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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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##### References:
 [1] A. Borel, Cohomology of arithmetic groups , Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, Canadian Mathematics Congress, Montreal, 1975, pp. 435-442. · Zbl 0338.20051 [2] A. Borel, Stable real cohomology of arithmetic groups , Ann. Sci. École Norm. Sup. (4) 7 (1974), 235-272. · Zbl 0316.57026 [3] A. Borel, Stable real cohomology of arithmetic groups II , Manifolds and Lie Groups (Notre Dame, Ind., 1980), Progr. Math., vol. 14, Birkhäuser, Boston, 1981, Papers in Honor of Yozo Matsushima, pp. 21-55. · Zbl 0483.57026 [4] A. Borel and J.-P. Serre, Corners and arithmetic groups , Comment. Math. Helv. 48 (1973), 436-491. · Zbl 0274.22011 [5] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups , Ann. of Math. Stud., vol. 94, Princeton Univ. Press, Princeton, 1980. · Zbl 0443.22010 [6] N. Bourbaki, Commutative Algebra , Hermann, Paris, 1961, 1964, and 1965. [7] G. E. Breadon, Sheaf Theory , McGraw-Hill, New York, 1967. · Zbl 0158.20505 [8] K. S. Brown, Cohomology of groups , Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1982. · Zbl 0584.20036 [9] M. Eichler, Eine Verallgemeinerung der Abelschen Integrale , Math. Z. 67 (1957), 267-298. · Zbl 0080.06003 [10] A. Guichardet, Cohomologie des groupes topologiques et des algèbres de Lie , Textes Mathématiques [Mathematical Texts], vol. 2, CEDIC, Paris, 1980. · Zbl 0464.22001 [11] G. Harder, Eisenstein cohomology of arithmetic groups. The case $$\mathrm GL_ 2$$ , Invent. Math. 89 (1987), no. 1, 37-118. · Zbl 0629.10023 [12] G. Harder, On the cohomology of discrete arithmetically defined groups , Discrete Subgroups of Lie Groups and Applications to Moduli (Internat. Colloq., Bombay, 1973), Oxford Univ. Press, Bombay, 1975, pp. 129-160. · Zbl 0317.57022 [13] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces , Pure Appl. Math., vol. 80, Academic Press, New York, 1978. · Zbl 0451.53038 [14] H. Hida, On abelian varieties with complex multiplication as factors of the abelian variety attached to Hilbert modular forms , Japan. J. Math. (N.S.) 5 (1979), no. 1, 157-208. · Zbl 0422.14026 [15] H. Hida, Galois representations into $$\mathrm GL_ 2(\mathbf Z_ p[[X]])$$ attached to ordinary cusp forms , Invent. Math. 85 (1986), no. 3, 545-613. · Zbl 0612.10021 [16] H. Hida, Modules of congruence of Hecke algebras and $$L$$-functions associated with cusp forms , Amer. J. Math. 110 (1988), no. 2, 323-382. JSTOR: · Zbl 0645.10029 [17] H. Hida, On $$p$$-adic Hecke algebras for $$\mathrm GL_ 2$$ over totally real fields , Ann. of Math. (2) 128 (1988), no. 2, 295-384. JSTOR: · Zbl 0658.10034 [18] H. Hida, On nearly ordinary Hecke algebras for $$\mathrm GL(2)$$ over totally real fields , Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, In Honor of K. Iwasawa, pp. 139-169. · Zbl 0742.11026 [19] H. Hida, On $$p$$-adic $$L$$-functions of $$\mathrm GL(2)\times \mathrm GL(2)$$ over totally real fields , Ann. Inst. Fourier (Grenoble) 41 (1991), no. 2, 311-391. · Zbl 0739.11019 [20] H. Hida, Modular $$p$$-adic $$L$$-functions and $$p$$-adic Hecke algebras , to appear in Sugaku Expositions. · Zbl 0811.11040 [21] P. J. Hilton and U. Stammbach, A Course in Homological Algebra , Grad. Texts in Math., vol. 4, Springer-Verlag, New York, 1971. · Zbl 0238.18006 [22] G. Hochschild and J.-P. Serre, Cohomology of group extensions , Trans. Amer. Math. Soc. 74 (1953), 110-134. JSTOR: · Zbl 0050.02104 [23] H. Jacquet and R. P. Langlands, Automorphic forms on $$\mathrm GL(2)$$ , Springer-Verlag, Berlin, 1970. · Zbl 0236.12010 [24] Y. Matsushima and S. Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds , Ann. of Math. (2) 78 (1963), 365-416. JSTOR: · Zbl 0125.10702 [25] Y. Matsushima and G. Shimura, On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes , Ann. of Math. (2) 78 (1963), 417-449. JSTOR: · Zbl 0141.38704 [26] G. Shimura, Sur les intégrales attachées aux formes automorphes , J. Math. Soc. Japan 11 (1959), 291-311. · Zbl 0090.05503 [27] G. Shimura, On Dirichlet series and abelian varieties attached to automorphic forms , Ann. of Math. (2) 76 (1962), 237-294. JSTOR: · Zbl 0142.05501 [28] G. Shimura, On the Eisenstein series of Hilbert modular groups , Rev. Mat. Iberoamericana 1 (1985), no. 3, 1-42. · Zbl 0608.10028 [29] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions , Kanô Memorial Lectures, vol. 1, Iwanami Shoten, Tokyo, 1971, Publ. Math. Soc. Japan 11, Princeton Univer. Press, Princeton, 1971. · Zbl 0221.10029 [30] H. M. Stark, $$L$$-functions at $$s=1$$. II. Artin $$L$$-functions with rational characters , Advances in Math. 17 (1975), no. 1, 60-92. · Zbl 0316.12007 [31] R. Taylor, Congruences between modular forms on $$\mathrmGL_2$$ over imaginary quadratic fields , [32] A. Weil, Dirichlet Series and Automorphic Forms , Lecture Notes in Math., vol. 189, Springer-verlag, Berlin, 1971. · Zbl 0218.10046 [33] A. Weil, On a certain type of characters of the idèle-class group of an algebraic number field , Proceedings of the International Symposium on Algebraic Number Theory, Tokyo and Nikko, 1955, Science Council of Japan, Tokyo, 1956, pp. 1-7. · Zbl 0073.26303 [34] S. Zucker, $$L^2$$-cohomology of Shimura varieties , Automorphic Forms, Shimura Varieties, and $$L$$-functions, Volume II (Ann Arbor, MI, 1988), Perspect. Math., vol. 11, Academic Press, Boston, 1990, pp. 377-391. · Zbl 0707.14018
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