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Journal articles on the topic 'Iwasawa's'

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Published: 4 June 2021

Last updated: 18 February 2022

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1

KADOKAMI, TERUHISA, and YASUSHI MIZUSAWA. "IWASAWA TYPE FORMULA FOR COVERS OF A LINK IN A RATIONAL HOMOLOGY SPHERE." Journal of Knot Theory and Its Ramifications 17, no. 10 (October 2008): 1199–221. http://dx.doi.org/10.1142/s0218216508006580.

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Based on the analogy between links and primes, we present an analogue of the Iwasawa's class number formula in a Zp-extension for the p-homology groups of pn-fold cyclic covers of a link in a rational homology 3-sphere. We also describe the associated Iwasawa invariants precisely for some examples and discuss analogies with the number field case.

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2

SCHETTLER, JORDAN. "GENERALIZATIONS OF IWASAWA'S "RIEMANN–HURWITZ" FORMULA FOR CYCLIC p-EXTENSIONS OF NUMBER FIELDS." International Journal of Number Theory 10, no. 01 (January 22, 2014): 219–33. http://dx.doi.org/10.1142/s1793042113500905.

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We produce generalizations of Iwasawa's "Riemann–Hurwitz" formula for number fields. These generalizations apply to cyclic extensions of number fields of degree pn for any positive integer n. We first deduce some congruences and inequalities and then use these formulas to establish a vanishing criterion for Iwasawa λ-invariants which generalizes a result of Fukuda et al. for totally real number fields.

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3

HAJIR, FARSHID, and CHRISTIAN MAIRE. "Prime decomposition and the Iwasawa MU-invariant." Mathematical Proceedings of the Cambridge Philosophical Society 166, no. 3 (April 26, 2018): 599–617. http://dx.doi.org/10.1017/s0305004118000191.

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AbstractFor Γ = ℤp, Iwasawa was the first to construct Γ-extensions over number fields with arbitrarily large μ-invariants. In this work, we investigate other uniform pro-p groups which are realisable as Galois groups of towers of number fields with arbitrarily large μ-invariant. For instance, we prove that this is the case if p is a regular prime and Γ is a uniform pro-p group admitting a fixed-point-free automorphism of odd order dividing p−1. Both in Iwasawa's work, and in the present one, the size of the μ-invariant appears to be intimately related to the existence of primes that split completely in the tower.

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4

Belliard, Jean-Robert. "Global Units Modulo Circular Units: Descent Without Iwasawa’s Main Conjecture." Canadian Journal of Mathematics 61, no. 3 (June 1, 2009): 518–33. http://dx.doi.org/10.4153/cjm-2009-027-0.

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Abstract.Iwasawa's classical asymptotical formula relates the orders of the p-parts Xn of the ideal class groups along a ℤp-extension F∞/F of a number field F to Iwasawa structural invariants ƛ and μ attached to the inverse limit X∞ = . It relies on “good” descent properties satisfied by Xn. If F is abelian and F∞ is cyclotomic, it is known that the p-parts of the orders of the global units modulo circular units Un/Cn are asymptotically equivalent to the p-parts of the ideal class numbers. This suggests that these quotientsUn/Cn, so to speak unit class groups, also satisfy good descent properties. We show this directly, i.e., without using Iwasawa's Main Conjecture.

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5

Sands, Jonathan W. "Kummer's and Iwasawa's Version of Leopoldt's Conjecture." Canadian Mathematical Bulletin 31, no. 3 (September 1, 1988): 338–46. http://dx.doi.org/10.4153/cmb-1988-049-0.

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AbstractWe present a refinement of Iwasawa's approach to Leopoldt's conjecture on the non-vanishing of the p-adic regulator of an algebraic number field K. As an application, the conjecture for K implies the conjecture for a solvable extension L of degree g over K if g is relatively prime to p — 1 and p does not divide g, the discriminant of K, and the quotient of class numbers where is a primitive pth root of unity. This can be viewed as generalizing a theorem of Kummer on cyclotomic units.

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6

Schindler, Werner. "Iwasawa's Theorem and Integrals on Lie Groups." Mathematische Nachrichten 162, no. 1 (1993): 315–27. http://dx.doi.org/10.1002/mana.19931620122.

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7

Fukuda, Takashi. "Iwasawa's $\lambda $-invariants of certain real quadratic fields." Proceedings of the Japan Academy, Series A, Mathematical Sciences 65, no. 7 (1989): 260–62. http://dx.doi.org/10.3792/pjaa.65.260.

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8

Ochiai, Tadashi, and Kazuma Shimomoto. "Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)." Nagoya Mathematical Journal 218 (June 2015): 125–73. http://dx.doi.org/10.1017/s0027763000027045.

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AbstractIn this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

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9

Ochiai, Tadashi, and Kazuma Shimomoto. "Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)." Nagoya Mathematical Journal 218 (June 2015): 125–73. http://dx.doi.org/10.1215/00277630-2891620.

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AbstractIn this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

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10

Sun, Liping, Wende Liu, Xiaocheng Gao, and Boying Wu. "Restricted Envelopes of Lie Superalgebras." Algebra Colloquium 22, no. 02 (April 15, 2015): 309–20. http://dx.doi.org/10.1142/s1005386715000279.

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Certain important results concerning p-envelopes of modular Lie algebras are generalized to the super-case. In particular, any p-envelope of the Lie algebra of a Lie superalgebra can be naturally extended to a restricted envelope of the Lie superalgebra. As an application, a theorem on the representations of Lie superalgebras is given, which is a super-version of Iwasawa's theorem in Lie algebra case. As an example, the minimal restricted envelopes are computed for three series of modular Lie superalgebras of Cartan type.

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11

Van Order, Jeanine. "On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms." Canadian Journal of Mathematics 65, no. 2 (April 1, 2013): 403–66. http://dx.doi.org/10.4153/cjm-2012-002-x.

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AbstractWe construct a bipartite Euler systemin the sense ofHoward forHilbertmodular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini–Darmon, Longo, Nekovar, Pollack–Weston, and others. The construction has direct applications to Iwasawa's main conjectures. For instance, it implies inmany cases one divisibility of the associated dihedral or anticyclotomicmain conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated p-adic L-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.

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12

Horie, Kuniaki. "Iwasawa's λ - -Invariant and a Supplementary Factor in an Algebraic Class Number Formula." Transactions of the American Mathematical Society 308, no. 1 (July 1988): 313. http://dx.doi.org/10.2307/2000965.

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13

BANDINI, A., and I. LONGHI. "CONTROL THEOREMS FOR ELLIPTIC CURVES OVER FUNCTION FIELDS." International Journal of Number Theory 05, no. 02 (March 2009): 229–56. http://dx.doi.org/10.1142/s1793042109002067.

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Let F be a global field of characteristic p > 0, 𝔽/F a Galois extension with [Formula: see text] and E/F a non-isotrivial elliptic curve. We study the behavior of Selmer groups SelE(L)l (l any prime) as L varies through the subextensions of 𝔽 via appropriate versions of Mazur's Control Theorem. In the case l = p, we let 𝔽 = ∪ 𝔽d where 𝔽d/F is a [Formula: see text]-extension. We prove that Sel E(𝔽d)p is a cofinitely generated ℤp[[ Gal (ℤd/F)]]-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic L-function associated to E in ℤp[[Gal(ℤ/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.

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14

Ueki, Jun. "On the Iwasawa invariants for links and Kida’s formula." International Journal of Mathematics 28, no. 06 (April 11, 2017): 1750035. http://dx.doi.org/10.1142/s0129167x17500355.

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Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M. Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida’s formula on [Formula: see text]-invariants in a [Formula: see text]-extension of [Formula: see text]-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa’s second proof, with use of [Formula: see text]-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched [Formula: see text]-cover as an inverse system of cyclic branched [Formula: see text]-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.

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15

Wingberg, Kay. "Galois groups of number fields generated by torsion points of elliptic curves." Nagoya Mathematical Journal 104 (December 1986): 43–53. http://dx.doi.org/10.1017/s0027763000022662.

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Coates and Wiles [1] and B. Perrin-Riou (see [2]) study the arithmetic of an elliptic curve E defined over a number field F with complex multiplication by an imaginary quadratic field K by using p-adic techniques, which combine the classical descent of Mordell and Weil with ideas of Iwasawa’s theory of Zp-extensions of number fields. In a special case they consider a non-cyclotomic Zp-extension F∞ defined via torsion points of E and a certain Iwasawa module attached to E/F, which can be interpreted as an abelian Galois group of an extension of F∞. We are interested in the corresponding non-abelian Galois group and we want to show that the whole situation is quite analogous to the case of the cyclotomic Zp-extension (which is generated by torsion points of Gm).

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16

Kleine, Sören. "Local behavior of Iwasawa’s invariants." International Journal of Number Theory 13, no. 04 (March 24, 2017): 1013–36. http://dx.doi.org/10.1142/s1793042117500543.

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Let [Formula: see text] be a number field, let [Formula: see text] denote a fixed rational prime. We study the local behavior of Iwasawa’s invariants as functions on the set [Formula: see text] of all [Formula: see text]-extensions of [Formula: see text]. With respect to a certain topology on [Formula: see text] that takes care of ramification, we prove that for each [Formula: see text] the [Formula: see text]-invariant of [Formula: see text] is locally maximal among the [Formula: see text]-invariants, and we give sufficient conditions for the [Formula: see text]-invariant to be locally maximal (e.g., a vanishing [Formula: see text]-invariant). This concerns a question raised by R. Greenberg in 1973. Our main result also provides information about [Formula: see text]- (and even [Formula: see text]-) invariants in the case of a nonvanishing [Formula: see text]-invariant. The main tool used in the proof is a new result based on the stabilization of certain ranks, which considerably generalizes a theorem of T. Fukuda.

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17

Rogov, Vasily. "Complex Geometry of Iwasawa Manifolds." International Mathematics Research Notices 2020, no. 23 (November 7, 2018): 9420–39. http://dx.doi.org/10.1093/imrn/rny230.

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Abstract An Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient $G/\Lambda $, where $G$ is the group of complex unipotent $3 \times 3$ matrices and $\Lambda \subset G$ is a cocompact lattice. In this work, we study holomorphic submanifolds in Iwasawa manifolds. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic subtorus. We also prove that any complex surface in an Iwasawa manifold is either an abelian surface or a Kähler non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix, we show that any subtorus in Iwasawa manifold carries complex multiplication.

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18

Büyükboduk, Kâzım. "Stark units and the main conjectures for totally real fields." Compositio Mathematica 145, no. 5 (September 2009): 1163–95. http://dx.doi.org/10.1112/s0010437x09004163.

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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.

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19

HOWSON, SUSAN. "EULER CHARACTERISTICS AS INVARIANTS OF IWASAWA MODULES." Proceedings of the London Mathematical Society 85, no. 3 (October 14, 2002): 634–58. http://dx.doi.org/10.1112/s0024611502013680.

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If $G$ is a pro-$p$, $p$-adic, Lie group containing no element of order $p$ and if $\Lambda (G)$ denotes the Iwasawa algebra of $G$ then we propose a number of invariants associated to finitely generated $\Lambda (G)$-modules, all given by various forms of Euler characteristic. The first turns out to be none other than the rank, and this gives a particularly convenient way of calculating the rank of Iwasawa modules. Others seem to play similar roles to the classical Iwasawa $\lambda $- and $\mu $-invariants. We explore some properties and give applications to the Iwasawa theory of elliptic curves.2000 Mathematical Subject Classification: primary 16E10; seconday 11R23.

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20

Gold, Robert, and Manohar Madan. "Iwasawa invariants." Communications in Algebra 13, no. 7 (January 1985): 1559–78. http://dx.doi.org/10.1080/00927878508823239.

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21

Friedman, Eduardo. "Iwasawa invariants." Mathematische Annalen 271, no. 1 (March 1985): 13–30. http://dx.doi.org/10.1007/bf01455793.

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22

BARMAN, RUPAM. "ANOTHER LOOK AT IWASAWA λ-INVARIANTS OF p-ADIC MEASURES ON $\mathbb {Z}_{p}^n$ AND Γ-TRANSFORMS." International Journal of Number Theory 09, no. 05 (June 16, 2013): 1289–99. http://dx.doi.org/10.1142/s1793042113500279.

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In [Iwasawa λ-invariants of p-adic measures on [Formula: see text] and their Γ-transforms, J. Number Theory132(10) (2012) 2258–2266; Iwasawa λ-invariants and Γ-transforms of p-adic measure on [Formula: see text], Int. J. Number Theory6(8) (2010) 1819–1829], the author and Saikia defined Iwasawa λ-invariants for multi-variable power series and proved a relation between the Iwasawa λ-invariant of a p-adic measure on [Formula: see text] and its Γ-transform. In this paper, we give two new definitions of λ-invariants for multi-variable power series which generalize the λ-invariant of single variable power series and prove that the new λ-invariants also satisfy the same relation. We also give algebraic interpretations of the new λ-invariants and discuss an application of our main results.

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23

Johnston, Henri, and Andreas Nickel. "Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture." American Journal of Mathematics 140, no. 1 (2018): 245–76. http://dx.doi.org/10.1353/ajm.2018.0005.

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24

Nickel, Andreas. "A Generalization of a Theorem of Swan with Applications to Iwasawa Theory." Canadian Journal of Mathematics 72, no. 3 (November 16, 2018): 656–75. http://dx.doi.org/10.4153/s0008414x18000093.

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AbstractLet $p$ be a prime and let $G$ be a finite group. By a celebrated theorem of Swan, two finitely generated projective $\mathbb{Z}_{p}[G]$-modules $P$ and $P^{\prime }$ are isomorphic if and only if $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P$ and $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P^{\prime }$ are isomorphic as $\mathbb{Q}_{p}[G]$-modules. We prove an Iwasawa-theoretic analogue of this result and apply this to the Iwasawa theory of local and global fields. We thereby determine the structure of natural Iwasawa modules up to (pseudo-)isomorphism.

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25

Barnes, Donald W. "Ado-Iwasawa extras." Journal of the Australian Mathematical Society 78, no. 3 (June 2005): 407–21. http://dx.doi.org/10.1017/s1446788700008600.

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AbstractLet L be a finite-dimensional Lie algebra over the field F. The Ado-Iwasawa Theorem asserts the existence of a finite-dimensional L-module which gives a faithful representation ρ of L. Let S be a subnormal subalgebra of L, let be a saturated formation of soluble Lie algebras and suppose that S ∈ . I show that there exists a module V with the extra property that it is -hypercentral as S-module. Further, there exists a module V which has this extra property simultaneously for every such S and , along with the Hochschild extra that ρ(x) is nilpotent for every x ∈ L with ad(x) nilpotent. In particular, if L is supersoluble, then it has a faithful representation by upper triangular matrices.

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26

Franzen, A., P. Kaura, A. Misra, and R. Ray. "Uplifting the Iwasawa." Fortschritte der Physik 54, no. 4 (April 3, 2006): 207–24. http://dx.doi.org/10.1002/prop.200510269.

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27

ZÁBRÁDI, GERGELY. "Characteristic elements, pairings and functional equations over the false Tate curve extension." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 3 (May 2008): 535–74. http://dx.doi.org/10.1017/s0305004108001114.

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AbstractWe construct a pairing on the dual Selmer group over false Tate curve extensions of an elliptic curve with good ordinary reduction at a primep≥5. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of thep-adicL-function. As an application we compute the characteristic elements of those modules – arising naturally in the Iwasawa-theory for elliptic curves over the false Tate curve extension – which have rank 1 over the subgroup of the Galois group fixing the cyclotomic extension of the ground field. We also show that the example of a non-principal reflexive left ideal of the Iwasawa algebra does not rule out the possibility that all torsion Iwasawa-modules are pseudo-isomorphic to the direct sum of quotients of the algebra by principal ideals.

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28

Tateno, Sohei. "ON IWASAWA’S CLASS NUMBER FORMULA FOR Zp*Zp-EXTENSIONS." JP Journal of Algebra, Number Theory and Applications 43, no. 1 (June 20, 2019): 77–99. http://dx.doi.org/10.17654/nt043010077.

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29

Han, Dong, and Feng Wei. "Normal elements in the mod-𝑝 Iwasawa algebra over SL𝑛(ℤ𝑝): A computational approach." Forum Mathematicum 31, no. 6 (November 1, 2019): 1417–46. http://dx.doi.org/10.1515/forum-2018-0260.

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AbstractThis is the last in a series of articles where we are concerned with normal elements of noncommutative Iwasawa algebras over {\mathrm{SL}_{n}(\mathbb{Z}_{p})}. Our goal in this portion is to give a positive answer to an open question in [D. Han and F. Wei, Normal elements of noncommutative Iwasawa algebras over \mathrm{SL}_{3}(\mathbb{Z}_{p}), Forum Math. 31 2019, 1, 111–147] and make up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over \mathrm{SL}_{n}(\mathbb{Z}_{p}), Internat. J. Algebra Comput. 20 2010, 8, 1021–1039] simultaneously. Let n ({n\geq 2}) be a positive integer. Let p ({p>2}) be a prime integer, {\mathbb{Z}_{p}} the ring of p-adic integers and {\mathbb{F}_{p}} the finite filed of p elements. Let {G=\Gamma_{1}(\mathrm{SL}_{n}(\mathbb{Z}_{p}))} be the first congruence subgroup of the special linear group {\mathrm{SL}_{n}(\mathbb{Z}_{p})} and {\Omega_{G}} the mod-p Iwasawa algebra of G defined over {\mathbb{F}_{p}}. By a purely computational approach, for each nonzero element {W\in\Omega_{G}}, we prove that W is a normal element if and only if W contains constant terms. In this case, W is a unit. Also, the main result has been already proved under “nice prime” condition by Ardakov, Wei and Zhang [Non-existence of reflexive ideals in Iwasawa algebras of Chevalley type, J. Algebra 320 2008, 1, 259–275; Reflexive ideals in Iwasawa algebras, Adv. Math. 218 2008, 3, 865–901]. This paper currently provides a new proof without the “nice prime” condition. As a consequence of the above-mentioned main result, we observe that the center of {\Omega_{G}} is trivial.

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30

Ji, Qingzhong, and Hourong Qin. "Iwasawa Theory for K2n." Journal of K-Theory 12, no. 1 (May 2, 2013): 115–23. http://dx.doi.org/10.1017/is013004020jkt225.

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AbstractGiven a number field F and a prime number p; let Fn denote the cyclotomic extension with [Fn : F] = pn; and let $\mathematical script capital(O)_F_n\$ denote its ring of integers. We establish an analogue of the classical Iwasawa theorem for the orders of K2i ($\mathematical script capital(O)_F_n\$){p}.

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31

Cacciatori, Sergio L., Bianca L. Cerchiai, and Alessio Marrani. "Iwasawa N=8 attractors." Journal of Mathematical Physics 51, no. 10 (October 2010): 102502. http://dx.doi.org/10.1063/1.3501024.

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32

Sujatha, R. "Iwasawa Theory and Modular." Pure and Applied Mathematics Quarterly 2, no. 2 (2006): 519–38. http://dx.doi.org/10.4310/pamq.2006.v2.n2.a6.

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33

Ritter, Jürgen, and Alfred Weiss. "Toward equivariant Iwasawa theory." manuscripta mathematica 109, no. 2 (October 1, 2002): 131–46. http://dx.doi.org/10.1007/s00229-002-0306-8.

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34

Madan, Manohar L., and Horst G. Zimmer. "Relations among Iwasawa invariants." Journal of Number Theory 25, no. 2 (February 1987): 213–19. http://dx.doi.org/10.1016/0022-314x(87)90027-8.

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35

Mejía Alemán, Carlos, and Mario Enrique Santiago Saldaña. "El Teorema de Iwasawa." Pesquimat 24, no. 1 (June 30, 2021): 1–8. http://dx.doi.org/10.15381/pesquimat.v24i1.20511.

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Sean G un grupo, Ω un conjunto y K = {g ∈ G | ω * g = ω, Ɐω ∈ Ω} el núcleo de Ω donde G actua sobre el conjunto Ω. Mostraremos que G/K es simple en el caso que el grupo G verifique ser primitivo sobre Ω, así como también que sea igual a su subgrupo derivado y por último si α ∈ Ω entonces Gα tiene un subgrupo A que es abeliano y normal tal que G =< Ag | g ∈ G >, donde Gα es el estabilizador de α en G. Para finalizar daremos una aplicación de que el grupo alternante A5 es simple.

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36

Büyükboduk, Kâzım, and Antonio Lei. "Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms." Forum Mathematicum 30, no. 4 (July 1, 2018): 887–913. http://dx.doi.org/10.1515/forum-2016-0189.

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Abstract This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic {\mathbb{Z}_{p}} -tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson–Flach elements.

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37

LEE, CHERN–YANG. "Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative reduction." Mathematical Proceedings of the Cambridge Philosophical Society 154, no. 2 (October 29, 2012): 303–24. http://dx.doi.org/10.1017/s0305004112000564.

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AbstractThis paper studies the compact p∞-Selmer Iwasawa module X(E/F∞) of an elliptic curve E over a False Tate curve extension F∞, where E is defined over ℚ, having multiplicative reduction at the odd prime p. We investigate the p∞-Selmer rank of E over intermediate fields and give the best lower bound of its growth under certain parity assumption on X(E/F∞), assuming this Iwasawa module satisfies the H(G)-Conjecture proposed by Coates–Fukaya–Kato–Sujatha–Venjakob.

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38

Burns, David, and Otmar Venjakob. "On descent theory and main conjectures in non-commutative Iwasawa theory." Journal of the Institute of Mathematics of Jussieu 10, no. 1 (April 26, 2010): 59–118. http://dx.doi.org/10.1017/s147474800900022x.

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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.

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39

ITOH, TSUYOSHI, YASUSHI MIZUSAWA, and MANABU OZAKI. "ON THE ℤp-RANKS OF TAMELY RAMIFIED IWASAWA MODULES." International Journal of Number Theory 09, no. 06 (September 2013): 1491–503. http://dx.doi.org/10.1142/s1793042113500395.

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For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension unramified outside S over the cyclotomic ℤp-extension of a number field k. In the case where S does not contain p and k is the rational number field or an imaginary quadratic field, we give the explicit formulae of the ℤp-ranks of the S-ramified Iwasawa modules by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of algebraic numbers.

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40

Kataoka, Takenori. "On Greenberg’s generalized conjecture for complex cubic fields." International Journal of Number Theory 13, no. 03 (February 9, 2017): 619–31. http://dx.doi.org/10.1142/s1793042117500312.

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Let [Formula: see text] be a prime number. For a number field [Formula: see text], let [Formula: see text] be the compositum of all [Formula: see text]-extensions of [Formula: see text]. Then Greenberg’s generalized conjecture (GGC) claims that the unramified Iwasawa module [Formula: see text] is pseudo-null over the Iwasawa algebra associated to the Galois group of [Formula: see text]. In this paper, we establish sufficient conditions of GGC when [Formula: see text] is a complex cubic field and give many examples which satisfy the conditions with the help of computer programs.

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41

Jha, Somnath, and Aprameyo Pal. "Algebraic functional equation for Hida family." International Journal of Number Theory 10, no. 07 (September 9, 2014): 1649–74. http://dx.doi.org/10.1142/s1793042114500493.

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We prove a functional equation for the characteristic ideal of the "big" Selmer group 𝒳(𝒯ℱ/F cyc ) associated to an ordinary Hida family of elliptic modular forms over the cyclotomic ℤp extension of a general number field F, under the assumption that there is at least one arithmetic specialization whose Selmer group is torsion over its Iwasawa algebra. For a general number field, the two-variable cyclotomic Iwasawa main conjecture for ordinary Hida family is not proved and this can be thought of as an evidence to the validity of the Iwasawa main conjecture. The central idea of the proof is to prove a variant of the result of Perrin-Riou [Groupes de Selmer et accouplements; cas particulier des courbes elliptiques, Doc. Math.2003 (2003) 725–760, Extra Volume: Kazuya Kato's fiftieth birthday] by constructing a generalized pairing on the individual Selmer groups corresponding to the arithmetic points and make use of the appropriate specialization techniques of Ochiai [Euler system for Galois deformations, Ann. Inst. Fourier (Grenoble)55(1) (2005) 113–146].

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42

OCHIAI, TADASHI, and FABIEN TRIHAN. "On the Selmer groups of abelian varieties over function fields of characteristic p > 0." Mathematical Proceedings of the Cambridge Philosophical Society 146, no. 1 (January 2009): 23–43. http://dx.doi.org/10.1017/s0305004108001801.

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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.

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43

Bertolini, Massimo, and Henri Darmon. "Iwasawa’s Main Conjecture for elliptic curves over anticyclotomic ℤp-extensions." Annals of Mathematics 162, no. 1 (July 1, 2005): 1–64. http://dx.doi.org/10.4007/annals.2005.162.1.

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44

Ueki, Jun. "On the homology of branched coverings of 3-manifolds." Nagoya Mathematical Journal 213 (March 2014): 21–39. http://dx.doi.org/10.1017/s0027763000026167.

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AbstractFollowing the analogies between 3-manifolds and number rings in arithmetic topology, we study the homology of branched covers of 3-manifolds. In particular, we show some analogues of Iwasawa’s theorems on ideal class groups and unit groups, Hilbert’s Satz 90, and some genus-theory–type results in the context of 3-dimensional topology. We also prove that the 2-cycles valued Tate cohomology of branched Galois covers is a topological invariant, and we give a new insight into the analogy between 2-cycle groups and unit groups.

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45

Ueki, Jun. "On the homology of branched coverings of 3-manifolds." Nagoya Mathematical Journal 213 (March 2014): 21–39. http://dx.doi.org/10.1215/00277630-2393795.

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AbstractFollowing the analogies between 3-manifolds and number rings in arithmetic topology, we study the homology of branched covers of 3-manifolds. In particular, we show some analogues of Iwasawa’s theorems on ideal class groups and unit groups, Hilbert’s Satz 90, and some genus-theory–type results in the context of 3-dimensional topology. We also prove that the 2-cycles valued Tate cohomology of branched Galois covers is a topological invariant, and we give a new insight into the analogy between 2-cycle groups and unit groups.

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46

Nickel, Andreas. "On the equivariant Tamagawa number conjecture in tame CM-extensions, II." Compositio Mathematica 147, no. 4 (May 4, 2011): 1179–204. http://dx.doi.org/10.1112/s0010437x11005331.

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AbstractWe use the notion of non-commutative Fitting invariants to give a reformulation of the equivariant Iwasawa main conjecture (EIMC) attached to an extension F/K of totally real fields with Galois group 𝒢, where K is a global number field and 𝒢 is a p-adic Lie group of dimension one for an odd prime p. We attach to each finite Galois CM-extension L/K with Galois group G a module SKu(L/K) over the center of the group ring ℤG which coincides with the Sinnott–Kurihara ideal if G is abelian. We state a conjecture on the integrality of SKu (L/K) which follows from the equivariant Tamagawa number conjecture (ETNC) in many cases, and is a theorem for abelian G. Assuming the vanishing of the Iwasawa μ-invariant, we compute Fitting invariants of certain Iwasawa modules via the EIMC, and we show that this implies the minus part of the ETNC at p for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above p, provided that (an appropriate p-part of) the integrality conjecture holds.

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47

Gold, Robert, and Manohar Madan. "Galois representations of Iwasawa modules." Acta Arithmetica 46, no. 3 (1986): 243–55. http://dx.doi.org/10.4064/aa-46-3-243-255.

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48

Bröker, Reinier, David Hubbard, and Lawrence Washington. "Explicit computations in Iwasawa theory." Open Book Series 2, no. 1 (January 28, 2019): 137–53. http://dx.doi.org/10.2140/obs.2019.2.137.

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49

Hu, Su, and Min-Soo Kim. "The (S,{2})-Iwasawa theory." Journal of Number Theory 158 (January 2016): 73–89. http://dx.doi.org/10.1016/j.jnt.2015.06.013.

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50

Kleine, Sören. "T-ranks of Iwasawa modules." Journal of Number Theory 196 (March 2019): 61–86. http://dx.doi.org/10.1016/j.jnt.2018.09.022.

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