Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T22:52:16Z http://mathoverflow.net/feeds/question/110273 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110273/can-one-ignore-primes-lying-over-l-in-the-fontaine-mazur-conjecture-counterexa Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples? Mikhail Bondarko 2012-10-21T22:08:32Z 2012-10-22T01:22:07Z <p>The Fontaine-Mazur conjecture predicts that an $l$-adic Galois representation of a number field is 'geometric' if it is unramified outside a finite set of primes and is De Rham for primes lying over $l$. Now, what happens if one forgets about the latter restriction; are there any counterexamples, and is there any (general?) way to understand that those are not geometric without using the De Rham restriction?</p> http://mathoverflow.net/questions/110273/can-one-ignore-primes-lying-over-l-in-the-fontaine-mazur-conjecture-counterexa/110275#110275 Answer by Kevin Ventullo for Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples? Kevin Ventullo 2012-10-21T23:04:51Z 2012-10-21T23:04:51Z <p>In fact there are one-dimensional counterexamples: if $\chi$ is the $l$-adic cyclotomic character, and $k\in \mathbb{Z}_l \backslash \mathbb{Z}$, then $\chi^{(l-1)k}$ is unramified outside $l$, but does not arise from geometry (in particular it is not de Rham).</p> <p>A general way of understanding 'all' Galois representations is via Mazur's universal deformation rings. These parametrize all $l$-adic Galois representations (unramified outside a prescribed set of primes and perhaps with extra conditions) whose reduction mod $l$ is isomorphic to a given mod $l$ representation. The $R=T$ theorems you may have heard of state that certain universal deformation rings are isomorphic to rings coming from the theory of modular forms.</p> <p>I don't know if it's expected in general whether all Galois representations are limits of deRham representations.</p> http://mathoverflow.net/questions/110273/can-one-ignore-primes-lying-over-l-in-the-fontaine-mazur-conjecture-counterexa/110288#110288 Answer by Joël for Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples? Joël 2012-10-22T01:22:07Z 2012-10-22T01:22:07Z <p>To complete Kevin's good answer: the number of $\ell$-adic representations (up to isomorphism) of a number field $K$ is countable, since so are varieties over a $K$. On the other hand, we know by Mazur's theory of deformations that representations of the type you consider that is, of the Galois group of the maximal extensions of a number field $K$ unramified outside a finite set of places $S$ containing places above $\ell$ and $\infty$) form multi-dimensional $\ell$-adic family (e.g. parametrized by spaces like $\mathbb{Z}_\ell^n$ for some $n>0$, hence are uncountable. Thus not only are there counter-examples to Fontaine-Mazur's conjecture without the de Rham hypothesis, but most examples of such representations are counter-examples.</p> <p>Among all representations, the de Rham (or geometric) representations are expected to be dense in certain cases (e.g. representation of dimension 2 of $\mathbb Q$) but not in general (e.g. representations of dimension $2$ of a non-totally real number field)</p>