DOCSLIB.ORG

ON the TATE MODULE of a NUMBER FIELD II 1. Introduction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
ON the TATE MODULE of a NUMBER FIELD II 1. Introduction ON THE TATE MODULE OF A NUMBER FIELD II SOOGIL SEO Abstract. We generalize a result of Kuz'min on a Tate module of a number field k. For a fixed prime p, Kuz'min described the inverse limit of the p part of the p-ideal class groups over the cyclotomic Zp-extension in terms of the global and local universal norm groups of p-units. This result plays a crucial rule in studying arithmetic of p-adic invariants especially the generalized Gross conjecture. We extend his result to the S-ideal class group for any finite set S of primes of k. We prove it in a completely different way and apply it to study the properties of various universal norm groups of the S-units. 1. Introduction S For a number field k and an odd prime p, let k1 = n kn be the cyclotomic n Zp-extension of k with kn the unique subfield of k1 of degree p over k. For m ≥ n, let Nm;n = Nkm=kn denote the norm map from km to kn and let Nn = Nn;0 denote the norm map from kn to the ground field k0 = k. H × ≥ Huniv For a subgroup n of kn ; n 0, let k be the universal norm subgroup of H k defined as follows \ Huniv H k = Nn n n≥0 univ and let (Hn ⊗Z Zp) be the universal norm subgroup of Hk ⊗Z Zp is defined as follows \ univ (Hk ⊗Z Zp) = Nn(Hn ⊗Z Zp): n≥0 The natural question whether the norm functor commutes with intersections of subgroups k× was already given in many articles and is related with algebraic and arithmetic problems including the generalized Gross conjecture and the Leopoldt conjecture. See for example, Bertrandias and Payan [1], Coleman [4], Gras [5], Gross [6], Jaulent [9], Kolster [10], Kuz'min [11] and [12]. Many applications are mainly for the global units and the global S-units for a set S of finite primes of k. In spite that it is very simple to describe the problem, it is very difficult to find solutions even in simpler cases. In [15], [16] and [17], we try to give certain informations on these problems up to torsion, that is, the ranks of various groups. The difficulties occur in the infinite class field theory with its complicate idele topology over k1=k. We will search for a criterion over which the two functors of the universal and the intersection commutes exactly for the global units and the global S-units of k. Firstly we will generalize a result of Kuz'min with a new and clear proof. Secondly we will apply this result to the question mentioned. For a finite cyclic extension K=k and for a finite set S of primes of k, let Uk(S) be the global S-units of k. Let S0 be the set of primes of K lying over each prime 1 2 SOOGIL SEO v 2 S, S0 = fwjv ; v 2 Sg: 0 0 We will often write UK (S) for the S -units UK (S ) of K. Let Y Y × × JK;S = Uv kv w2 =S0 w2S0 be the S-idele group where we identify UK (S) with a subgroup of JK;S via the diagonal imbedding ϕK;S : UK (S) −! JK;S. For K = kn, we write Un(S) = Ukn (S); ϕn = ϕkn;S: We have the following exact sequence −! −! −!ϕn b 0 1 ker(ϕn) Uk(S) H (Gn;Jkn;S) where Gn = G(kn=k) denotes the Galois group of kn=k. Then Hasse's local-global norm theorem for k× and the fact that every global norm is a local norm imply that ⊂ \ × NnUn(S) ker(ϕn) = Uk(S) Nnkn : By tensoring with Zp, the exact sequence induces −! ⊗ Z −! ⊗ Z −!ϕn b 0 1 ker(ϕn) = ker(ϕn) p Uk(S) p H (Gn;Jkn;S) together with ⊗ Z ⊂ ⊗ Z \ × ⊗ Z Nn(Un(S) p) ker(ϕn) p = (Uk(S) Nnkn ) p: We let \ \ ⊗ Z ker(ϕ1(S)) = ker(ϕn) = (ker(ϕn) p) n≥0 n≥0 for the intersection over all layers. Let Γ denote the procyclic group G(k1=k) and for each n ≥ 0, let Γn = n G(k1=kn) be the unique subgroup of Γ with index p . Let S = Sn be a finite set of primes of kn. Let Cln be the ideal class group of kn and let Cln(S) = Cln=hcl(v)iv2S be the S-ideal class group of kn where cl(v) 2 Cln is the ideal class containing v. We define the S-Tate module Tk(S) of k following Kuz'min as the inverse limit of Cln(S) with respect to the norm maps Tk(S) = lim− An(S) n where An(S) = Cln(S) ⊗ Zp. For the group In(S) of fractional S-ideals of kn, let ⊗ Z I1(S) = lim− im(In(S) p) be the inverse limit of the image im(In(S) ⊗ Zp) of (In(S) ⊗ Zp) with respect to the norm maps inside An(S) induced from the exact sequence 1 −! Pn(S) ⊗ Zp −! In(S) ⊗ Zp −! An(S) −! 1 where Pn(S) denotes the S-principal ideals of kn. The following theorem is a generalized version of Kuz'min's result for any set S of primes of k. Hence when S is empty, then the Tate module can be described in terms of the global units. We will prove the main result in a totally different way ON THE TATE MODULE OF A NUMBER FIELD II 3 using genus theory for p-ideal class group via cohomology theory rather than the infinite class field theory as was used by Kuz'min. The advantage of our proof is that it describes clearly the isomorphism and it is easy to generalize for an arbitrary set S compared to Kuz'min's proof. Let On(S) be the ring of S-integers of kn. For each generating class (cl(an) ⊗ 2 Γ 2 × − O 1)n Tk(S) and αn kn such that (γ 1)an = αn n(S), let × ⊗ Z univ \ ⊗ Z Γ ker(ϕ1(S)) (k p) (Uk(S) p) 1 ! (S): Tk(S) univ = univ (Uk(S) ⊗ Zp) (Uk(S) ⊗ Zp) be defined as 1(S)((cl(an) ⊗ 1)) = Nn(αn) ⊗ 1 which is well defined modulo Nn(Un(S) ⊗ Zp) for all n. Theorem 1.1. Let k be a number field and let S be any finite set of primes of k. Then 1(S) defines an isomorphism Γ ∼ × univ Tk(S) = (k ⊗ Zp) \ (Uk(S) ⊗ Zp) 1 ! (S): Γ univ : I1(S) (Uk(S) ⊗ Zp) Let Sk(p) be the set of primes of k dividing p. Since the universal norm group and the norm comparable group are identical over the compact subgroups, Theorem 1.1 can be written as the following corollary. Corollary 1.2. For a finite set S of primes of k, Γ ∼ × ⊗ Z univ \ \ ⊗ Z Tk(S) != (k p) (Uk(S Sp) p) Γ univ I1(S) (Uk(S \ Sp) ⊗ Zp) Γ Tk(S \ Sp) = Γ : I1(S \ Sp) Γ We remark that the statement that for the set S of the primes dividing p, Tk(S) is finite is an equivalent version of the generalized Gross conjecture (see [6], [9] and [10]). Note that for a set S with S ⊃ Sk(p), it recovers the result of Kuz'min ∼ × ⊗ Z univ Γ != (k p) Tk(Sp) univ : (Uk(Sp) ⊗ Zp) When S \ Sp is the empty set ϕ, Corollary 1.2 contains information on the global units Uk since Uk(S \ Sp) is equal to Uk. By comparing these cases that S \ Sp = ϕ and that S ⊃ Sp, we have the following proposition. Proposition 1.3. Let S be a finite set of primes containing all the primes dividing p. If the primes in S are principal over all intermediate layers, then the identity map induces the following isomorphism × univ × univ (k ⊗ Zp) \ (Uk ⊗ Zp) ∼ (k ⊗ Zp) univ = univ : (Uk ⊗ Zp) (Uk(S) ⊗ Zp) In particular, it follows that univ univ (Uk(S) ⊗ Zp) \ (Uk ⊗ Zp) = (Uk ⊗ Zp) and univ \ univ Uk(S) Uk = Uk : 4 SOOGIL SEO 2. Proofs of the main results Let L=k be a finite Galois extension. For a finite set S of finite primes of k, let Ik(S);Pk(S) and Uk(S) be the group of S-ideals of k, the subgroup of principal S-ideals in Ik(S) and the S-units of k respectively. Similarly we define IL(S);PL(S) 0 0 and UL(S) as the the group of S -ideals of L, the subgroup of principal S -ideals 0 0 in IL(S) and the S -units of L respectively where S denotes the prime ideals of 0 L dividing the prime ideals of S. Let ClL(S) = IL(S)=PL(S) be the S -ideal class group of L and A (S) = Cl (S) ⊗ Z : L L S p Z Over the cyclotomic p-extension k1 = n kn of k, let Tk(S) = lim− An(S) the inverse limit of An(S) with respect to the norm maps which will be called the S-Tate module. For the group In(S) of fractional S-ideals of kn, let ⊗ Z I1(S) = lim− im(In(S) p) be the inverse limit of the image im(In(S) ⊗ Zp) of (In(S) ⊗ Zp) with respect to the norm maps inside An(S) induced from the exact sequence 1 −! Pn(S) ⊗ Zp −! In(S) ⊗ Zp −! An(S) −! 1 where Pn(S) denotes the S-principal ideals of kn.
Load more

Recommended publications
  • Generalization of Anderson's T-Motives and Tate Conjecture
    数理解析研究所講究録 第 884 巻 1994 年 154-159 154 Generalization of Anderson’s t-motives and Tate conjecture AKIO TAMAGAWA $(\exists\backslash i]||^{\wedge}x\ovalbox{\tt\small REJECT} F)$ RIMS, Kyoto Univ. $(\hat{P\backslash }\#\beta\lambda\doteqdot\ovalbox{\tt\small REJECT}\Phi\Phi\Re\Re_{X}^{*}ffi)$ \S 0. Introduction. First we shall recall the Tate conjecture for abelian varieties. Let $B$ and $B’$ be abelian varieties over a field $k$ , and $l$ a prime number $\neq$ char $(k)$ . In [Tat], Tate asked: If $k$ is finitely genera $tedo1^{\gamma}er$ a prime field, is the natural homomorphism $Hom_{k}(B', B)\bigotimes_{Z}Z_{l}arrow H_{0}m_{Z_{l}[Ga1(k^{sep}/k)](T_{l}(B'),T_{l}(B))}$ an isomorphism? This is so-called Tate conjecture. It has been solved by Tate ([Tat]) for $k$ finite, by Zarhin ([Zl], [Z2], [Z3]) and Mori ([Ml], [M2]) in the case char$(k)>0$ , and, finally, by Faltings ([F], [FW]) in the case char$(k)=0$ . We can formulate an analogue of the Tate conjecture for Drinfeld modules $(=e1-$ liptic modules ([Dl]) $)$ as follows. Let $C$ be a proper smooth geometrically connected $C$ curve over the finite field $F_{q},$ $\infty$ a closed point of , and put $A=\Gamma(C-\{\infty\}, \mathcal{O}_{C})$ . Let $k$ be an A-field, i.e. a field equipped with an A-algebra structure $\iota$ : $Aarrow k$ . Let $k$ $G$ and $G’$ be Drinfeld A-modules over (with respect to $\iota$ ), and $v$ a maximal ideal $\neq Ker(\iota)$ . Then, if $k$ is finitely generated over $F_{q}$ , is the natural homomorphism $Hom_{k}(G', G)\bigotimes_{A}\hat{A}_{v}arrow Hom_{\hat{A}_{v}[Ga1(k^{sep}/k)]}(T_{v}(G'), T_{v}(G))$ an isomorphism? We can also formulate a similar conjecture for Anderson’s abelian t-modules ([Al]) in the case $A=F_{q}[t]$ .
    [Show full text]
  • Arxiv:2012.03076V1 [Math.NT]
    ODONI’S CONJECTURE ON ARBOREAL GALOIS REPRESENTATIONS IS FALSE PHILIP DITTMANN AND BORYS KADETS Abstract. Suppose f K[x] is a polynomial. The absolute Galois group of K acts on the ∈ preimage tree T of 0 under f. The resulting homomorphism ρf : GalK Aut T is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian→ fields K there exists a polynomial f for which ρf is surjective. We show that this conjecture is false. 1. Introduction Suppose that K is a field and f K[x] is a polynomial of degree d. Suppose additionally that f and all of its iterates f ◦k(x)∈:= f f f are separable. To f we can associate the arboreal Galois representation – a natural◦ ◦···◦ dynamical analogue of the Tate module – as − ◦k 1 follows. Define a graph structure on the set of vertices V := Fk>0 f (0) by drawing an edge from α to β whenever f(α) = β. The resulting graph is a complete rooted d-ary ◦k tree T∞(d). The Galois group GalK acts on the roots of the polynomials f and preserves the tree structure; this defines a morphism φf : GalK Aut T∞(d) known as the arboreal representation attached to f. → ❚ ✐✐✐✐ 0 ❚❚❚ ✐✐✐✐ ❚❚❚❚ ✐✐✐✐ ❚❚❚❚ ✐✐✐✐ ❚❚❚ ✐✐✐✐ ❚❚❚❚ ✐✐✐✐ ❚❚❚ √3 √3 ❏ t ❑❑ ✈ ❏❏ tt − ❑❑ ✈✈ ❏❏ tt ❑❑ ✈✈ ❏❏ tt ❑❑❑ ✈✈ ❏❏ tt ❑ ✈✈ ❏ p3 √3 p3 √3 p3+ √3 p3+ √3 − − − − Figure 1. First two levels of the tree T∞(2) associated with the polynomial arXiv:2012.03076v1 [math.NT] 5 Dec 2020 f = x2 3 − This definition is analogous to that of the Tate module of an elliptic curve, where the polynomial f is replaced by the multiplication-by-p morphism.
    [Show full text]
  • Modular Galois Represemtations
    Modular Galois Represemtations Manal Alzahrani November 9, 2015 Contents 1 Introduction: Last Formulation of QA 1 1.1 Absolute Galois Group of Q :..................2 1.2 Absolute Frobenius Element over p 2 Q :...........2 1.3 Galois Representations : . .4 2 Modular Galois Representation 5 3 Modular Galois Representations and FLT: 6 4 Modular Artin Representations 8 1 Introduction: Last Formulation of QA Recall that the goal of Weinstein's paper was to find the solution to the following simple equation: QA: Let f(x) 2 Z[x] irreducible. Is there a "rule" which determine whether f(x) split modulo p, for any prime p 2 Z? This question can be reformulated using algebraic number theory, since ∼ there is a relation between the splitting of fp(x) = f(x)(mod p) and the splitting of p in L = Q(α), where α is a root of f(x). Therefore, we can ask the following question instead: QB: Let L=Q a number field. Is there a "rule" determining when a prime in Q split in L? 1 0 Let L =Q be a Galois closure of L=Q. Since a prime in Q split in L if 0 and only if it splits in L , then to answer QB we can assume that L=Q is Galois. Recall that if p 2 Z is a prime, and P is a maximal ideal of OL, then a Frobenius element of Gal(L=Q) is any element of FrobP satisfying the following condition, FrobP p x ≡ x (mod P); 8x 2 OL: If p is unramifed in L, then FrobP element is unique.
    [Show full text]
  • The Geometry of Lubin-Tate Spaces (Weinstein) Lecture 1: Formal Groups and Formal Modules
    The Geometry of Lubin-Tate spaces (Weinstein) Lecture 1: Formal groups and formal modules References. Milne's notes on class field theory are an invaluable resource for the subject. The original paper of Lubin and Tate, Formal Complex Multiplication in Local Fields, is concise and beautifully written. For an overview of Dieudonn´etheory, please see Katz' paper Crystalline Cohomology, Dieudonn´eModules, and Jacobi Sums. We also recommend the notes of Brinon and Conrad on p-adic Hodge theory1. Motivation: the local Kronecker-Weber theorem. Already by 1930 a great deal was known about class field theory. By work of Kronecker, Weber, Hilbert, Takagi, Artin, Hasse, and others, one could classify the abelian extensions of a local or global field F , in terms of data which are intrinsic to F . In the case of global fields, abelian extensions correspond to subgroups of ray class groups. In the case of local fields, which is the setting for most of these lectures, abelian extensions correspond to subgroups of F ×. In modern language there exists a local reciprocity map (or Artin map) × ab recF : F ! Gal(F =F ) which is continuous, has dense image, and has kernel equal to the connected component of the identity in F × (the kernel being trivial if F is nonarchimedean). Nonetheless the situation with local class field theory as of 1930 was not completely satisfying. One reason was that the construction of recF was global in nature: it required embedding F into a global field and appealing to the existence of a global Artin map. Another problem was that even though the abelian extensions of F are classified in a simple way, there wasn't any explicit construction of those extensions, save for the unramified ones.
    [Show full text]
  • On the Structure of Galois Groups As Galois Modules
    ON THE STRUCTURE OF GALOIS GROUPS AS GALOIS MODULES Uwe Jannsen Fakult~t f~r Mathematik Universit~tsstr. 31, 8400 Regensburg Bundesrepublik Deutschland Classical class field theory tells us about the structure of the Galois groups of the abelian extensions of a global or local field. One obvious next step is to take a Galois extension K/k with Galois group G (to be thought of as given and known) and then to investigate the structure of the Galois groups of abelian extensions of K as G-modules. This has been done by several authors, mainly for tame extensions or p-extensions of local fields (see [10],[12],[3] and [13] for example and further literature) and for some infinite extensions of global fields, where the group algebra has some nice structure (Iwasawa theory). The aim of these notes is to show that one can get some results for arbitrary Galois groups by using the purely algebraic concept of class formations introduced by Tate. i. Relation modules. Given a presentation 1 + R ÷ F ÷ G~ 1 m m of a finite group G by a (discrete) free group F on m free generators, m the factor commutator group Rabm = Rm/[Rm'R m] becomes a finitely generated Z[G]-module via the conjugation in F . By Lyndon [19] and m Gruenberg [8]§2 we have 1.1. PROPOSITION. a) There is an exact sequence of ~[G]-modules (I) 0 ÷ R ab + ~[G] m ÷ I(G) + 0 , m where I(G) is the augmentation ideal, defined by the exact sequence (2) 0 + I(G) ÷ Z[G] aug> ~ ÷ 0, aug( ~ aoo) = ~ a .
    [Show full text]
  • Galois Module Structure of Lubin-Tate Modules
    GALOIS MODULE STRUCTURE OF LUBIN-TATE MODULES Sebastian Tomaskovic-Moore A DISSERTATION in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2017 Supervisor of Dissertation Ted Chinburg, Professor of Mathematics Graduate Group Chairperson Wolfgang Ziller, Professor of Mathematics Dissertation Committee: Ted Chinburg, Professor of Mathematics Ching-Li Chai, Professor of Mathematics Philip Gressman, Professor of Mathematics Acknowledgments I would like to express my deepest gratitude to all of the people who guided me along the doctoral path and who gave me the will and the ability to follow it. First, to Ted Chinburg, who directed me along the trail and stayed with me even when I failed, and who provided me with a wealth of opportunities. To the Penn mathematics faculty, especially Ching-Li Chai, David Harbater, Phil Gressman, Zach Scherr, and Bharath Palvannan. And to all my teachers, especially Y. S. Tai, Lynne Butler, and Josh Sabloff. I came to you a student and you turned me into a mathematician. To Philippe Cassou-Nogu`es,Martin Taylor, Nigel Byott, and Antonio Lei, whose encouragement and interest in my work gave me the confidence to proceed. To Monica Pallanti, Reshma Tanna, Paula Scarborough, and Robin Toney, who make the road passable using paperwork and cheer. To my fellow grad students, especially Brett Frankel, for company while walking and for when we stopped to rest. To Barbara Kail for sage advice on surviving academia. ii To Dan Copel, Aaron Segal, Tolly Moore, and Ally Moore, with whom I feel at home and at ease.
    [Show full text]
  • The Mumford-Tate Conjecture for Drinfeld-Modules
    MUMFORD-TATE CONJECTURE 1 THE MUMFORD-TATE CONJECTURE FOR DRINFELD-MODULES by Richard PINK* Abstract Consider the Galois representation on the Tate module of a Drinfeld module over a finitely generated field in generic characteristic. The main object of this paper is to determine the image of Galois in this representation, up to commensurability. We also determine the Dirichlet density of the set of places of prescribed reduction type, such as places of ordinary reduction. §0. Introduction Let F be a finitely generated field of transcendence degree 1 over a finite field of characteristic p. Fix a place ∞ of F , and let A be the ring of elements of F which are regular outside ∞. Consider a finitely generated extension K of F and a Drinfeld module ϕ : A → EndK (Ga) of rank n ≥ 1 (cf. Drinfeld [10]). In other words K is a finitely generated field of transcendence degree ≥ 1 over Fp, and ϕ has “generic characteristic”. sep Let K ⊂ K¯ denote a separable, respectively algebraic closure of K. Let Fλ denote the completion of F at a place λ. If λ 6= ∞ we have a continuous representation sep ρλ : Gal(K /K) −→ GLn(Fλ) Communicated by Y. Ihara, September 20, 1996. 1991 Mathematics Subject Classifications: 11G09, 11R58, 11R45 * Fakult¨at f¨ur Mathematik und Informatik, Universit¨at Mannheim, D-68131 Mannheim, Germany 2 RICHARD PINK which describes the Galois action on the λ-adic Tate module of ϕ. The main goal of this article is to give a qualitative characterization of the image of ρλ. Here the term “qualitative” refers to properties that are shared by all open subgroups, i.e.
    [Show full text]
  • Sato-Tate Distributions
    SATO-TATE DISTRIBUTIONS ANDREW V.SUTHERLAND ABSTRACT. In this expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and we give an explicit formulation of the Sato- Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension g 3 and compute some of the corresponding trace distributions. This article is based on a series of lectures≤ presented at the 2016 Arizona Winter School held at the Southwest Center for Arithmetic Geometry. 1. AN INTRODUCTION TO SATO-TATE DISTRIBUTIONS Before discussing the Sato-Tate conjecture and Sato-Tate distributions in the context of abelian vari- eties, let us first consider the more familiar setting of Artin motives (varieties of dimension zero). 1.1. A first example. Let f Z[x] be a squarefree polynomial of degree d. For each prime p, let 2 fp (Z=pZ)[x] Fp[x] denote the reduction of f modulo p, and define 2 ' Nf (p) := # x Fp : fp(x) = 0 , f 2 g which we note is an integer between 0 and d. We would like to understand how Nf (p) varies with p. 3 The table below shows the values of Nf (p) when f (x) = x x + 1 for primes p 60: − ≤ p : 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 Nf (p) 00111011200101013 There does not appear to be any obvious pattern (and we should know not to expect one, because the Galois group of f is nonabelian).
    [Show full text]
  • Galois Representations
    Galois representations London Taught Course Centre Lecture 3, 30 January 2017 5. Local Galois representations 5.1. Restricting global Galois representations For L=Q finite Galois, Pjp, we defined IP ⊂ GP ⊂ Gal(L=Q); depending, up to conjugacy, only on p , where: I IP = inertia group, trivial if p - Disc(L), ∼ I DP =IP = Gal(Fpf =Fp) = hFrobP i, where Fpf = OL=P. An Artin representation: Gal(Q=Q) ! Gal(L=Q) ! GLm(C); is determined (up to equivalence) by the “local data” p 7! ρ(Frobp) := [ρ(FrobP )]; for p - Disc(L). An `-adic representation (e.g., `-adic cyclotomic character): ρ : Gal(Q=Q) ! GLm(Q`); might not factor through Gal(L=Q) for a finite L=Q. Q ⊂ Q The inclusions \\ induce a homomorphism: Qp ⊂ Qp Gal(Qp=Qp) ! Gal(Q=Q); I well-defined up to conjugacy in Gal(Q=Q), I injective by Krasner’s Lemma (which ) Qp = QQp). As “local data” consider restrictions ρj . Gal(Qp=Qp) For finite F over Qp, the p-adic metric extends uniquely to F (so also uniquely to Qp). Let OF = f α 2 F j jαj ≤ 1 g, complete DVR, with maximal ideal PF = f α 2 F j jαj < 1 g: e ∼ Let pOF = PF and OF =PF = Fpf , so [F : Qp] = ef . F is unramified if e = 1, define F0 ⊂ F, maximal unramified. For L finite over Q, L ⊂ Q ⊂ Qp determines Pjp. = = O = n Let LP closure of L in Qp, field of fractions of lim− L P , Taking F = LP (finite over Qp, as above), we have e = eP ; OL=P = OF =PF ; f = fP : Every finite (Galois) F=Qp is LP for some finite (Galois) L=Q.
    [Show full text]
  • THE THEOREM of HONDA and TATE 11 We Have 1/2 [Lw : Fv] = [L0,W0 : F0,V0 ] = [L0 : F0] = [E : F ]
    THE THEOREM OF HONDA AND TATE KIRSTEN EISENTRAGER¨ 1. Motivation Consider the following theorem of Tate (proved in [Mum70, Theo- rems 2–3, Appendix I]): Theorem 1.1. Let A and B be abelian varieties over a finite field k of size q, and let fA, fB ∈ Z[T ] be the characteristic polynomials of their q-Frobenius endomorphisms, so fA and fB have degrees 2 · dimA and 2 · dimB with constant terms qdimA and qdimB respectively. The following statements are equivalent: (1) B is k-isogenous to an abelian subvariety of A. (2) fB|fA in Q[T ]. In particular, A is k-isogenous to B if and only if fA = fB. Moreover, A is k-simple if and only if fA is a power of an irreducible polynomial in Q[T ]. Recall that the roots of fA in C are “Weil q-integers”: algebraic integers whose images in C all have absolute value q1/2. By Theorem 1.1 to describe the isogeny classes of abelian varieties over finite fields we would like to know for which Weil q-integers π in C we can find a k- simple abelian variety A/k such that π is a root of fA. We will show that for each π, we can find a k-simple abelian variety A and an embedding 0 Q[π] ,→ Endk(A) such that π is the q-Frobenius endomorphism of A. We will follow the discussion in [Tat68] and [MW71]. Brian Conrad heavily revised Section 8 to simplify the computation of the invariant invv(E) for v | p and he provided the proofs for Theorem 8.3 and the results in the appendix.
    [Show full text]
  • P-Adic Hodge Theory (Math 679)
    MATH 679: INTRODUCTION TO p-ADIC HODGE THEORY LECTURES BY SERIN HONG; NOTES BY ALEKSANDER HORAWA These are notes from Math 679 taught by Serin Hong in Winter 2020, LATEX'ed by Aleksander Horawa (who is the only person responsible for any mistakes that may be found in them). Official notes for the class are available here: http: //www-personal.umich.edu/~serinh/Notes%20on%20p-adic%20Hodge%20theory.pdf They contain many more details than these lecture notes. This version is from July 28, 2020. Check for the latest version of these notes at http://www-personal.umich.edu/~ahorawa/index.html If you find any typos or mistakes, please let me know at [email protected]. The class will consist of 4 chapters: (1) Introduction, Section1, (2) Finite group schemes and p-divisible groups, Section2, (3) Fontaine's formalism, (4) The Fargues{Fontaine curve. Contents 1. Introduction2 1.1. A first glimpse of p-adic Hodge theory2 1.2. A first glimpse of the Fargues{Fontaine curve8 1.3. Geometrization of p-adic representations 10 2. Foundations of p-adic Hodge theory 11 2.1. Finite flat group schemes 11 2.2. Finite ´etalegroup schemes 18 2.3. The connected ´etalesequence 20 2.4. The Frobenius morphism 22 2.5. p-divisible groups 25 Date: July 28, 2020. 1 2 SERIN HONG 2.6. Serre{Tate equivalence for connected p-divisible groups 29 2.7. Dieudonn´e{Maninclassification 37 2.8. Hodge{Tate decomposition 40 2.9. Generic fibers of p-divisible groups 57 3. Period rings and functors 60 3.1.
    [Show full text]
  • Arxiv:1907.05144V3 [Math.NT] 5 Feb 2021 Edaaouso Litccre N Bla Aite.In Varieties
    ANDERSON T-MODULES WITH THIN T-ADIC GALOIS REPRESENTATIONS ANDREAS MAURISCHAT Abstract. Pink has given a qualitative answer to the Mumford-Tate conjecture for Drin- feld modules in the 90s. He showed that the image of the p-adic Galois representation is p-adically open in the motivic Galois group for any prime p. In contrast to this result, we provide a family of uniformizable Anderson t-modules for which the Galois representations of their t-adic Tate-modules are “far from” having t-adically open image in their motivic Galois groups. Nevertheless, the image is still Zariski-dense in the motivic Galois group which is in accordance to the Mumford-Tate conjecture. For the proof, we explicitly de- termine the motivic Galois group as well as the Galois representation for these t-modules. Contents 1. Introduction 1 2. Notation 3 3. Density of Galois representations 6 4. Prolongations of the Carlitz module 7 5. The Motivic Galois group 8 6. The Galois representation 9 References 12 1. Introduction Given an abelian variety of dimension d over a number field F , there are several groups attached to it. AmongA others, there is the associated Hodge group (also called Mumford-Tate group) which is an algebraic subgroup GA of GL(V ) ∼= GL2d,Q where V := H1( (C), Q) denotes the singular homology of (C) with respect to some embedding F A A n+1 → C. Furthermore, the rational ℓ-adic Tate module V ( ) = lim [ℓ ] Z Q of , ℓ A n A ⊗ ℓ ℓ A is naturally isomorphic to V Q Qℓ, and carries an action of←− the absolute Galois group Gal(F sep/F ) via the action on⊗ the torsion points, i.e.
    [Show full text]