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Geometric Deformations of Orthogonal and Symplectic Galois Representations

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Geometric Deformations of Orthogonal and Symplectic Galois Representations Geometric Deformations of Orthogonal and Symplectic Galois Representations Jeremy Booher submitted July 2016, corrections from paper version incorporated October 2016 Contents 1 Introduction 3 1.1 Motivation and Results . 4 1.1.1 Background about Galois Representations . 4 1.1.2 Geometric Lifts . 5 1.1.3 Relation to Previous Work . 6 1.2 Overview of the Proof . 6 1.2.1 Generalizing Ramakrishna's Method . 6 1.2.2 Fontaine-Laffaille Deformation Condition . 8 1.2.3 Minimally Ramified Deformation Condition . 9 1.3 Acknowledgements . 11 2 Producing Geometric Deformations 12 2.1 Preliminaries about Algebraic Groups . 12 2.1.1 Good and Very Good Primes . 12 2.1.2 Finite Groups of Lie Type . 13 2.2 Review of Galois Cohomology and Deformations . 15 2.2.1 Results about Galois Cohomology . 15 2.2.2 Deformations of Galois Representations . 17 2.3 Big Representations . 20 2.3.1 Big Representations . 20 2.3.2 Checking the Assumptions . 22 2.4 Ramakrishna's Deformation Condition . 23 2.4.1 Constructing the Deformation Condition . 24 2.4.2 Shrinking the Dual Selmer Group . 27 2.5 Generalizing Ramakrishna's Method . 31 2.5.1 Odd Galois Representations . 31 2.5.2 Local Lifting Implies Global Lifting . 32 2.5.3 Choosing Deformation Conditions . 33 2.5.4 Lifting Representations valued in Tori . 35 3 Fontaine-Laffaille Deformations with Pairings 37 3.1 Fontaine-Laffaille Theory with Pairings . 37 3.1.1 Covariant Fontaine-Laffaille Theory . 38 3.1.2 Tensor Products and Freeness . 40 3.1.3 Duality . 43 3.2 Fontaine-Laffaille Deformation Condition . 46 3.2.1 Definitions and Basic Properties . 46 3.2.2 Liftability . 48 3.2.3 Tangent Space . 52 1 4 Minimally Ramified Deformations 56 4.1 Representatives for Nilpotent Orbits . 57 4.1.1 Algebraically Closed Fields . 57 4.1.2 Representatives of Nilpotent Orbits . 58 4.2 Smoothness of Centralizers of Pure Nilpotents . 62 4.2.1 Centralizers over Fields . 62 4.2.2 Centralizers in Classical Cases . 64 4.2.3 Checking Flatness over a Dedekind Base . 66 4.2.4 Smooth Centralizers . 67 4.3 Deformations of Nilpotent Elements . 72 4.3.1 Pure Nilpotent Lifts . 72 4.4 The Minimally Ramified Deformation Condition for Tamely Ramified Representations . 73 4.4.1 Passing between Unipotents and Nilpotents . 73 4.4.2 Minimally Ramified Deformations . 74 4.4.3 Deformation Conditions Based on Parabolic Subgroups . 76 4.5 Minimally Ramified Deformations of Symplectic and Orthogonal Groups . 77 4.5.1 Decomposing Representations . 77 4.5.2 Extension of Representations . 80 4.5.3 Lifts with Pairings . 83 2 Chapter 1 Introduction Galois representations are central in modern number theory, perhaps most famously in the proof of Fermat's Last Theorem. There are natural sources of Galois representations in algebraic geometry, and the Langlands program conjecturally connects them with automorphic forms. For example, given a classical modular eigenform there is an associated two-dimensional Galois representation which we say is modular. Another example comes from an elliptic curve defined over Q: the absolute Galois group of the rationals acts on the p-adic Tate module, giving a two-dimensional Galois representation over the p-adic integers. Fermat's Last Theorem follows from the fact that the Galois representations arising from elliptic curves are modular. More generally, Galois representations arise from algebraic geometry through the action of the Galois group on the p-adic ´etalecohomology of a variety defined over a number field. Such representations are ramified at finitely many primes and are potentially semistable above p. An absolutely irreducible p-adic Galois representation satisfying these latter properties is said to be geometric. The Fontaine-Mazur conjecture asserts that all geometric Galois representations arise as subsets of Tate twists of p-adic ´etalecohomology. A more refined version has been proven in the two-dimensional case: two-dimensional geometric Galois representations of Gal(Q=Q) are modular. A key technique in answering these questions is the deformation theory of a residual Galois representation ρ : Gal(Q=Q) ! GLn(Fq). Understanding the universal deformation ring (the ring Rρ which parametrizes lifts of ρ to complete Noetherian local rings with residue field Fq) gives results about Galois representations. Residual representations are also interesting in their own right. In the two-dimensional case, Serre's con- jecture states that every odd irreducible representation is the reduction of a modular Galois representation. This was proven by Khare and Winterberger. Generalizations of Serre's conjecture to groups other than GL2 have been proposed, most recently by Gee, Herzig, and Savitt [GHS15]. The combination of Serre's conjecture and the Fontaine-Mazur conjecture motivates the study of when a residual representation admits a geometric lift to characteristic zero. In particular, we are interested in the following situation. Let K be a finite extension of Q with absolute Galois group ΓK . Suppose k is a finite field of characteristic p, O the ring of integers in a p-adic field with residue field O=m = k, and G is a connected reductive group defined over O. For a continuous representation ρ :ΓK ! G(k), in light of these conjectures it is important to study when there exists a 1 continuous representation ρ :ΓK ! G(O) lifting ρ that is geometric in a suitable sense over O[ p ] (using G,! GLn). Suppose K = Q. When G = GL2, Ramakrishna developed a technique to produce geometric lifts [Ram02]. His results provided evidence for Serre's conjecture. In the symplectic and orthogonal cases, we generalize this method and prove the existence of geometric lifts in favorable conditions. Two essential hypotheses are that ρ is odd (as defined in x2.5.1) and that ρ restricted to the decomposition group at p looks like the reduction of a crystalline representation with distinct Hodge-Tate weights. For precise statements, see Theorem 1.1.2.2 and Theorem 2.5.3.4. This provides evidence for generalizations of Serre's conjecture. In contrast, when G = GLn with n > 2, the representation ρ cannot be odd, and the method does not apply. In such cases, there is no expectation that such lifts exist. Ramakrishna's method works by establishing a local-to-global result for lifting Galois representations subject to local constraints. Let ρ be a lift of ρ to O=mn. Provided a cohomological obstruction vanishes, 3 n+1 it is possible to lift ρ over O=m subject to local constraints if (and only if) it possible to lift ρjΓv over O=mn+1 for all v in a fixed set of places of Q containing p and the places where ρ is ramified. In Chapter 2, we generalize Ramakrishna's argument so it can be applied to any connected reductive group and show that the cohomological obstruction vanishes if we allow controlled ramification at additional places at which ρ is unramified. It remains to pick local deformation conditions at p and at places where ρ is ramified which are liftable in the sense that it is always possible to suitably lift ρjΓv . At p, we define a Fontaine-Laffaille deformation condition in Chapter 3 by using deformations arising from Fontaine-Laffaille modules that carry extra data corresponding to a symmetric or alternating pairing. At a prime ` 6= p where ρ is ramified, we generalize the minimally ramified deformation condition defined for GLn in [CHT08, x2.4.4]. In simple cases, this deformation condition controls the ramification of ρ by controlling deformations of a unipotent element u of GLn(k). There is a natural parabolic k-subgroup containing u, and the deformation condition is analyzed by deforming this parabolic subgroup and then lifting u inside this subgroup. This idea does not work for other algebraic groups. In Example 1.2.3.2 and x4.4.3, we discuss an explicit example in GSp4 where the analogous deformation based on parabolics is provably not liftable. In Chapter 4, we define a minimally ramified deformation condition by instead requiring that u deform so that \it lies in the same unipotent orbit as u," and explain that this is a general phenomenon. The discovery and study of this deformation condition at ramified places ` 6= p (see x1.2.3) is the key innovation in this thesis. For GLn, our notion agrees with minimally ramified deformation of [CHT08], but for other groups it is a genuinely different (liftable) deformation condition. 1.1 Motivation and Results We will now review the background and state the results in more detail. 1.1.1 Background about Galois Representations Let p be a prime. Definition 1.1.1.1. A Galois representation ρ :ΓQ ! GL2(Qp) is modular if there exists a modular eigenform f such that the Galois representation associated to f at the prime p is isomorphic to ρ. A residual Galois representation ρ :ΓQ ! GL2(Fp) is modular if it is the reduction of a modular representation ρ. Serre's conjecture is one of the ingredients which motivated Ramakrishna's lifting result for two dimen- sional representations. Fact 1.1.1.2 (Serre's Conjecture). Let ρ :ΓQ ! GL2(Fp) be an odd irreducible representation. Then ρ is modular. Remark 1.1.1.3. More precise versions give the weight and level of an associated modular form. This conjecture was proven by Khare and Winterberger [KW09a] [KW09b]. Let K be a number field with absolute Galois group ΓK . Next we define two notions necessary to state the Fontaine-Mazur conjecture. Definition 1.1.1.4. An irreducible representation of ΓK over Qp comes from geometry if it is isomorphic i to a sub-quotient of H´et(XK ; Qp(r)) for some variety X over K and some r 2 Z.
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