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There is no abelian scheme over Z Contents Introduction 1 1 Nonexistence of Certain Abelian Varieties 3 1.1 Overview ........................................ 3 1.2 Preliminaries ................................... ... 3 1.2.1 FiniteFlatGroupSchemes . 3 1.2.1.1 AffineGroupSchemes. 3 1.2.1.2 FiniteFlatGroupSchemes . 5 1.2.1.3 K¨ahler Differentials on Affine Group Schemes . .... 6 1.2.1.4 Finite EtaleGroupSchemes´ .................... 6 1.2.1.5 Quotients, Cokernels and Exact Sequences . .... 7 1.2.1.6 Classification of Finite Flat Group Schemes . ..... 9 1.2.1.7 Prolongations of Commutative p-GroupSchemes . 12 1.2.2 p-divisibleGroups ............................... 13 1.2.2.1 BasicDefinitionsandProperties . 13 1.2.2.2 FormalLieGroups. 15 1.2.2.3 Passage to Special Fibers, Generic Fibers and Tate Modules . 17 1.2.2.4 Deformation of p-divisibleGroups . 19 1.2.2.5 Classification of p-divisibleGroups. 19 1.2.3 AbelianVarietiesandAbelianSchemes . ...... 21 1.2.3.1 RigidityandCommutativity . 21 1.2.3.2 Picard Schemes and Existence of Dual Abelian Schemes..... 23 1.2.3.3 IsogeniesandPolarizations . 26 1.2.3.4 DualityandtheWeilPairing . 29 1.2.3.5 N´eronModelsandReductions . 29 1.2.3.6 JacobiansofRelativeCurves . 32 1.2.3.7 Reduction Types Via p-divisibleGroups. 34 1.2.3.8 Tate Modules and Faltings’ Finiteness Theorems . ...... 35 1.3 Nonexistence of Abelian Scheme over Z ....................... 37 1.3.1 Fontaine’sRamificationBound . 37 1.3.1.1 Ramification of Complete Intersection Algebra . ...... 38 1.3.1.2 ConversetoKrasner’sLemma . 39 1.3.1.3 RamificationBound . 40 1.3.2 Constraints on p-groups and p-divisibleGroups . 43 1.3.2.1 ResultsofFontaine . 43 1.3.2.2 Restrictionson2-Groups . 49 1.4 Nonexistence of Certain Semi-stable Abelian Varieties over Q ........... 50 1.4.1 ResultsofSchoof............................... 51 1.4.1.1 The category p ........................... 51 Dℓ 1.4.1.2 Criterion for Appropriate Choice of Primes ℓ = p ......... 52 1 6 1.4.1.3 Calculation of ExtZ[1/ℓ](µp, Z/pZ) ................. 53 1.4.1.4 Simple Objects of p ........................ 56 Dℓ 1.4.2 ResultsofBrumer-Kramer. 57 1.4.2.1 IncreasingEffective Stage of Inertia . 58 1.4.2.2 FinishingtheProof . 59 2 Nonexistence of Certain Proper Schemes 61 2.1 Overview ........................................ 61 2.2 Preliminaries ................................... 62 2.2.1 EtaleCohomologyandtheWeilConjectures´ . 62 2.2.1.1 SitesandTopoi ........................... 62 2.2.1.2 Etale´ Site, Etale´ Sheaves and Etale´ Cohomology . 67 2.2.1.3 Ga, Gm, µn and Z/nZ ........................ 69 2.2.1.4 FinitenessConditionsonSheaves. 70 2.2.1.5 BaseChangeTheorems . 72 2.2.1.6 Cohomology with Proper Support and Finiteness Theorems. 73 2.2.1.7 K¨unneth Formula, Poincar´eDuality . 74 2.2.1.8 ℓ-adicCohomology.......................... 75 2.2.1.9 Weil Conjectures and the Hard Lefschetz Theorem . ..... 76 2.2.2 p-adicHodgeTheory.............................. 77 2.2.2.1 Ax-Sen-Tate Theorem and Galois Cohomology of Cp ....... 77 2.2.2.2 Hodge-TateRepresentations . 79 2.2.2.3 AdmissibleRepresentations . 81 2.2.2.4 DeRhamRepresentations. 83 2.2.2.5 Crystalline and Semi-stable Period Rings . ..... 87 2.2.2.6 Filtered (ϕ,N)-modules....................... 90 2.2.2.7 Etale´ ϕ-modules ........................... 92 2.2.2.8 Comparison Theorems I: Generalities, de Rham Cohomology . 94 2.2.2.9 Comparison Theorems II: Crystalline and Semi-stable Conjectures 97 2.2.3 Integral p-adicHodgeTheory . .100 2.2.3.1 Fontaine-LaffailleTheory . 100 2.2.3.2 BreuilModules. .102 2.2.3.3 KisinModules ............................105 2.2.3.4 (ϕ, G)-modules............................106 2.2.3.5 Torsion Kisin Modules and Torsion (ϕ, G)-modules. 108 2.3 Vanishing of Low-Degree“ Hodge Cohomologies: Good Reduction Case . 110 2.3.1 ResultsofFontaine. .. .. .. .. .“ .. .. .. 110 2.3.2 Ramification Bounds for Crystalline Representations ............113 2.4 Vanishing of Low-Degree Hodge Cohomologies: Semi-stable Reduction Case . 114 2.4.1 Ramification Bounds for Semi-stable Representations ............114 2.4.2 ResultsofAbrashkin. 116 References 116 Gyujin Oh There is no abelian scheme over Z Introduction In his 1962 ICM talk [Sh], Shafarevich suggested several conjectures regarding the finiteness of isomorphism classes of arithmetic objects having good reduction almost everywhere. Such problems can find their origins from basic finiteness theorems in algebraic number theory, es- pecially the Hermite-Minkowski theorem: for any integer N > 0 and a number field K, there are only finitely many number fields L such that the discriminant of L/K is at most N. A more geometric re-statement of the theorem is as follows. Theorem (Hermite-Minkowski). For any number field K, a finite set of primes S of K and an integer N > 0, there are only finitely many isomorphism classes of zero-dimensional varieties of degree at most N over K which possess a smooth model over Spec( ), where is the OK,S OK,S ring of S-integers in K. In this regard, we can state the Shafarevich conjectures in the following form. Conjecture (Shafarevich). Let K be a number field and S be a finite set of primes of K. Let g 2 be an integer. ≥ (a) (Shafarevich conjecture for curves) There are only finitely many isomorphism classes of smooth curves over of genus g. Equivalently, there are only finitely many isomorphism OK,S classes of curves over K of genus g having good reduction outside S. (b) (Shafarevich conjecture for abelian varieties) There are only finitely many isomorphism classes of abelian schemes over of dimension g. Equivalently, there are only finitely many OK,S isomorphism classes of abelian varieties over K of dimension g having good reduction outside S. In particular, Faltings [Fa] proved the Shafarevich conjectures in conjunction with various other finiteness results, including the finiteness of isogeny classes and the Mordell’s conjecture. On the other hand, there are some special cases where one can suspect whether the set of isomorphism classes of arithmetic object is actually empty. This can be motivated from the classic theorem of Minkowski that there is no nontrivial unramified extension of Q. We can as well geometrically re-interpret the statement as follows. Theorem (Minkowski). The only connected zero-dimensional variety over Q admitting a smooth model over Spec(Z) is Spec(Q). From this theorem, Shafarevich further conjectured that the sets of isomorphism classes considered in the Shafarevich conjectures are empty if K = Q and S = . In other words, ∅ Conjecture (Shafarevich). There is no nontrivial abelian scheme over Z. Equivalently, there is no nontrivial abelian variety over Q with everywhere good reduction. This conjecture is established independently by Fontaine [Fo1] and Abrashkin [Ab1], and this is the direction we will mostly focus on amongst many Shafarevich conjectures. The basic strategy behind the first proofs is to study ramification of finite flat group schemes and p-divisible groups. Specifically, if there is an abelian scheme A over Z, then for a prime p, the collection of pn-torsions A[pn] forms an object called a p-divisible group over Z. By { }n≥1 studying the ramification bounds on such objects, just like the proof of Minkowski’s theorem, the proofs show that, for a small prime p, a p-divisible group over Z is of very simple form, so simple that it cannot arise as a p-divisible group from p-power torsions of an abelian variety. Somehow in a different flavor, Fontaine [Fo2] and Abrashkin [Ab2] later revisited the nonex- istence of abelian scheme over Z. Instead of analyzing the ramification behavior of p-divisible 1 Gyujin Oh There is no abelian scheme over Z groups and torsion subgroups, which are objects only available to group schemes, they instead analyzed the ramification of p-adic ´etale cohomology as a Galois representation. This strategy became possible via the development of p-adic Hodge theory and its integral counterpart. This strategy enabled them to generalize the nonexistence results to certain smooth proper schemes with no group structure. In particular, they proved the following. Theorem (Fontaine, [Fo2, Th´eor`eme 1], [Ab2, 7.6]). Let X be a smooth proper variety over Q with everywhere good reduction. Then, Hi(X, Ωj ) = 0 for i = j, i + j 3. X 6 ≤ In particular, this implies the nonexistence of abelian scheme over Z as corollary. In this essay, we review the both approaches towards the proof of nonexistence of abelian variety over Q with everywhere good reduction. The first chapter will focus on the analysis of finite flat group schemes and p-divisible groups. In the chapter, we will go through the details of Fontaine’s original proof. In the chapter, we will also review some extensions of the result using the same kind of technique, most notably the one by Schoof [Sc1]; it gives the nonexistence of abelian varieties over a small number field with semi-stable reduction at one small prime and good reduction at everywhere else. We briefly examine the results due to Brumer-Kramer [BK], who used a different approach more in conjunction with the Shafarevich conjecture (or, Faltings’ Finiteness Theorem). The second chapter will be aiming for p-adic Hodge theoretic proofs of nonexistence of abelian schemes over Z. In particular, we will observe various classes of p-adic Galois represen- tations, and examine how to classify those representations in a different way. In particular, we will see p-adic Galois representations and their integral sublattices can be classified by mod- ules with various (semi)linear structures attached. Such modules then will have a similar kind of discriminant bound as p-divisible groups and finite flat group schemes have. In particu- lar, using integral p-adic Hodge theoretic constructions, including Fontaine-Laffaille modules, Breuil-Kisin modules and (ϕ, G)-modules, we give discriminant bounds for torsion crystalline and semi-stable representations.
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