Narrow Results

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 $\lambda$-invariants in a $p$-extension of ${\mathbb{Z}}_p$-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa’s second proof, with use of $p$-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched ${\mathbb{Z}}_p$-cover as an inverse system of cyclic branched $p$-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.

For real subfields K of a cyclotomic field ℚ(ς) we remove the tameness assumption at a given odd prime number l, which was needed in [11] in order to establish the equivalence of the Lifted Root Number Conjecture at l and an equivariant main conjecture of Iwasawa theory for abelian extensions of totally real number fields K/k.

We recover a result of Iwasawa on the p-adic logarithm of principal units of ℚ_p(ζ_pⁿ⁺¹) by studying the value at s = 1 of p-adic L-functions.

Let F be a global field of characteristic p > 0, 𝔽/F a Galois extension with and E/F a non-isotrivial elliptic curve. We study the behavior of Selmer groups Sel_E(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 = ∪ 𝔽_d where 𝔽_d/F is a -extension. We prove that Sel_E(𝔽_d)_p is a cofinitely generated ℤ_p[[Gal(ℤ_d/F)]]-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic L-function associated to E in ℤ_p[[Gal(ℤ/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.

In this paper, we study the p-adic behavior of Weil numbers in the cyclotomic ℤ_p-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.

Let F be a number field, abelian over ℚ, and fix a prime p ≠ 2. Consider the cyclotomic ℤ_p-extension F_∞/F and denote F_n the nth finite subfield and C_n its group of circular units as defined by Sinnott. Then the Galois groups G_m,n = Gal(F_m/F_n) act naturally on the C_m's (for any m ≥ n ≫ 0). We compute the Tate cohomology groups for i = -1,0 without assuming anything else neither on F nor on p.

Let p be a prime number, and let k be an imaginary quadratic number field in which p decomposes into two distinct primes 𝔭 and . Let k_∞ be the unique ℤ_p-extension of k which is unramified outside of 𝔭, and let K_∞ be a finite extension of k_∞, abelian over k. Following closely the ideas of Belliard in [1], we prove that in K_∞, the projective limit of the p-class group and the projective limit of units modulo elliptic units share the same μ-invariant and the same λ-invariant. We deduce that a version of the classical main conjecture, which is known to be true for p ∉ {2, 3}, holds also for p ∈ {2, 3} once we neglect the μ-invariants.

This paper proves a control theorem for the p-primary Selmer group of an abelian variety with respect to extensions of the form: Maximal pro-p extension of a number field unramified outside a finite set of primes R which does not include any primes dividing p in which another finite set of primes S splits completely. When the Galois group of the extension is not p-adic analytic, the control theorem gives information about p-ranks of Selmer and Tate–Shafarevich groups of the abelian variety. The paper also discusses what can be said in regards to a control theorem when the set R contains all the primes of the number field dividing p.

We construct p-adic L-functions associated to cuspidal Hilbert modular eigenforms of parallel weight two in certain dihedral or anticyclotomic extensions via the Jacquet–Langlands correspondence, generalizing works of Bertolini–Darmon, Vatsal and others. The construction given here is adelic, which allows us to deduce a precise interpolation formula from a Waldspurger-type theorem, as well as a formula for the dihedral μ-invariant. We also make a note of Howard's non-vanishing criterion for these p-adic L-functions, which can be used to reduce the associated Iwasawa main conjecture to a certain non-triviality criterion for families of p-adic L-functions.

In [A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory, in Proc. St. Petersburg Mathematical Society, Vol. 12, American Mathematical Society Translations, Series 2, Vol. 219 (American Mathematical Society, Providence, RI, 2006), pp. 1–85] Fukaya and Kato presented equivariant Tamagawa number conjectures that implied a very general (non-commutative) Iwasawa main conjecture for rather general motives. In this article we apply their methods to the case of one-parameter families of motives to derive a main conjecture for such families. On our way there we get some unconditional results on the variation of the (algebraic) λ- and μ-invariant. We focus on the results dealing with Selmer complexes instead of the more classical notion of Selmer groups. However, where possible we give the connection to the classical notions.

Let p be a prime number, and let 𝔹_p,n be the nth layer in the cyclotomic ℤ_p-extension of ℚ with ring of integers . In this paper we show that for each even integer m ≥ 2 and each prime number p the orders of the Quillen K-groups are unbounded, and that there are in fact infinitely many different prime numbers dividing the order of for some n.

In this paper we apply methods from the number field case of Perrin-Riou [20] and Zábrádi [32] in the function field setup. In ℤ_ℓ- and GL₂-cases (ℓ ≠ p), we prove algebraic functional equations of the Pontryagin dual of Selmer group which give further evidence of the main conjectures of Iwasawa theory. We also prove some parity conjectures in commutative and non-commutative cases. As a consequence, we also get results on the growth behavior of Selmer groups in commutative and non-commutative extension of function fields.

We prove a functional equation for the characteristic ideal of the "big" Selmer group 𝒳(𝒯_ℱ/F_cyc) associated to an ordinary Hida family of elliptic modular forms over the cyclotomic ℤ_p 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].

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚ_p, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

In this paper, we prove that there exist infinitely many elliptic curves whose Mordell–Weil ranks increase at the fourth layer of the cyclotomic ${\mathbb{Z}}_2$-extension of $\mathbb{Q}$.

Let $p$ be a prime number. For a number field $k$, let $\tilde{k}$ be the compositum of all ${\mathbb{Z}}_p$-extensions of $k$. Then Greenberg’s generalized conjecture (GGC) claims that the unramified Iwasawa module $X(\tilde{k})$ is pseudo-null over the Iwasawa algebra associated to the Galois group of $\tilde{k}/k$. In this paper, we establish sufficient conditions of GGC when $k$ is a complex cubic field and give many examples which satisfy the conditions with the help of computer programs.

For a totally real number field which is abelian over Q, in which an odd prime $p$ is totally split, and which verifies certain rather mild conditions on the cohomology of the circular units, we show that a weak form of Greenberg’s conjecture for $p$ holds true. This fills a gap — pointed out by René Schoof — in a previous proof (see Sur la conjecture faible de Greenberg dans le cas abélien $p$-décomposé, Int. J. Number Theory2(1) (2006) 49–64), and also extends the original result (semi-simplicity is no longer required).

Let $K$ be a number field and let $p$ be an odd rational prime. Let $K_{\infty }$ be a ${\mathbb{Z}}_p$-extension of $K$ and let $L_{\infty }$ be a finite extension of $K_{\infty }$, abelian over $K$. In this paper we extend the classical results, e.g. [16], relating characteristic ideal of the $\chi$-quotient of the projective limit of the ideal class groups to the $\chi$-quotient of the projective limit of units modulo Stark units, in the non-semi-simple case, for some $\overline{{\mathbb{Q}}_p}$-irreductible characters $\chi$ of $\mathrm{\text{Gal}}(L_{\infty }/K_{\infty })$. The proof essentially uses the theory of Euler systems.

We establish the freeness of certain anticyclotomic Selmer groups of modular forms. The freeness of these Selmer groups plays a key role in the Euler system arguments introduced by Bertolini and Darmon in their work on the anticyclotomic main conjecture for modular forms. In particular, our result fills some implicit gaps which appeared in generalizations of the Bertolini-Darmon result to the case where the associated residual representation is not minimally ramified. The removal of such a minimal ramification hypothesis is essential for applications involving congruences of modular forms.

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime and $K_{\infty }/K$ the anticyclotomic ${\mathbb{Z}}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. In this paper, we make a conjecture about the fine Selmer group over $K_{\infty }$. We also make a conjecture about the structure of the module of Heegner points in $E(K_{{𝔭}_{\infty }})/p$ where $K_{{𝔭}_{\infty }}$ is the union of the completions of the fields $K_n$ at a prime of $K_{\infty }$ above $p$. We prove that these conjectures are equivalent. When $E$ has supersingular reduction at $p$ we also show that these conjectures are equivalent to the conjecture in our earlier work. Assuming these conjectures when $E$ has supersingular reduction at $p$, we prove various results about the structure of the Selmer group over $K_{\infty }$.