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I have one issue with the Jacquet Langlands correspondence. 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 correspondence explicitly 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 a 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:

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

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    $\begingroup$ 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). $\endgroup$
    – Rob Harron
    Jun 26, 2011 at 2:04
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    $\begingroup$ For a quick statement about the level, see Chenevier's IHP notes (The infinite fern and families of quaternionic modular forms). $\endgroup$
    – Kimball
    Jun 26, 2011 at 3:02

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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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  • $\begingroup$ There appears an additional $T log T$ term for congruence subgroups. $\endgroup$
    – Marc Palm
    Jun 25, 2011 at 21:53
  • $\begingroup$ 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. $\endgroup$
    – GH from MO
    Jun 25, 2011 at 22:00
  • $\begingroup$ 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. $\endgroup$
    – Marc Palm
    Jun 25, 2011 at 22:35
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    $\begingroup$ @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. $\endgroup$
    – GH from MO
    Jun 25, 2011 at 23:59
  • $\begingroup$ Discrete series at non-archimedean places means here supercuspidal, right? Or are Steinberg representations also allowed? $\endgroup$
    – Marc Palm
    Jul 19, 2012 at 14:56

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