Level one automorphic forms and modular forms
All data given below rely on J. Arthur's book The Endoscopic Classification of
Representations: Orthogonal and Symplectic Groups (not on its last chapter
though), and thus on the stabilization of the twisted trace formula.
The source code for the computer program which has produced these tables can be
found here.
Masses
Here are tables of masses, from which one can compute the elliptic terms of the geometric side of the trace formula, as explained in the paper.
Selfdual automorphic cuspidal representations for the general linear group
- For $n \geq 1$ and integers $w_1 > \dots > w_n > 0$, consider the finite set $O_o(w_1, \dots, w_n)$ of automorphic cuspidal representations $\pi = \bigotimes'_v \pi_v$ for $\mathrm{GL}_{2n+1}$ over $\mathbb{Q}$ such that
- $\pi$ is self-dual, i.e. $\pi^{\vee} \simeq \pi$,
- for any prime number $p$, the representation $\pi_p$ is unramified.
- the Langlands parameter of $\pi_{\infty}$ is
$$\epsilon^n \oplus \bigoplus_{r=1}^n \mathrm{Ind}_{W_{\mathbb{R}}}^{W_{\mathbb{C}}} \left( z \mapsto (z/\bar{z})^{w_r} \right)$$
where $W_{\mathbb{C}} \simeq \mathbb{C}^{\times}$ is the Weil group of $\mathbb{C}$, $W_{\mathbb{R}} = W_{\mathbb{C}} \sqcup j W_{\mathbb{C}}$ the Weil group of $\mathbb{R}$ and $\epsilon : W_{\mathbb{R}} \rightarrow \{ \pm 1 \}$ has kernel $W_{\mathbb{C}}$.
The notation $O_o$ stands for "odd orthogonal".
- Similarly, for $n \geq 1$ and half-integers $w_1 > \dots > w_n > 0$ (i.e. each $2w_r$ is an odd integer), consider the finite set $S(w_1, \dots , w_n)$ ($S$ stands for "symplectic")
of self-dual everywhere unramified automorphic cuspidal representations for $\mathrm{GL}_{2n}$ over $\mathbb{Q}$ such that the Langlands parameter of $\pi_{\infty}$ is
$$\bigoplus_{r=1}^n \mathrm{Ind}_{W_{\mathbb{R}}}^{W_{\mathbb{C}}} \left( z \mapsto (z/|z|)^{2w_r} \right).$$
- Finally, for $n \geq 1$ and integers $w_1 > \dots > w_{2n} \geq 0$ consider the finite set $O_e(w_1, \dots, w_{2n})$ ($O_e$ stands for "even orthogonal") of self-dual
everywhere unramified automorphic cuspidal representations for $\mathrm{GL}_{4n}$ over $\mathbb{Q}$ such that the Langlands parameter of $\pi_{\infty}$ is
$$\bigoplus_{r=1}^{2n} \mathrm{Ind}_{W_{\mathbb{R}}}^{W_{\mathbb{C}}} \left( z \mapsto (z/\bar{z})^{w_r} \right).$$
These three families partition the set of self-dual everywhere unramified automorphic cuspidal representations $\pi$ of a general linear group over $\mathbb{Q}$ such that
$\pi$ or $\pi \otimes |\det|^{1/2}$ is algebraic regular (see L. Clozel's Ann Arbor paper for a definition), plus a slightly irregular case (third case with $w_{2n}=0$).
- Table of $[w_1], \mathrm{card}\left(O_o(w_1)\right)$ when this last number is nonzero.
Table of $[w_1,w_2], \mathrm{card}\left(O_o(w_1,w_2)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3], \mathrm{card}\left(O_o(w_1,w_2,w_3)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3,w_4], \mathrm{card}\left(O_o(w_1,w_2,w_3,w_4)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3,w_4,w_5], \mathrm{card}\left(O_o(w_1,w_2,w_3,w_4,w_5)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3,w_4,w_5,w_6], \mathrm{card}\left(O_o(w_1,w_2,w_3,w_4,w_5,w_6)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3,w_4,w_5,w_6,w_7], \mathrm{card}\left(O_o(w_1,w_2,w_3,w_4,w_5,w_6,w_7)\right)$ when this last number is nonzero.
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Table of $[w_1], \mathrm{card}\left(S(w_1)\right)$ when this last number is nonzero.
Table of $[w_1,w_2], \mathrm{card}\left(S(w_1,w_2)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3], \mathrm{card}\left(S(w_1,w_2,w_3)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3,w_4], \mathrm{card}\left(S(w_1,w_2,w_3,w_4)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3,w_4,w_5], \mathrm{card}\left(S(w_1,w_2,w_3,w_4,w_5)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3,w_4,w_5,w_6], \mathrm{card}\left(S(w_1,w_2,w_3,w_4,w_5,w_6)\right)$ when this last number is nonzero.
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Table of $[w_1,w_2], \mathrm{card}\left(O_e(w_1,w_2)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3,w_4], \mathrm{card}\left(O_e(w_1,w_2,w_3,w_4)\right)$ when this last number is nonzero.
Table of $[w_1,w_2,w_3,w_4,w_5,w_6], \mathrm{card}\left(O_e(w_1,w_2,w_3,w_4,w_5,w_6)\right)$ when this last number is nonzero.
Compressed archive with more data (39MB)
Application: Siegel modular forms
For $n \geq 1$ and $\rho$ an algebraic representation of $\mathrm{GL}_n(\mathbb{C})$ denote $S_{\rho}(\Gamma_n)$ the space of cuspidal Siegel modular forms of weight $\rho$ for the
group $\Gamma_n = \mathrm{Sp}_{2n}(\mathbb{Z})$ (see van der Geer's lecture notes for a precise definition).
If $\rho$ is irreducible and $k_1 \geq \dots \geq k_n$ is its highest weight
we also denote $S_{k_1, \dots, k_n}(\Gamma_n)$ for $S_{\rho}(\Gamma_n)$, and if $k_1 = \dots = k_n$ (i.e. $\rho = \det^{k_1}$) we simply write $S_{k_1}(\Gamma_n)$ (scalar Siegel modular forms of weight $k_1$).
Below we give the value of $\dim S_{k_1, \dots, k_n}(\Gamma_n)$ for weights $\underline{k}$ satisfying $k_n \geq n+1$.
Some values and lower bounds in the scalar case. The "new" eigenforms are the tempered non-endoscopic ones, they correspond to $O_o(k-1, \dots, k-n)$.
Table of $[k], \dim S_k(\Gamma_1)$ when this last number is nonzero. These are the usual modular forms.
Table of $[k_1,k_2], \dim S_{k_1,k_2}(\Gamma_2)$ when this last number is nonzero.
Table of $[k_1,k_2,k_3], \dim S_{k_1,k_2,k_3}(\Gamma_3)$ when this last number is nonzero.
Table of $[k_1,k_2,k_3,k_4], \dim S_{k_1,k_2,k_3,k_4}(\Gamma_4)$ when this last number is nonzero.
Table of $[k_1,k_2,k_3,k_4,k_5], \dim S_{k_1,k_2,k_3,k_4,k_5}(\Gamma_5)$ when this last number is nonzero.
Table of $[k_1,k_2,k_3,k_4,k_5,k_6], \dim S_{k_1,k_2,k_3,k_4,k_5,k_6}(\Gamma_6)$ when this last number is nonzero.
Table of $[k_1,k_2,k_3,k_4,k_5,k_6,k_7], \dim S_{k_1,k_2,k_3,k_4,k_5,k_6,k_7}(\Gamma_7)$ when this last number is nonzero.
Compressed archive with more data (11MB)