Abstract
We first introduce the Hecke ring of a \(\mathbb {Z}\)-group G and discuss it basic properties (local-global structure, compatibility with isogenies, criterion for commutativity…). An elementary description of the Hecke rings of classical groups is given. Then, we recall the notion of a square integrable automorphic form for G, and that of a discrete automorphic representation of G. When G is the symplectic group Sp2g, we explain how the theory of Siegel modular forms fits into this picture. We also show how the p-neighbor problem for even unimodular lattices in rank n may be viewed as a question about automorphic representations for the orthogonal \(\mathbb {Z}\)-group On.
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Notes
- 1.
The assertions h(SLn) = h(Sp2g) = 1 recalled above are also very particular cases of Kneser’s strong approximation theorem (see [123], [162, Theorem 7.12]). It asserts that we have h(G) = 1 whenever the \(\mathbb {C}\)-group \(G_{\mathbb {C}}\) is semisimple and simply connected and the topological group \(G(\mathbb {R})\) does not have a nontrivial connected, compact, normal subgroup.
- 2.
This property is not automatic if X is infinite. Consider, for example, the group
of affine transformations of \(\mathbb {Q}\) and the Γ-set X consisting of the subsets of \(\mathbb {Q}\) of the form \(a\mathbb {Z}+b\) with \(a \in \mathbb {Q}^\times \) and \(b \in \mathbb {Q}\). - 3.
We refer to the article of Satake for a variant without the injectivity assumption on g. The reader will not miss much in the current discussion by assuming Γ ⊂ Γ′ and X ⊂ X′, with f and g the corresponding inclusions.
- 4.
At this point, it is useful to recall the following version of Schur’s lemma. Let U and V be Hilbert spaces endowed with unitary representations of a group Γ. We assume that U is topologically irreducible and that u: U → V is a nonzero, Γ-equivariant, continuous linear map. Then the adjoint u ∗: V → U (which is Γ-equivariant) satisfies u ∗∘ u = λIdU for some \(\lambda \in \mathbb {R}^\times \). Indeed, u ∗∘ u ∈End(U) is Hermitian and nonzero and commutes with Γ; by the spectral theorem, its spectrum is therefore reduced to a point {λ}. It follows that V is the orthogonal sum of Im(u) (which is closed) and Ker(u ∗).
- 5.
The reader should be aware that the definition we use here depends not only on \(G_{\mathbb {Q}}\) but also on G as a \(\mathbb {Z}\)-group. In the literature, our discrete automorphic representations are more commonly called “discrete automorphic representations of \(G(\mathbb {A})\) that are spherical (or unramified) with respect to \(G(\widehat {\mathbb {Z}})\).” The apparent loss of generality in our presentation is, however, at this point illusory, because every open compact subgroup of \(G(\mathbb {A}_f)\) is of the form \(G'(\widehat {\mathbb {Z}})\) for a well-chosen \(\mathbb {Z}\)-group G′ with \(G^{\prime }_{\mathbb {Q}} \simeq G_{\mathbb {Q}}\).
- 6.
In fact, a famous result of Godement shows that under this same hypothesis on G, the group \(G(\mathbb {Q})\) is cocompact in \(G(\mathbb {A})\), which implies the equality \(\mathcal {A}_{\mathrm{disc}}(G)={\mathcal {A}^2}(G)\) more directly in this specific case (see, for example, [35, Lemma 16.1]).
- 7.
A principal polarization on a lattice \(L \subset \mathbb {C}^g\) consists of a nondegenerate alternating bilinear form \(\eta \colon L \times L \rightarrow \mathbb {Z}\) whose extension of scalars \(\eta _{\mathbb {R}}\) to \(L \otimes \mathbb {R} = \mathbb {C}^g\) satisfies \(\eta _{\mathbb {R}}(ix,iy)=\eta _{\mathbb {R}}(x,y)\) for every \(x,y \in \mathbb {C}^g\) and whose associated Hermitian form \((x,y) \mapsto \eta _{\mathbb {R}}(ix,y)+i\eta _{\mathbb {R}}(x,y)\) on \(\mathbb {C}^g\) is positive definite. Riemann’s theory allows us to naturally identify \({\mathrm{Sp}}_{2g}(\mathbb {Z})\backslash \mathbb {H}_g\) with the set of \(\mathrm {GL}_g(\mathbb {C})\)-orbits of pairs (L, η), where \(L \subset \mathbb {C}^g\) is a lattice and η is a principal polarization on L.
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Chenevier, G., Lannes, J. (2019). Automorphic Forms and Hecke Operators. In: Automorphic Forms and Even Unimodular Lattices. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-319-95891-0_4
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