Jacquet Langlands correspondance - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:14:49Z http://mathoverflow.net/feeds/question/68813 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68813/jacquet-langlands-correspondance Jacquet Langlands correspondance Marc Palm 2011-06-25T20:45:48Z 2012-04-13T08:37:54Z <p>I have one issue with the Jacquet Langlands correspondance. The Weyl law for $H$ modulo a congruence subgroup and the Weyl law for cocompact groups are different. So why does this not contradict this functoriality? What am I missing?</p> <p>I have not yet studied the Jacquet Langlands correspondance explicitely yet. How explicit are the lifts, about the level etc.? I know that there is not an expansion formula for cocompact groups available as we have it for groups with an parabolic element.</p> <p>Update: After a reading a little bit, I found a paper which focuses exactly on the first part of the question and also gives references for the second part of the question, i.e.</p> <p>Risager, Morten S. Asymptotic densities of Maass newforms. J. Number Theory 109 (2004), no. 1, 96–119.</p> http://mathoverflow.net/questions/68813/jacquet-langlands-correspondance/68815#68815 Answer by GH for Jacquet Langlands correspondance GH 2011-06-25T21:46:08Z 2011-06-25T21:46:08Z <p>In what sense is the Weyl law different for congruence subgroups and cocompact groups? </p> <p>At any rate, the Jacquet-Langlands correspondence is not a bijection between the two cuspidal spectra. More precisely, let $D$ be a quaternion algebra over a number field $F$, and consider the groups $G=PD^\times$ and $G'=PGL_2$. Then the Jacquet-Langlands correspondence injects the automorphic representations of $G(\mathbb{A}_F)$ into those of $G'(\mathbb{A}_F)$. A cuspidal representation $\pi$ of $G'(\mathbb{A}_F)$ lies in the image of this map if and only if $\pi_v$ is a discrete series representation of $G'(F_v)$ at all places $v$ where $D$ ramifies. So unless $G'=G$, the image will miss several cuspidal representations of $G'(\mathbb{A}_F)$.</p> <p>I think the lifts are not explicit in the sense that they are not given by an explicit construction.</p>