Potential Automorphy and the Leopoldt conjecture
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POTENTIAL AUTOMORPHY AND THE LEOPOLDT CONJECTURE By CHANDRASHEKHAR B. KHARE and JACK A. THORNE Abstract. We study in this paper Hida’s p-adic Hecke algebra for GLn over a CM field F . Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his conjecture in the case F = Q implies the classical Leopoldt conjecture for a number field K of degree n over Q, if one assumes further the existence of automorphic induction of characters for the extension K/Q. We study Hida’s conjecture using the automorphy lifting techniques adapted to the GLn setting by Calegari-Geraghty. We prove an automorphy lifting result in this setting, conditional on existence and local-global compatibility of Galois representations arising from torsion classes in the cohomology of the corresponding symmetric manifolds. Under the same conditions we show that one can deduce the classical (abelian) Leopoldt conjectures for a totally real number field K and a prime p using Hida’s non-abelian Leopoldt conjecture for p-adic Hecke algebra for GLn over CM fields without needing to assume automorphic induction of characters for the extension K/Q. For this methods of potential automorphy results are used. Contents. 1. Introduction. 1.1. Notation. 2. Preliminaries in commutative algebra. 2.1. Change of coefficients. 2.2. Minimal complexes and the derived category. 2.3. Dimension. 2.4. Ordinary part. 3. Abstract patching. 4. Galois deformation theory. 4.1. Galois cohomology. 4.2. Local deformation problems. 4.2.1. Ordinary deformations. 4.2.2. Taylor-Wiles deformations. 4.3. Auxiliary places. 5. Interlude on q-adic Iwahori Hecke algebras. 6. Ordinary completed cohomology. 6.1. Set-up. 6.2. Cohomology and Hecke operators. 6.3. Ordinary part of homology. Manuscript received February 17, 2015; revised October 7, 2015. Research of the first author supported by NSF grant DMS-1161671 and by a Humboldt Research Award. In the period during which this research was conducted, the second author served as a Clay Research Fellow. American Journal of Mathematics 139 (2017), 1205–1273. c  2017 by Johns Hopkins University Press. 1205 1206 C. B. KHARE AND J. A. THORNE 6.4. Independence of weight. 6.5. Hecke algebras. 6.6. Auxiliary primes. 6.7. The Taylor-Wiles argument. 7. Potential automorphy and Leopoldt’s conjecture. References. 1. Introduction. Let F be a number field, and let p be a prime. If U is a sufficiently small open compact subgroup of GLn(A∞ F ), then the double quotient XU = GLn(F)\GLn(A)/(U ×K∞Z∞) has a natural structure of smooth manifold. (We write G∞ = GLn(F ⊗Q R), K∞ ⊂ G∞ for a choice of maximal compact subgroup, and Z∞ for the center of G∞.) This paper is an exploration of the ordinary part of the p-adic completed homology groups of the manifolds XU . More precisely, if U factors as U = UpUp, then for each integer c ≥ 1, we define Up (c) = Up·  v|p Iv(c,c), where Iv(c,c) ⊂ GLn(OFv ) is the subgroup of matrices which, modulo c v, are upper-triangular with constant diagonal entry (see Equation (6.8)). Then the manifolds XUp(c) fit into a tower··· −→ XUp(c) −→ ··· −→ XUp(2) −→ XUp(1), and there is a corresponding tower of homology groups (which are finite Zpmodules ):··· −→ H∗(XUp(c),Zp) −→ ··· −→ H∗(XUp(2),Zp) −→ H∗(XUp(1),Zp). There is a Hecke operator Up which acts compatibly on the whole tower of homology groups, and we define H∗(XUp(c))ord ⊂ H∗(XUp(c)) to be the ordinary part of H∗(XUp(c),Zp) with respect to Up, i.e., the maximal direct summand Zp-module on which Up acts invertibly. We then define H∗ ord(U) = lim ← − c Hd−∗(XUp(c),Zp)ord, where d = dimXU . These are Hida’s ordinary cohomology groups. They have a natural structure of finite Λ-module, where Λ = Zp(OF ⊗ Zp)×(p)n−1 is the completed group ring of the pro-p part of the p-adic points of a maximal torus of PGLn(F ⊗Q Qp). POTENTIAL AUTOMORPHY AND THE LEOPOLDT CONJECTURE 1207 A basic question is: what is...