Iwasawa Theory Research Articles

An R = T theorem for imaginary quadratic fields

We prove the modularity of certain residually reducible p-adic Galois representations of an imaginary quadratic field assuming the uniqueness of the residual representation. We obtain an R = T theorem using a new commutative algebra criterion that might be of independent interest. To apply the criterion, one needs to show that the quotient of the universal deformation ring R by its ideal of reducibility is cyclic Artinian of order no greater than the order of the congruence module T/J, where J is an Eisenstein ideal in the local Hecke algebra T. The inequality is proven by applying the Main conjecture of Iwasawa Theory for Hecke characters and using a result of Berger [Compos Math 145(3):603–632, 2009]. This strengthens our previous result [Berger and Klosin, J Inst Math Jussieu 8(4):669–692, 2009] to include the cases of an arbitrary p-adic valuation of the L-value, in particular, cases when R is not a discrete valuation ring. As a consequence we show that the Eisenstein ideal is principal and that T is a complete intersection.

On descent theory and main conjectures in non-commutative Iwasawa theory

AbstractWe develop an explicit descent theory in the context of Whitehead groups of non-commutative Iwasawa algebras. We apply this theory to describe the precise connection between main conjectures of non-commutative Iwasawa theory (in the spirit of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture and proving special cases of the equivariant Tamagawa number conjecture.

The integral logarithm in Iwasawa theory : an exercise

Soient l un nombre premier impair et H un groupe fini abelien. Nous decrivons le groupe d'unites de Λ Λ [H] (la completion du localise de ℤ[[T]][H] en l) ainsi que le noyau et le conoyau du logarithme integral L: Λ Λ [H] × → Λ Λ [H], qui apparait dans la theorie d'Iwasawa non-commutative.

Selmer varieties for curves with CM Jacobians

We study the Selmer variety associated to a canonical quotient of the $\Q_p$-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over $\Q$ whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multi-variable Iwasawa theory is used to prove dimension bounds, which, in turn, lead to a new proof of Diophantine finiteness over $\Q$ for such curves.

Bernoulli–Hurwitz numbers, Wieferich primes and Galois representations

Let K be a quadratic imaginary number field with discriminant D K ≠ − 3 , − 4 and class number one. Fix a prime p ⩾ 7 which is unramified in K. Given an elliptic curve A / Q with complex multiplication by K, let ρ A ¯ : Gal ( K ¯ / K ( μ p ∞ ) ) → SL ( 2 , Z p ) be the representation which arises from the action of Galois on the Tate module. Herein it is shown that, for all but finitely many inert primes p, the image of a certain deformation ρ A : Gal ( K ¯ / K ( μ p ∞ ) ) → SL ( 2 , Z p 〚 X 〛 ) of ρ A ¯ is “as large as possible”, that is, it is the full inverse image of a Cartan subgroup of SL ( 2 , Z p ) . If p splits in K, then the same result holds as long as a certain Bernoulli–Hurwitz number is a p-adic unit which, in turn, is equivalent to a prime ideal not being a Wieferich place. The proof rests on the theory of elliptic units of Robert and Kubert–Lang, and on the two-variable main conjecture of Iwasawa theory for quadratic imaginary fields.

Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms

Let Goo be a semisimple real Lie group with unitary dual Goo- We produce new upper bounds for the multiplicities with which representations ^ e of cohomological type appear in certain spaces of cusp forms on Goo* The main new idea is to apply noncommutative Iwasawa theory to certain p-adic completions of the cohomology of locally symmetric spaces.

Iwasawa theory of totally real fields for certain non-commutative p-extensions

In this paper, we will prove the non-commutative Iwasawa main conjecture—formulated by John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha and Otmar Venjakob (2005)—for certain specific non-commutative p-adic Lie extensions of totally real fields by using theory on integral logarithms introduced by Robert Oliver and Laurence R. Taylor, theory on Hilbert modular forms introduced by Pierre Deligne and Kenneth A. Ribet, and so on. Our results give certain generalization of the recent work of Kazuya Kato on the proof of the main conjecture for Galois extensions of Heisenberg type.

Stark units and the main conjectures for totally real fields

AbstractThe main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement in the context of local Iwasawa theory, assuming the truth of the Rubin–Stark conjecture and Leopoldt’s conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

ON WEIL NUMBERS IN CYCLOTOMIC FIELDS

In this paper, we study the p-adic behavior of Weil numbers in the cyclotomic ℤp-extension of the pth cyclotomic field. We determine the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in the Iwasawa theory. In a recent preprint [9] by Nguyen Quang Do and Nicolas, a generalization of this result to a semi-simple case was obtained.

Iwasawa invariants for the False–Tate extension and congruences between modular forms

For an ordinary prime p ⩾ 3 , we consider the Hida family associated to modular forms of a fixed tame level, and their Selmer groups defined over certain Galois extensions of Q ( μ p ) whose Galois group is G ≅ Z p ⋊ Z p . For Selmer groups defined over the cyclotomic Z p -extension of Q ( μ p ) , we show that if the μ-invariant of one member of the Hida family is zero, then so are the μ-invariants of the other members, while the λ-invariants remain the same only in a branch of the Hida family. We use these results to study the behavior of some invariants from non-commutative Iwasawa theory in the Hida family.

Root numbers, Selmer groups, and non-commutative Iwasawa theory

Let E E be an elliptic curve over a number field F F , and let F ∞ F_\infty be a Galois extension of F F whose Galois group G G is a p p -adic Lie group. The aim of the present paper is to provide some evidence that, in accordance with the main conjectures of Iwasawa theory, there is a close connection between the action of the Selmer group of E E over F ∞ F_\infty , and the global root numbers attached to the twists of the complex L L -function of E E by Artin representations of G G .

Algebraic $p$-Adic $L$-Functions in Non-Commutative Iwasawa Theory

We define canonical algebraic p -adic L -functions in non-commutative Iwasawa theory and establish some of their basic properties.

On the Selmer groups of abelian varieties over function fields of characteristic p > 0

AbstractWe study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

Truncated Euler Systems over Imaginary Quadratic Fields

AbstractLet K be an imaginary quadratic field and let F be an abelian extension of K. It is known that the order of the class group ClF of F is equal to the order of the quotient UF/ElF of the group of global units UF by the group of elliptic units ElF of F. We introduce a filtration on UF/ElF made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group. We provide evidence for the conjecture using Iwasawa theory.

Iwasawa Theory and Motivic L-functions

We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture.

Sur la dualité et la descente d’Iwasawa

Guided by the concrete examples of cyclotomic units and the ideal class group in cyclotomic Iwasawa theory, we develop a general tool for studying descent and codescent, with a special interest in relating the two of them.

P-adic weight pairings on pro-Jacobians

Let E denote an elliptic curve defined over the rational numbers. We outline a method of proving the statement [Display omitted] using properties of p-adic modular forms, i.e. no Iwasawa theory whatsoever. The proof employs a version of Kato's zeta-elements with Λ-adic coefficients. To cite this article: D. Delbourgo, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Algebraische Zahlentheorie

Please note the corrected of U. Hartl. The conference brought together researchers from Europe, the US, and Japan who reported on various recent and ongoing developments in algebraic number theory and related fields. As at previous meetings, organized by Deninger, Schneider and Scholl, one of the clearest themes was the prevalence of p -adic methods across a range of areas. A notable difference with previous years was the number of younger people both as speakers and participants. Colmez reported on his work relating unitary admissible \mathrm{GL}_2(\mathbb Q_p) -representations to local Galois representations. This realizes a program of Breuil and stands at the crossroads of p -adic Hodge theory, representations of p -adic reductive groups and explicit reciprocity laws, as well as having applications to modularity of global Galois representations. Related talks were given by Schneider who explained on going work with Vigneras, attempting to generalize some of Colmez' constructions to higher rank, as well as Orlik who discussed the construction of locally analytic representations from equivariant vector bundles on symmetric spaces. L. Berger reported on an extension of his earlier work on classification of local Galois representations. Hartl explained how these ideas could be used to give a description of the image of the Rapoport–Zink period morphism. This was a satisfying complement to his talk at the previous meeting where he had sketched some of these ideas. There were several talks related to Iwasawa theory and reciprocity laws. Zerbes reported on her work on reciprocity laws for higher dimensional local fields. Fukaya reported on joint work with Coates, Kato, Sujatha and Venjakob in non-abelian Iwasawa theory, and Ochiai discussed the Iwasawa theory of ordinary Hida families. The talk by Sharifi was also related to this area. It described a fascinating relation between Galois cohomology and modular symbols, which seems to be closely related to the Main conjecture of Iwasawa theory. There were a number of talks dealing with congruences between automorphic forms, and applications. The most exciting of these was by Fujiwara who outlined how Taylor–Wiles systems could be used, in certain circumstances, to prove the Leopoldt conjecture for totally real fields. Sorensen discussed his work on level raising for \mathrm{GSp}_4 and some applications to Selmer groups. T. Berger explained how to construct Galois representations attached to cusp forms on \mathrm{GL}_2 over an imaginary field. These had been constructed by Taylor about 15 years ago, but were previously known to have the correct L -factors only at a set of primes of density 1 . Berger also explained ongoing work on modularity lifting theorems in this situation. This would be an exciting advance since such theorems are currently available only over totally real fields. There were two talks on polylogarithms. Blottiere explained his results on the Eisenstein classes on Hilbert modular varieties and applications to special values of L -functions. Bannai discussed the crystalline realization of the elliptic polylogarithm. Somewhat related to this was the talk of Huber on the p -adic Borel regulator. Other talks were given by Yoshida who explained a computation of vanishing cycles on Shimura varieties, realizing the local Langlands and Jacquet–Langlands correspondences, Görtz who spoke on affine Deligne–Lusztig varieties, Saito who outlined his construction of the characteristic cycle of an l -adic sheaf, and Schmidt who discussed his work on integer rings of type K(π,1) .

Complex multiplication and parity in Iwasawa theory

We give a new, somewhat elementary method for proving parity results about Iwasawa-theoretic Selmer groups and apply our method to certain Galois representations which are not self-dual. The main result is essentially that Iwasawa's λ-invariants for these representations over dihedral Z p d -extensions are even. Our approach is a specialization argument and does not make use of Nekovář's deformation-theoretic Cassels pairing, though Nekovář's theory implies our results. Examples of the representations we consider arise naturally in the study of CM abelian varieties defined over the totally real subfield of the reflex field of the CM type. We also discuss connections with “large Selmer rank” in the sense of Mazur–Rubin and give several examples in the context of abelian varieties and modular forms.

On the integrality of characteristic elements in nonabelian Iwasawa theory

This paper is concerned with a conjecture formulated by Coates et al in [4] which describes the relation between the integrality of the characteristic elements and their evaluations in the noncommutative Iwasawa theory. We give an almost equivalent description of the conjecture and prove a certain part of it.