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Let G be a connected reductive group over a nonarchimedean field k and let \Gamma<\Gamma_1 be congruences subgroup of G(k). Let \pi be an irreducible smooth admissible representation of G. Suppose \pi is supercuspidal and \pi^{\Gamma_1}\neq 0.

Then \pi|_\Gamma contains a cuspidal representation \rho of \Gamma/\Gamma_1.

Therefore \pi contains an extension of \rho to Z\Gamma, \tilde{\rho}. So {\rm Hom}_{Z\Gamma}(\tilde{\rho},\pi|_{Z\Gamma})\neq 0. By Frobenius reciprocity,

{\rm Hom}_G({\rm ind}_{Z\Gamma}^G \tilde{\rho},\pi)\cong {\rm Hom}_{Z\Gamma}(\tilde{\rho},\pi|_{Z\Gamma}).

Since {\rm ind}_{Z\Gamma}^G\tilde{\rho} is irreducible, it must be isomorphic to \pi.

Now consider the decomposition of the induced representation into irreducible components

{\rm ind}_{Z\Gamma}^G\tilde{\rho}|_{\Gamma_1}=\oplus_{x\in Z\Gamma\backslash G/\Gamma_1} {\rm ind}_{Z\Gamma^x\cap\Gamma_1}^{\Gamma_1} (\tilde{\rho}^x|_{Z\Gamma^x\cap\Gamma_1}).

where Z\Gamma^x=x^{-1}Z\Gamma x and \tilde{\rho}^x is the obvious representation of Z\Gamma^x (transport \tilde{\rho} through conjugation by x, i.e., \tilde{\rho}^x(x^{-1}hx) = \tilde{\rho}(h) for h\in Z\Gamma.)

When does {\rm ind}_{Z\Gamma^x\cap\Gamma_1}^{\Gamma_1} (\tilde{\rho}^x|_{Z\Gamma^x\cap\Gamma_1}) contain the trivial representation of \Gamma_1? Equivalent questions are

When is {\rm Hom}_{\Gamma_1}({\rm ind}_{Z\Gamma^x\cap\Gamma_1}^{\Gamma_1}(\tilde{\rho}^x), \textbf{1}_{\Gamma_1})\neq 0?

When is {\rm Hom}_{Z\Gamma^x\cap\Gamma_1}(\tilde{\rho}^x|_{Z\Gamma^x\cap\Gamma_1},\textbf{1}_{Z\Gamma^x\cap\Gamma_1})\neq 0?

One can show x\in Z\Gamma by considering {\rm ind}_{Z\Gamma}^G\tilde{\rho}|_{\Gamma_1}. If this contains the trivial representation, then {\rm ind}_{Z\Gamma}^G\tilde{\rho}|_\Gamma contains an irreducible representation, say \tau, such that \tau_{\Gamma_1}\supset\textbf{1}_{\Gamma_1}.

This implies that \tau is trivial on \Gamma_1. So \pi contains \rho and \tau. (A weaker approach is to use explicit representatives for Z\Gamma\backslash G/\Gamma_1.)

General theory implies that \rho and \tau intertwine, i.e., there exist x\in G such that

{\rm Hom}_{\Gamma^x\cap\Gamma}(\rho^x,\tau)\neq 0.

This implies x\in Z\Gamma. Using cuspidality of \rho plus reps for \Gamma\backslash G/\Gamma, we have that \rho\cong\tau.

It follows that \pi^{\Gamma_1}=\tilde{\rho}|_{\Gamma_1}. In particular,

{\rm dim}\, \pi^{\Gamma_1}={\rm dim}\, \tilde{\rho}={\rm dim}\, \rho.

L. Morris, Tamely ramified supercuspidal representations, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 5, 639–667.
A. Moy and G. Prasad, Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), no. 1-3, 393–408.

Jacquet and Langlands have shown how to associate classical holomorphic modular forms f to automorphic representations \pi_f of {\rm GL}(2,\mathbb{A}). These representations can be written in terms of local components. In this case, the local components, which are infinite dimensional, are complex representations of {\rm GL}(2,\mathbb{Q}_v). Also, the representation is realized in the action of {\rm GL}(2,\mathbb{A}) by right translation on a certain space of functions on {\rm GL}(2,\mathbb{Q})\backslash{\rm GL}(2,\mathbb{A}).

The classical theory of the passage of modular forms to automorphic representations suggests how it may be extended to Siegel modular forms of higher degree. Spaces of cuspidal Siegel modular forms of degree n are associated to cuspidal automorphic representations of {\rm GSp}(2n,\mathbb{A}). These cuspidal automorphic representations \pi can be written in terms of local components \pi_v, where v is a place of \mathbb{Q}. Many cusp forms are associated to a single such representation \pi, but among them is a unique primitive f known as a newform.

In addition to representation theory, modular forms are also related to certain abelian varieties. The nature of this relationship is made precise by the famous Taniyama-Shimura conjecture, proven in 1999.

Taniyama-Shimura Conjecture: Let E: y^2=x^3+ax+b be an elliptic curve with integral coefficients, conductor N, and L-series


Then there is a cusp form of weight 2 and level N which is a Hecke eigenform with Fourier series

f(\tau)=\sum_{n=1}^\infty a_nq^n,\quad {\rm where}\, q=e^{2\pi i\tau}

The Langlands philosophy suggests that there should be abelian varieties associated with degree 2 Siegel modular forms. Brumer and Kramer have recently stated a conjectured extension of the Taniyama-Shimura conjecture to the degree 2 case for paramodular forms.

In the GL(2) case, cuspidal eigenforms determine Galois representations as well as automorphic representations. By a theorem of Deligne and Carayol, Galois representations and automorphic representations both have local components which determine each other. This theorem tells us that a modular elliptic curve has the same conductor as the level of the corresponding cusp form.

We would like to do the same thing for the GSp(4) case, but we need to know the corresponding abelian varieties. Unfortunately, these are unknown in general. But there is evidence supporting a conjectured correspondence for the paramodular group K(p) of GSp(4) due to Brumer and Kramer.

The Paramodular Conjecture: There is a one-to-one correspondence between isogeny classes of rational abelian surfaces A of conductor N with {\rm End}_\mathbb{Q} A = \mathbb{Z} and weight 2 newforms f on K(N) with rational eigenvalues, not in the span of the Gritsenko lifts, such that L(A,s) = L(f,s). The \ell-adic representations associated to f should be isomorphic to those of the Tate module of A for any \ell prime to N.

The paramodular conjecture is consistent with known examples from Brumer and Kramer and also in Poor and Yuen’s work on weight 2 Siegel paramodular forms. There are many open problems related to this conjecture, such as translating the different actions one can perform on abelian varieties to actions on spaces of paramodular forms.

More generally, Yoshida conjectured that for any rational abelian surface, there exists a discrete subgroup \Gamma' of {\rm Sp}(4,\mathbb{Q}) and a degree 2 Siegel modular form of weight 2, f\in\mathcal{M}_2(\Gamma'), with the same L-series.

A. Brumer and K. Kramer, Paramodular abelian varieties of odd conductor, arXiv:1004.4699v2 (2010).

C. Poor and D. Yuen, Paramodular Cusp Forms, arXiv:0912.0049v1 (2009).

H. Yoshida, Siegel modular forms and the arithmetic of quadratic forms, Invent. Math. 60 (1980), no. 3, 193-248.

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