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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?

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.

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.

Risager, Morten S. Asymptotic densities of Maass newforms. J. Number Theory 109 (2004), no. 1, 96–119.

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What do you mean by "explicit"? You know the local components (using the Weil representation) and Shimizu gave an explicit global realization of the correspondence using theta series (see ams.org/mathscinet-getitem?mr=333081). –  Rob Harron Jun 26 '11 at 2:04
For a quick statement about the level, see Chenevier's IHP notes (The infinite fern and families of quaternionic modular forms). –  Kimball Jun 26 '11 at 3:02
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1 Answer

up vote 3 down vote accepted

In what sense is the Weyl law different for congruence subgroups and cocompact groups?

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 $G'(\mathbb{A}_F)$.

I think the lifts are not explicit in the sense that they are not given by an explicit construction.

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There appears an additional $T log T$ term for congruence subgroups. –  plusepsilon.de Jun 25 '11 at 21:53
The $T\log T$ term is an error term, i.e. it is really $O(T\log T)$. There is an error term for the cocompact case as well, perhaps it is not spelled out so explicitly. –  GH from MO Jun 25 '11 at 22:00
no, it is a main term, they Weyl law is $C T^2 + D T log T + O( T / log T)$ for congruence subgroups and $C' T^2 + O( T / log T)$ for cocompact groups. –  plusepsilon.de Jun 25 '11 at 22:35
@pm: Very interesting. Do you know the value of $D$ for $\mathrm{SL}_2(\mathbb{Z})$? Also, there seems to be a main term $ET$ as well. –  GH from MO Jun 25 '11 at 23:59
Discrete series at non-archimedean places means here supercuspidal, right? Or are Steinberg representations also allowed? –  plusepsilon.de Jul 19 '12 at 14:56
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