On geometric density of Hecke eigenvalues for certain cusp forms
Construction of cusp forms is an important problem in the theory of automorphic forms. This paper continues the author's study of the construction of cuspidal automorphic forms via Poincaré series. \textit{G. Muić} [Math. Ann. 343, No. 1, 207--227 (2009; Zbl 1185.11037)]; [Compos. Math. 146, No. 1, 1--20 (2010; Zbl 1248.11038)]; [J. Lie Theory 18, No. 4, 937--949 (2008; Zbl 1163.22005)]; [Ramanujan J. 21, No. 2, 223--239 (2010; Zbl 1237.11023)]; [Proc. Am. Math. Soc. 138, No. 9, 3147--3158 (2010; Zbl 1208.11069)]; [Int. J. Number Theory 7, No. 2, 351--370 (2011; Zbl 1267.11039)]; [J. Number Theory 130, No. 7, 1488--1511 (2010; Zbl 1206.11053)]. Work of other authors dealing with the method of Poincaré series includes \textit{W. L. Baily jun.} and \textit{A. Borel} [Ann. Math. (2) 84, 442--528 (1966; Zbl 0154.08602)], \textit{A. Borel} [Publ. Math., Inst. Hautes Étud. Sci. 16, 101--126 (1963; Zbl 0135.08902)], [Automorphic forms on \(\mathrm{SL}_2(\mathbb R)\). Cambridge: Cambridge University Press (2008; Zbl 1142.11035)], and work discussing other approaches to the problem includes [\textit{J. Arthur},n Clay Math. Proc. 4, 1--263 (2005; Zbl 1152.11021)], [\textit{D. Goldfeld}, Automorphic forms and \(L\)-functions for the group \(\text{GL}(n,\mathbb R)\). Cambridge: Cambridge University Press (2006; Zbl 1108.11039)], [\textit{E. Lindenstrauss} and \textit{A. Venkatesh}, Geom. Funct. Anal. 17, No. 1, 220--251 (2007; Zbl 1137.22011)], [\textit{W. Müller}, Groups and analysis. The legacy of Hermann Weyl. Lond. Math. Soc. Lect. Note Ser. 354, 133--163 (2008; Zbl 1230.11067)], [\textit{T. Miyake}, Modular forms. Berlin: Springer (2006; Zbl 1159.11014)], [\textit{A. Selberg}, J. Indian Math. Soc., N. Ser. 20, 47--87 (1956; Zbl 0072.08201)]. In this paper, the author takes up the study of the Hecke module structure of the space of cusp forms constructed via his refinement of the Poincaré series method. Let \(G\) be a semisimple algebraic group defined over a number field \(k\), and let \(S\) be a finite set of places of \(k\) containing all infinite places and all finite places \(v\) such that \(G\) is not unramified over \(k_v\). Then fix a second finite set of places \(T\), disjoint from \(S.\) For each \(v \in T\), the place \(v\) is finite, the group \(G\) is defined over the ring \(\mathcal O_v\) of integers of \(k_v\), and \(K_v := G( \mathcal O_v)\) is a hyperspecial maximal compact subgroup of \(G(k_v)\). One has the local Hecke algebra \(\mathcal H_v = C_c^\infty(K_v \backslash G(k_v)/K_v)\), and can also consider the finite product \(\mathcal H_T := \otimes_{v\in T}\mathcal H_v\). These are the Hecke algebras under consideration. To describe the space of Poincaré series considered, we first fix data at the places of \(S\). Write \(G_\infty = \prod_{v \mid \infty} G(k_v)\) as usual. Fix a maximal compact subgroup \(K_\infty\) of \(G_\infty\) and a smooth, integrable function \(\varphi_\infty: G_\infty \to \mathbb{C}\) which generates a finite dimensional space under the action of \(K_\infty\) and the center of the universal enveloping algebra of the Lie algebra of \(G_\infty\). Note that assuming the existence of such a function places a condition on \(G\). It is equivalent to assuming that \(G_\infty\) has discrete series representations. (See the author [Math. Ann. 343, No. 1, 207--227 (2009; Zbl 1185.11037)]). Now for \(v \in S\) finite, choose \(\varphi_v \in C_c^\infty( G( k_v))\) and set \(\varphi_S = \prod_{v\in S} \varphi_v: G_S \to \mathbb{C}\). Here \(G_S\) is the product of the groups \(G(k_v)\) for \(v \in S\) and \(G^S\) is the restricted product (relative to \(G( \mathcal O_v)\)) of the groups \(G(k_v)\) for \(v \notin S\), as usual. Next, fix an open compact subgroup \(L'\) of \(G^{S\cup T}\) and set \(L = K_T \times L' \subset G^S\), where \(K_T=\prod_{v \in T} K_v\). The Poincaré series considered is \[ P( \varphi_S \otimes \text{char}_L)(g) = \sum_{\gamma \in G(k) } ( \varphi_s \otimes \text{char}_L)(\gamma \cdot g), \] where \(\text{char}_L\) is the characteristic function of \(L\). It is a right-\(K_T\)-invariant cuspidal automorphic form defined on the group \(G(\mathbb{A})\). Let \(U_{P( \varphi_S \otimes \text{char}_L)}\) denote the \((\text{Lie}(G_\infty), K_\infty ) \times G(\mathbb{A}_f)\)-module it generates, and let \(J( \varphi_S, L)^T\) denote the \(\mathcal H_T\) module it generates. Then \(U_{P( \varphi_S \otimes \text{char}_L)}\) is a finite direct sum of irreducible cuspidal automorphic representations of \(G(\mathbb{A})\), each of which has a \(K_T\)-fixed vector, and hence \(J( \varphi_S, L)^T\) is isomorphic to a direct sum of finitely many one-dimensional representations of \(\mathcal H_T\). The paper addresses the following problems: \(\bullet\) how many distinct isomorphism classes of irreducible cuspidal automorphic representations of \(G(\mathbb{A})\), appear in the decomposition of \(U_{P( \varphi_S \otimes \text{char}_L)}\)? \(\bullet\) how many distinct characters of \(\mathcal H_T\) appear in the decomposition of \(J( \varphi_S, L)^T\)? The answers, of course, are given in terms of an appropriate measure of the size of the group \(L'\). Moreover, the author shows that as \(L'\) becomes very small, the set of distinct representations/ characters becomes not only infinite, but also Zariski dense in an appropriate variety. As alluded to in the previous paragraph, an appropriate measure of the size of \(L'\) plays a crucial role. To define this measure, let \(\Gamma_{L', S \cup T}\) be the subgroup of \(G(k)\) consisting of those elements whose images in \(G^{S\cup T}\) lie in \(L\). Identify it with its image in \(G_{S\cup T}\), which is a discrete subgroup of \(G_{S\cup T}\). Take \(C_S\) a compact subset of \(G_S\) and \(C_T\) a union of double \(K_T\)-cosets in \(G_T\) such that \(C_S \times C_T\) is contained in a fundamental domain for the action of \(\Gamma_{L', S \cup T}\) on \(G_{S\cup T}\). The author shows that the number of distinct double \(K_T\)-cosets in \(C_T\) is bounded by a constant \(k_{C_S}(L')\). This gives a measure of the size of \(L'\) (depending on the choice of \(C_S\)). The first main result is: Theorem. Write \(l(L)\) for the number of distinct characters of \(\mathcal H_T\) appear in the decomposition of \(J( \varphi_S, L)^T\). If \[ \int_{C_S}|\varphi_S(g_S)|\, dg_S > \frac 12 \int_{G_S} |\varphi_S(g_S)|\, dg_S, \] then \(l(L) \geq k_{C_S}(L')\). From this it follows that \(l(L)\) may be made arbitrarily large by taking \(L'\) sufficiently small. The next main result is a refinement of this. In order to state it, one must first recall that the space of characters of \(\mathcal H_T\) may be identified with \(\text{Spec}_{\max} \mathcal H_T\), or, via the Satake isomorphism with the product \(\prod_{v\in T} A_v/W_v\) of several copies of the variety of Weyl orbits in the maximal torus of the \(L\) group of \(G\). Theorem. Choose \(C_S\) as above and a decreasing sequence \(L_1' \supset L_2' \supset \dots\) of open compact subgroups of \(G^{S\cup T}\) with \(\bigcap_{n=1}^\infty \Gamma_{L_n', S \cup T} = \{ 1\}\). Then \(\bullet\) there exists \(n_0\geq 1\) such that \(P( \varphi_S \otimes \text{char}_L)\neq 0 \) for \(n \geq n_0,\) \(\bullet\) if \(\mathcal V( I( \varphi_S, L_n)^T)\) is the set of characters appearing in \(J( \varphi_S, L)^T\) (regarded as a subvariety of \(\text{Spec}_{\max} \mathcal H_T\)) then \(\bigcup_{n = n_0}^\infty \mathcal V( I( \varphi_S, L_n)^T)\) is Zariski dense in \(\text{Spec}_{\max}\mathcal H_T\). \(\bullet\) \(\lim_{n\to \infty} l(L_n) = \infty\). Of course, these facts about the \(\mathcal H_T\)-module generated by \(P(\varphi_S \otimes \text{char}_L)\) imply corresponding facts about the \((\text{Lie}(G_\infty), K_\infty)\times G(\mathbb{A}_f)\)-module it generates. Indeed, if \(m(L)\) is the number of distinct isomorphism classes of irreducible cuspidal automorphic representations of \(G(\mathbb{A})\) which appear in the decomposition of \(U_{P( \varphi_S \otimes \text{char}_L)}\) then \(m(L) \geq l(L)\). Further, in the situation of the second theorem, the Satake parameters the places in \(T\) become dense in \(\prod_{v \in T}A_v/W_v\) as \(n\to \infty\). The author also shows that if one chooses a representative in \(A_v\) for each element of \(A_v/W_v\), then the resulting elements of \(\prod_{v \in T}A_v\) become Zariski dense as \(n \to \infty\). In particular infinitely many of them will lie outside of any given Zariski closed set, such as the set of reducibility points for the unramified principal series, or the image of a functorial lifting.
scientific article; zbMATH DE number 5700562
scientific article