Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Arithmetic duality theorems

We prove the one-, two-, and three-variable Iwasawa-Greenberg Main Conjectures for a large class of modular forms that are ordinary with respect to an odd prime p. The method of proof involves an analysis of an Eisenstein ideal for ordinary Hida families for GU(2,2).

/pdf/the-iwasawa-main-conjectures-for-gl2-27is3jcxn0.pdf

The Iwasawa Main Conjectures for GL2

In this paper, we associate to every p-adic representation V a p-adic differential equation D

 †

 rig(V), that is to say a module with a connection over the Robba ring. We do this via the theory of Fontaine’s (ϕ,Γ
 K
 )-modules.¶This construction enables us to relate the theory of (ϕ,Γ
 K
 )-modules to p-adic Hodge theory. We explain how to construct D

 cris(V) and D

 st(V) from D

 †

 rig(V), which allows us to recognize semi-stable or crystalline representations; the connection is then unipotent or trivial on D

 †

 rig(V)[1/t].¶In general, the connection has an infinite number of regular singularities, but V is de Rham if and only if those are apparent singularities. A structure theorem for modules over the Robba ring allows us to get rid of all singularities at once, and to obtain a “classical” differential equation, with a Frobenius structure.¶Using this, we construct a functor from the category of de Rham representations to that of classical p-adic differential equations with Frobenius structure. A recent theorem of Y. Andre gives a complete description of the structure of the latter object. This allows us to prove Fontaine’s p-adic monodromy conjecture: every de Rham representation is potentially semi-stable.¶As an application, we can extend to the case of arbitrary perfect residue fields some results of Hyodo (H

 1

 g
 =H

 1

 st
 ), of Perrin-Riou (the semi-stability of ordinary representations), of Colmez (absolutely crystalline representations are of finite height), and of Bloch and Kato (if the weights of V are ≥2, then Bloch-Kato’s exponential exp
 V
 is an isomorphism).

Représentations p-adiques et équations différentielles

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/Q is abelian and K^\infty is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of Gal(K^\infty/Q). We also give analogous results when K/Q is non-abelian, the base field is not Q and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their "regulator constants", and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.

Regulator constants and the parity conjecture

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p
 ∞-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/ℚ is semistable at 2 and 3, K/ℚ is abelian and K
 ∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of $\mathop{\mathrm{Gal}}(K^{\infty}/{\mathbb{Q}})$
 
. We also give analogous results when K/ℚ is non-abelian, the base field is not ℚ and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.

We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over $p$-adic rings extends uniquely to a cohomology theory for varieties over $p$-adic fields that satisfies $h$-descent. This new cohomology - syntomic cohomology - is a Bloch-Ogus cohomology theory, admits period map to \'etale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the \'etale side. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soul\'e's \'etale regulators land in the potentially semistable Selmer groups. 
Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on $p$-adic comparison

/pdf/syntomic-cohomology-and-regulators-for-varieties-over-p-adic-4rcobiassk.pdf

Syntomic cohomology and regulators for varieties over p-adic fields

We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over p-adic rings extends uniquely to a cohomology theory for varieties over p-adic fields that satisfies h-descent. This new cohomology-syntomic cohomology-is a Bloch-Ogus cohomology theory, admits period map to etale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the etale side and is related to the Bloch-Kato exponential map. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soule's ´ etale regulators land in the potentially semistable Selmer groups. Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on p-adic comparison theorems.

/pdf/syntomic-cohomology-and-p-adic-regulators-for-varieties-over-ieispyo5br.pdf

Syntomic cohomology and p-adic regulators for varieties over p-adic fields

Note that potential modularity of E [Wi, Thm. A.1] implies that the L-function L(E/F, s) has a meromorphic continuation to C and satisfies the expected functional equation ([T2, proof of Cor. 2.2]; [N2, 12.11.6]). As a result, the integer ords=1 L(E/F, s) ∈ Z is well-defined. Various special cases of Theorem A (for F 6= Q) were proved in [N2], [Ki] and [N6]. If the p-primary part of X(E/F ) is finite for some prime number p, then sp(E/F ) = rkZE(F ) and the statement of Theorem A is the conjecture of Birch and Swinnerton-Dyer for E over F modulo 2. Theorem B. Let g = ∑∞ n=1 anq n ∈ S2r(Γ0(N)) (r ≥ 1) be a normalised (a1 = 1) newform, let L = Q(a1, a2, . . .) be the (totally real) number field generated by its coefficients. For any prime p of L above a rational prime p 6= 2, denote by Vp(g) the two-dimensional representation of GQ = Gal(Q/Q) over Lp attached to g: det (1−X Frgeom(l) | Vp(g)) = 1− alX + l2r−1X2 (l pN).

/pdf/some-consequences-of-a-formula-of-mazur-and-rubin-for-24e1vn9i1l.pdf

Some consequences of a formula of Mazur and Rubin for arithmetic local constants

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$
 are compatible with the usual local constants at all primes not dividing $p$
 and in two special cases also at primes dividing $p$
 . We deduce new cases of the $p$
 -parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

Compatibility of arithmetic and algebraic local constants (the case )

Abstract:We show that a certain part of the middle degree $\\ell$-adic cohomology of an arbitrary simple unitary Shimura variety is a semisimple Galois representation.

Semisimplicity of certain galois representations occurring in étale cohomology of unitary shimura varieties

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