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\(p\)-ordinary cohomology groups for \(\text{SL}(2)\) over number fields. (English) Zbl 0941.11024
The author extends his \(\Lambda\)-adic theory of ordinary automorphic forms to the setting of cohomology of arithmetic subgroups of SL(2) over a number field. This is a remarkably beautiful theory, but its description requires a fair amount of notation. Our description below is a simplified account of the contents of the paper.
The author works with central simple algebra \(B\) of dimension 4 over a number field \(F\), while our account is confined to \(B=M_2(F)\). Indeed, the author imposes enough local splitting conditions on \(B\) so that no genuinely new ideas are needed to pass from \(M_2(F)\) to the general case. Let \(F\) be a number field and let \(R\) be the ring of integers of \(F\). Fix a rational prime \(p\) and an ideal \(N\) of \(R\) that is relatively prime to \(p\). Let \(\Delta \subseteq \text{SL}_2(R)\) be a congruence subgroup of level \(N\) and consider the groups \(\Delta_1 (p^\alpha)=\Delta \cap \Gamma_1 (p^\alpha)\), \(\alpha >0\). Fix an embedding \(\overline\mathbb{Q}\subseteq \overline\mathbb{Q}_p\) and let \(I\) be the set of embeddings of \(F\) into \(\overline\mathbb{Q}\). Let \(K\) be a finite extension of \(\mathbb{Q}_p\) large enough to contain all of the conjugates of \(F\) and let \(\mathcal O\) be the ring of integers of \(K\). Fix \(n=\sum_{\alpha \in I}n_\sigma \sigma \in \mathbb{Z}[I]\) with all \(n_\sigma \geq 0\) and consider the module \(L(n;\mathcal O)\) of polynomials over \(\mathcal O\) in \(2(\#I)\) variables \(\{(X_\sigma, Y_\sigma)\}_{\sigma \in I}\) which are homogeneous of degree \(n_\sigma\) in \((X_\sigma, Y_\sigma)\) for each \(\sigma \in I\). Then \(L(n;\mathcal O)\) is endowed with a natural action of the semigroup of \(R\)-integral matrices in \(\text{GL}_2(F)\) and this action extends naturally to an action on \(L(n;A):= L(n;\mathcal O)\otimes_{\mathcal O} A\) for any \(\mathcal O\)-module \(A\). We may therefore form the cohomology and homology groups \(H^q(\Delta_1 (p^\alpha),L(n,A))\) and \(H_q(\Delta_1 (p^\alpha),L(n,A))\) and equip each of these groups with a standard action of the Hecke operators \(T(\xi_0)\) for \(\xi_0 \in F^\times\). The Hecke operator \(T(p)\) and the nebentype operators play a special role in the author’s theory.
Letting \(H\) denote either of the above cohomology or homology groups with \(A=\mathcal O\) or \(A=K/\mathcal O\), \(H\) then decomposes naturally into a direct sum \(H=H^{\text{ord}}\oplus H^{\text{ss}}\), where \(T(p)\) acts invertibly on \(H^{\text{ord}}\) and acts topologically nilpotently on \(H^{\text{ss}}\). Define the ”\(\Lambda\)-adic” [co]homology groups \[ H^q_{\text{ord}} (\Delta_1(p^\infty),L(n;A)):= \varinjlim_\alpha H^q_{\text{ord}} (\Delta_1 (p^\alpha), L(n;A))\quad \text{and} \]
\[ H^{\text{ord}}_q (\Delta_1(p^\infty),L(n;A)):= \varprojlim_\alpha H^{\text{ord}}_q (\Delta_1(p^\alpha),L(n;A)), \] where the connecting morphisms are the restriction maps [resp. transfer maps]. The Hecke operators act naturally on these groups and the nebentype operators induce a continuous action of the group \(R^\times_p\) of units in \(R_p:= R\otimes \mathbb{Z}_p\). Thus the \(\Lambda\)-adic [co]homology groups are naturally equipped with structures as \(\mathcal O[ [R^\times_p] ]\)-modules.
There are two main theorems in the paper. Theorem I asserts that \(H^{\text{ord}}_q(\Delta_1(p^\infty),L(n;\mathcal O))\) is of finite type over \(\mathcal O[ [R^\times_p] ]\) and hence by duality that \(H^q_{\text{ord}}(\Delta_1(p^\infty),L(n;K/\mathcal O))\) is of cofinite type. Moreover, these groups are independent of \(n\). Theorem II is a version of the author’s ”control theory”. Fix a character \(\epsilon \colon R^\times_p \to \mathcal O^\times\) defined modulo \(p^\alpha\). If \(H\) is a cohomology group for \(\Delta_1(p^\alpha)\) with coefficients in an \(\mathcal O\)-module, let \(H_\epsilon\) denote the submodule on which the nebentype operators act via \(\epsilon\). Let \(P_{n,\epsilon}\) be the kernel of the homomorphism \(\mathcal O[ [R^\times_p] ]\to \mathcal O\) induced by the character \(R^\times_p \to \mathcal O^\times\), \(t\mapsto \epsilon (t)t^n\).
Theorem II then asserts that for \(0\leq q\leq r+r^2\) \((r:= [F\colon \mathbb{Q}], r_2\colon= \#\{\text{complex places\;of} F\})\), the natural homomorphisms \[ (H^{\text{ord}}_q (\Delta_1(p^\infty),L(n;\mathcal O))/P_{n,\epsilon} \to H^{\text{ord}}_q (\Delta_1 (p^\alpha),L(n;\mathcal O))_\epsilon\quad\text{ and} \]
\[ (H^q_{\text{ord}}(\Delta_1 (p^\alpha),L(n;K/\mathcal O))_\epsilon \to H^q_{\text{ord}}(\Delta_1 (p^\alpha),L(n;K/\mathcal O))[P_{n,\epsilon}] \] (the kernel of \(P_{n,\epsilon}\)) have finite kernel and cokernel. If \(q=0\) or 1 these maps are isomorphisms. If \(q>r+r_2\) then the result remains true for \(P_{n,\epsilon}\) outside a closed subscheme of codimension \(\geq 1\).
Similar assertions are proved for other cohomology groups: compactly supported cohomology, parabolic cohomology, and boundary cohomology.

MSC:
11F75 Cohomology of arithmetic groups
11F33 Congruences for modular and \(p\)-adic modular forms
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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