Etale cohomology

We show that if is a pseudo-proper smooth noetherian formal scheme over a positive characteristic p field k then its De Rham complex τ ≤p (F /k * Ω • /k) is decomposable. Along the way we establish the Cartier isomorphism Ω i (p) / γ → H i (F / * Ω • /) associated to a map f : → of positive characteristic p noetherian formal schemes where (p) denotes the base change of along the Frobenius morphism of and F / denotes the relative Frobenius of over .

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In this paper we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [BK13] G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle forétale K-theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases this result is the best possible i.e if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle forétale K-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by S. Barańczuk in [B17]. We show that all our results remain valid for Quillen K-theory of X if the Bass and Quillen-Lichtenbaum conjectures hold true for X .

Let G be a connected reductive group defined over a local non-Archimedean field F with residue field F q ; let P be a parahoric subgroup with associated reductive quotient M. If σ is an irreducible cuspidal representation of M(F q) it provides an irreducible representation of P by inflation. We show that the pair (P , σ) is an S-type as defined by Bushnell and Kutzko. The cardinality of S can be bigger than one; we show that if one replaces P by the full centraliser P of the associated facet in the enlarged affine building of G, and σ by any irreducible smooth representationσ ofP which contains σ on restriction then (P ,σ) is an s-type for a singleton set s. Our methods employ invertible elements in the associated Hecke algebra H (σ) and they imply that the appropriate parabolic induction functor and its left adjoint can be realised algebraically via pullbacks from ring homomorphisms.

The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian prop Galois groups with ramification allowed at a maximal set of primes over p such that the module is torsion. A main conjecture for such an Iwasawa module describes its codimension one support in terms of a p-adic L-function attached to the primes of ramification. In this paper, we study more general and potentially much smaller modules that are quotients of exterior powers of Iwasawa modules with ramification at a set of primes over p by sums of exterior powers of inertia subgroups. We show that the higher codimension support of such quotients can be measured by finite collections of characteristic ideals of classical Iwasawa modules, hence by p-adic L-functions under the relevant CM main conjectures. 1 Introduction Iwasawa theory studies the growth of Selmer groups in towers of number fields. In the commutative setting, these towers have Galois groups isomorphic to Z r p for some r ≥ 1, and their Iwasawa algebras are isomorphic to a power series ring in r variables over Z p. The Selmer groups are typically attached to Galois-stable lattices in p-adic Galois representations that come from geometry. The local conditions defining the Selmer groups F. M.

γ = c0 + c1p + c2p + · · · = (. . . c3c2c1c0)p, with ci ∈ Z, 0 ≤ ci ≤ p− 1, called the digits of γ. This ring has a topology given by a restriction of the product topology—we will see this below. The ring Zp can be viewed as Z/pZ for an ‘infinitely high’ power n. This is a useful idea, for example, in the study of Diophantine equations: if such an equation has a solution in the integers, then it must have a solution modulo p for all n: to prove it does not have a solution, therefore, it suffices to show that it does not have a solution in Zp for some prime p. We can express the expansion of elements in Zp as Zp = lim ←− n Z/pZ

We show that if X is a pseudo-proper smooth noetherian formal scheme over a positive characteristic p field k then its De Rham complex τ ≤p (F X/k * Ω • X/k) is decomposable. Along the way we establish the Cartier isomorphism Ω i X (p) /Y γ → H i (F X/Y * Ω • X/Y) associated to a map f : X → Y of positive characteristic p noetherian formal schemes where X (p) denotes the base change of X along the Frobenius morphism of Y and F X/Y denotes the relative Frobenius of X over Y. Contents 17 6. Decomposition at p 24 References 27

We show that if X is a pseudo-proper smooth noetherian formal scheme over a positive characteristic p field k then its De Rham complex τ ≤p (F X/k * Ω • X/k) is decomposable. Along the way we establish the Cartier isomorphism Ω i X (p) /Y γ → H i (F X/Y * Ω • X/Y) associated to a map f : X → Y of positive characteristic p noetherian formal schemes where X (p) denotes the base change of X along the Frobenius morphism of Y and F X/Y denotes the relative Frobenius of X over Y. Contents 17 6. Decomposition at p 24 References 27

We show that if is a pseudo-proper smooth noetherian formal scheme over a positive characteristic p field k then its De Rham complex τ ≤p (F /k * Ω • /k) is decomposable. Along the way we establish the Cartier isomorphism Ω i (p) / γ → H i (F / * Ω • /) associated to a map f : → of positive characteristic p noetherian formal schemes where (p) denotes the base change of along the Frobenius morphism of and F / denotes the relative Frobenius of over .

For a site S (with enough points), we construct a topological space X (S) and a full embedding ϕ * of the category of sheaves on S into those on X (S) (i.e., a morphism of toposes ϕ: Sh(X (S)) → Sh(S)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms H * (S, A) = H * (X (S) , ϕ * A) for any Abelian sheaf A on S. As a particular case, this will give for any scheme Y a topological space X (Y) and a functorial isomorphism between the étale cohomology H * (Yé t , A) and the ordinary sheaf cohomology H * (X (Y) , ϕ * A), for any sheaf A for the étale topology on Y .

For the cyclotomic extension F (µ∞)= S m≥1 F (µm) of a number field F, we prove that the reduction map K 2n+1 (F (µ∞)) −→ K 2n+1 (κṽ), when restricted to nontorsion elements, is surjective. Here κṽ denotes the residue field at a primeṽ of F (µ∞).

We apply the recently proven compatibility of Beilinson and Soulé elements in K-theory to investigate density of rational primes p, for which the reduction map K 2n+1 (Z) → K 2n+1 (F p) is nontrivial. Here n is an even, positive integer and F p denotes the field of p elements. In the proof we use arithmetic of cyclotomic numbers which come from Soulé elements. Divisibility properties of the numbers are related to the Vandiver conjecture on the class group of cyclotomic fields. Using the K-theory of the integers, we compute an upper bound on the divisibility of these cyclotomic numbers.

Using results of Greither-Popescu [19] on the Brumer-Stark conjecture we construct l-adic imprimitive versions of these characters, for primes l > 2. Further, the special values of these l-adic Hecke characters are used to construct G(F/K)-equivariant Stickelberger-splitting maps in the Quillen localization sequence for F , extending the results obtained in [1] for K = Q. We also apply the Stickelberger-splitting maps to construct special elements in K 2n (F) l and analyze the Galois module structure of the group D(n) l of divisible elements in K 2n (F) l. If n is odd, l n, and F = K is a fairly general totally real number field, we study the cyclicity of D(n) l in relation to the classical conjecture of Iwasawa on class groups of cyclotomic fields and its potential generalization to a wider class of number fields. Finally, if F is CM, special values of our l-adic Hecke characters are used to construct Euler systems in odd K-groups K 2n+1 (F, Z/l k). These are vast generalizations of Kolyvagin's Euler system of Gauss sums [33] and of the K-theoretic Euler systems constructed in [4] when K = Q.

The Stickelberger splitting map in the case of abelian extensions $F / \Q$ was defined in [Ba1, Chap. IV]. The construction used Stickelebrger's theorem. For abelian extensions $F / K$ with an arbitrary totally real base field $K$ the construction of \cite{Ba1} cannot be generalized since Brumer's conjecture (the analogue of Stickelberger's theorem) is not proved yet at that level

Euler systems introduced by A.Kolyvagin and K.Rubin may be used to construct interesting systems of elements in algebraic K-theory. Let F/Q be an abelian field extension. We fix an odd prime p and a natural number m. Let S be the set of square free natural numbers

In this paper we study the divisibility and the wild kernels in algebraic K-theory of global fields F. We extend the notion of the wild kernel to all K-groups of global fields and prove that Quillen-Lichtenbaum conjecture for F is equivalent to the equality of wild kernels with corresponding groups of divisible elements in K-groups of F. We show that there exist generalized Moore exact sequences for even K-groups of global fields. Without appealing to the Quillen-Lichtenbaum conjecture we show that the group of divisible elements is isomorphic to the corresponding group of étale divisible elements and we apply this result for the proof of the lim 1 analogue of Quillen-Lichtenbaum conjecture. We also apply this isomorphism to investigate: the imbedding obstructions in homology of GL, the splitting obstructions for the Quillen localization sequence, the order of the group of divisible elements via special values of ζ F (s). Using the motivic cohomology results due to Bloch, Friedlander, Levine, Lichtenbaum, Morel, Rost, Suslin, Voevodsky and Weibel, which established the Quillen-Lichtenbaum conjecture, we conclude that wild kernels are equal to corresponding groups of divisible elements.

In this paper we de ne 2-adic cyclotomic elements in K-theory and etale cohomology of the integers. We construct a comparison map which sends the 2-adic elements in K-theory onto 2-adic elements in cohomology. We also compute explicitly some of the product maps in K-theory of Z at the prime 2: 1. 2-adic cyclotomic elements in K 2n?1 (Z 1 2 ]) Z2 De nition 1.1. For a commutative ring with identity R we denote by K i (R; Z2) = lim ? K i (R; Z=2 m)

For a CM abelian extension F/K of an arbitrary totally real number field K, we construct the Stickelberger splitting maps (in the sense of [1]) for both theétale and the Quillen K-theory of F and we use these maps to construct Euler systems in the even Quillen K-theory of F. The Stickelberger splitting maps give an immediate proof of the annihilation of the groups of divisible elements divK 2n (F) l of the even K-theory of the top field by higher Stickelberger elements, for all odd primes l. This generalizes the results of [1], which only deals with CM abelian extensions of Q. The techniques involved in constructing our Euler systems at this level of generality are quite different from those used in [3], where an Euler system in the odd K-theory with finite coefficients of abelian CM extensions of Q was given. We work under the assumption that the Iwasawa µ-invariant conjecture holds. This permits us to make use of the recent results of Greither-Popescu [16] on thé etale Coates-Sinnott conjecture for arbitrary abelian extensions of totally real number fields, which are conditional upon this assumption. In upcoming work, we will use the Euler systems constructed in this paper to obtain information on the groups of divisible elements divK 2n (F) l , for all n > 0 and odd l. It is known that the structure of these groups is intimately related to some of the deepest unsolved problems in algebraic number theory, e.g. the Kummer-Vandiver and Iwasawa conjectures on class groups of cyclotomic fields. We make these connections explicit in the introduction.

We prove a prop version of the classical decomposition of a Z p-torsion free Z p C p-module into indecomposable modules. We also describe some prop Z p C p nmodules obtained from a semidirect product of a free prop group F and a cyclic group C p n of automorphisms by factoring out the (closed) commutator subgroup [F, F ].

In this paper we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [BK13] G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle forétale K-theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases this result is the best possible i.e if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle forétale K-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by S. Barańczuk in [B17]. We show that all our results remain valid for Quillen K-theory of X if the Bass and Quillen-Lichtenbaum conjectures hold true for X .

We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for {é}tale $K$-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by S. Bara{ń}czuk in \\cite{b17}. We show that all our results remain valid for Quillen $K$-theory of ${\\cal X}$ if the Bass and Quillen-Lichtenbaum conjectures hold true for ${\\cal X}.$

In this paper we consider the Quillen-Lichtenbaum conjecture for number fields using description of the higher K-theory in terms of SK 1 groups. 1991 Mathematics Subject Classification. Primary 19F27; Secondary 19Fxx. This work has been sponsored in part by a KBN grant 2P03A 005 11.

We introduce the notion of a p-Cartier smooth algebra. It generalises that of a smooth algebra and includes valuation rings over a perfectoid base. We give several characterisations of p-Cartier smoothness in terms of prismatic cohomology, and deduce a comparison theorem between syntomic and étale cohomologies under this hypothesis. Contents 1 Introduction 1 2 Prismatic cohomology of Cartier smooth algebras 4 3 Valuation rings are Cartier smooth 18 4 The syntomic-étale comparison theorem 25

The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian prop Galois groups with ramification allowed at a maximal set of primes over p such that the module is torsion. A main conjecture for such an Iwasawa module describes its codimension one support in terms of a p-adic L-function attached to the primes of ramification. In this paper, we study more general and potentially much smaller modules that are quotients of exterior powers of Iwasawa modules with ramification at a set of primes over p by sums of exterior powers of inertia subgroups. We show that the higher codimension support of such quotients can be measured by finite collections of characteristic ideals of classical Iwasawa modules, hence by p-adic L-functions under the relevant CM main conjectures. 1 Introduction Iwasawa theory studies the growth of Selmer groups in towers of number fields. In the commutative setting, these towers have Galois groups isomorphic to Z r p for some r ≥ 1, and their Iwasawa algebras are isomorphic to a power series ring in r variables over Z p. The Selmer groups are typically attached to Galois-stable lattices in p-adic Galois representations that come from geometry. The local conditions defining the Selmer groups F. M.

We begin a study of mth Chern classes and mth characteristic symbols for Iwasawa modules which are supported in codimension at least m. This extends the classical theory of characteristic ideals and their generators for Iwasawa modules which are torsion, i.e., supported in codimension at least 1. We apply this to an Iwasawa module constructed from an inverse limit of p-parts of ideal class groups of abelian extensions of an imaginary quadratic field. When this module is pseudo-null, which is conjecturally always the case, we determine its second Chern class and show that it has a characteristic symbol given by the Steinberg symbol of two Katz p-adic L-functions.

Introduction 1 2. Algebricity of the stack of log maps 4 3. Minimal logarithmic maps to rank one Deligne-Faltings log pairs 16 4. The stack of minimal log maps as category fibered over LogSch f s 27 5. The boundedness theorem for minimal stable log maps 31 6. The weak valuative criterion for minimal stable log maps 36 Appendix A. Prerequisites on logarithmic geometry 45 Appendix B. Logarithmic curves and their stacks 49 References 52

Let R be a finite commutative Frobenius ring and S a Galois extension of R of degree m. For positive integers k and k , we determine the number of free S-submodules B of S with the property k = rank S (B) and k = rank R (B ∩ R). This corrects the wrong result (Bill in Linear Algebr Appl 22:223-233, 1978, Theorem 6) which was given in the language of codes over finite fields.

Let K be a p-adic local field with residue field k such that [k : k p ] = p e < ∞ and V be a p-adic representation of Gal(K/K). Then, by using the theory of p-adic differential modules, we show that V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(K/K) if and only if V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(K pf /K pf) where K pf /K is a certain p-adic local field whose residue field is the smallest perfect field k pf containing k. As an application, we prove the p-adic monodromy theorem of Fontaine in the imperfect residue field case. 1/p m i) m∈N. Note that, if V is a p-adic representation of G K , it can be restricted to a p-adic representation of G K pf = Gal(K pf /K pf) where we choose an algebraic closure K pf of K pf containing K. Since K pf is a complete discrete valuation field with perfect residue field, we can apply the classical theory (i.e. in the perfect residue field case) to p-adic representations of G K pf. Our main results are the following. Theorem 1.1. With notation as above, we have the following equivalences.

Let p be an odd prime. Let K p = Q(ζ p) be the p-cyclotomic field and Z[ζ p ] be the ring of integers of K p. Let π be the prime ideal of K p lying over p. Let G be the Galois group of K p. Let v be a primitive root mod p. Let σ be a Q-isomorphism of K p defined by σ : ζ p → ζ v p. Let P (σ) = σ p−2 v −(p−2) + ... + σv −1 + 1 ∈ Z[G], where v n is understood (mod p). Let C p be the p-class group of K p. Let q be a prime ideal of Z[ζ p ] with Cl(q) ∈ C p. Let q the prime number lying above q. Let A be a singular number defined by AZ[ζ p ] = q p. From Stickelberger relation we prove the π-adic congruences: 1. π 2p−1 | A P (σ) if q ≡ 1 mod p, 2. π 2p−1 A P (σ) if q ≡ 1 mod p and p (q−1)/p ≡ 1(mod q). 3. π 2p | A P (σ) if q ≡ 1 mod p. These results allow us to connect the structure of the p-class group C p with π-adic expression of singular numbers A and with solutions of some explicit congruences mod p in Z[X]. The last section connect Stickelberger relation with complete class group of K p. This paper is at elementary level in Classical Algebraic Number Theory.

Introduction 1 2. Algebricity of the stack of log maps 4 3. Minimal logarithmic maps to rank one Deligne-Faltings log pairs 16 4. The stack of minimal log maps as category fibered over LogSch f s 27 5. The boundedness theorem for minimal stable log maps 31 6. The weak valuative criterion for minimal stable log maps 36 Appendix A. Prerequisites on logarithmic geometry 45 Appendix B. Logarithmic curves and their stacks 49 References 52

It is well-known that the Wüstholz' analytic subgroup theorem is one of the most powerful theorems in transcendence theory. The theorem gives in a very systematic and conceptual way the transcendence of a large class of complex numbers, e.g. the transcendence of π which is originally due to Lindemann. In this paper we revisit the p-adic analogue of the analytic subgroup theorem and present a proof based on the method described and developed by the authors in a recent related paper.

We develop the local-global theory of blocks for profinite groups. Given a field k of characteristic p and a profinite group G, one may express the completed group algebra krrGss as a product ś iPI Bi of closed indecomposable algebras, called the blocks of G. To each block B of G we associate a pro-p subgroup of G, called the defect group of B, unique up to conjugacy in G. We give several characterizations of the defect group in analogy with defect groups of blocks of finite groups. Our main theorem is a Brauer correspondence between the blocks of G with defect group D and the blocks of the normalizer NGpDq with defect group D.

We develop the local-global theory of blocks for profinite groups. Given a field k of characteristic p and a profinite group G, one may express the completed group algebra krrGss as a product ś iPI B i of closed indecomposable algebras, called the blocks of G. To each block B of G we associate a prop subgroup of G, called the defect group of B, unique up to conjugacy in G. We give several characterizations of the defect group in analogy with defect groups of blocks of finite groups. Our main theorem is a version of Brauer's first main theorem: a correspondence between the blocks of G with defect group D and the blocks of the normalizer N G pDq with defect group D.

A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a much more general one, namely that the absolute Galois group of K acts with finitely many fixed points on the étale cohomology with 𝐐 / 𝐙 {\mathbf{Q}/\mathbf{Z}} -coefficients of a smooth proper K ¯ {\overline{K}} -variety defined over K. We also present a conjectural generalization of Ribet’s theorem to torsion cycles of higher codimension. We offer supporting evidence for the conjecture in codimension 2, as well as an analogue in positive characteristic.

For a prime number ℓ we say that an oriented pro-ℓ group (G,θ) has the Bogomolov-Positselski property if the kernel of the canonical projection on its maximal θ-abelian quotient π^ab_G,θ G→ G(θ) is a free pro-ℓ group contained in the Frattini subgroup of G. We show that oriented pro-ℓ groups of elementary type have the Bogomolov-Positselski property. This shows that Efrat&#39;s Elementary Type Conjecture implies a positive answer to Positselski&#39;s version of Bogomolov&#39;s Conjecture on maximal pro-ℓ Galois groups of a field K in case that K^×/(K^×)^ℓ is finite. Secondly, it is shown that for an H^∙-quadratic oriented pro-ℓ group (G,θ) the Bogomolov-Positselski property can be expressed by the injectivity of the transgression map d_2^2,1 in the Hochschild-Serre spectral sequence.

For a number field F and an odd prime number p, letF be the compositum of all Z p-extensions of F andΛ the associated Iwasawa algebra. Let G S (F) be the Galois group overF of the maximal extension which is unramified outside p-adic and infinite places. In this paper we study theΛ-module X (−i) S (F) := H 1 (G S (F), Z p (−i)) and its relationship with X(F (µ p))(i − 1) ∆ , the ∆ := Gal(F (µ p)/F)-invariant of the Galois group over F (µ p) of the maximal abelian unramified prop extension ofF (µ p). More precisely, we show that under a decomposition condition, the pseudo-nullity of thẽ Λ-module X(F (µ p))(i − 1) ∆ is implied by the existence of a Z d p-extension L with X (−i) S (L) := H 1 (G S (L), Z p (−i)) being without torsion over the Iwasawa algebra associated to L, and which contains a Z p-extension F ∞ satisfying H 2 (G S (F ∞), Q p /Z p (i)) = 0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i ≡ 1 mod [F (µ p) : F ]. This existence is fulfilled for (p, i)-regular fields.

For a number field F and an odd prime number p, letF be the compositum of all Z p-extensions of F,Λ the associated Iwasawa algebra and X(F) the Galois group overF of the maximal abelian unramified prop-extension of F. In this paper we show that under a decomposition condition, the pseudo-nullity of theΛ-module X(F (µ p))(i − 1) ∆ is implied by the existence of a Z d p-extension L with X (−i) S (L) := H 1 (G S (L), Z p (−i)) is without torsion over the Iwasawa algebra associated to L, and which contains a Z p-extension satisfying W LC i. This existence is fulfilled for (p, i)-regular fields. In particular, when the integer i ≡ 1 mod [F (µ p) : F ] we have a sufficient condition for the validity of Greenberg's generalized conjecture.