements in group theory which imply deep results in number theory Guillermo Mantilla 2009-12-03T23:22:35Z 2009-12-03T23:22:35Z <p>I'd say that classification of subgroups of GL$_2(F_p)$ plays a big part in Serre's result about the almost surjectivity of $\ell$-adic Galois representations of CM Elliptic curves. </p> <p>Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. (French) Invent. Math. 15 (1972), no. 4, 259-331.</p> http://mathoverflow.net/questions/7712/statements-in-group-theory-which-imply-deep-results-in-number-theory/7730#7730 Answer by Guillermo Mantilla for Statements in group theory which imply deep results in number theory Guillermo Mantilla 2009-12-04T00:09:03Z 2009-12-04T00:09:03Z <p>The notion of arithmetically equivalent number fields is a good example of a connection between group theory and number theory, see for example: <a href="http://sbseminar.wordpress.com/2007/08/29/zeta-function-relations-and-linearly-equivalent-group-actions/" rel="nofollow">http://sbseminar.wordpress.com/2007/08/29/zeta-function-relations-and-linearly-equivalent-group-actions/</a></p> <p>a couple of specific applications: </p> <p>Lemma: Let $G$ be a finite $p$-group. Any two subgroups of index $p$ are quasi-conjugated. </p> <p>Corollary: Two number fields $K$, $L$ of degree $p$ prime are arithmetically equivalent if and only if $[KL:Q] \neq p^2$ See "A remark about zeta functions of number fields of prime degree" by R. Perlis. </p> <p>Also by doing some basic group theory one can prove that any two arithmetically equivalent number fields of degree less than $7$ must be isomorphic.(This is also proven in a paper by Perlis but I don't remember what paper.) </p> <p>Another result that comes to my mind with this question (totally unrelated to arithmetical equivalence) is that every group of odd order can be realized as a Galois group over Q(odd order theorem plus Shafarevich).</p> http://mathoverflow.net/questions/7712/statements-in-group-theory-which-imply-deep-results-in-number-theory/7763#7763 Answer by Pete L. Clark for Statements in group theory which imply deep results in number theory Pete L. Clark 2009-12-04T07:37:03Z 2009-12-04T07:37:03Z <p>The fundamental theorem of arithmetic (uniqueness of factorization of integers into primes) is an immediate consequence of the Jordan-Holder theorem on uniqueness of composition factors of finite groups. </p> http://mathoverflow.net/questions/7712/statements-in-group-theory-which-imply-deep-results-in-number-theory/7784#7784 Answer by TG for Statements in group theory which imply deep results in number theory TG 2009-12-04T17:01:04Z 2009-12-04T17:01:04Z <p>Brauer's theorem implies meromorphic continuation of Artin L-functions (indeed, I believe that was Brauer's motivation).</p> <p>http://en.wikipedia.org/wiki/Brauer's_theorem_on_induced_characters</p>