Minus Part Research Articles

Minimal resolutions of Iwasawa modules

In this paper, we study the module-theoretic structure of classical Iwasawa modules. More precisely, for a finite abelian p-extension K/k of totally real fields and the cyclotomic Zp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_p$$\\end{document}-extension K∞/K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{\\infty }/K$$\\end{document}, we consider XK∞,S=Gal(MK∞,S/K∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{K_{\\infty },S}={{\\,\ extrm{Gal}\\,}}(M_{K_{\\infty },S}/K_{\\infty })$$\\end{document} where S is a finite set of places of k containing all ramifying places in K∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{\\infty }$$\\end{document} and archimedean places, and MK∞,S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{K_{\\infty },S}$$\\end{document} is the maximal abelian pro-p-extension of K∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{\\infty }$$\\end{document} unramified outside S. We give lower and upper bounds of the minimal numbers of generators and of relations of XK∞,S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{K_{\\infty },S}$$\\end{document} as a Zp[[Gal(K∞/k)]]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_p[[{{\\,\ extrm{Gal}\\,}}(K_{\\infty }/k)]]$$\\end{document}-module, using the p-rank of Gal(K/k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Gal}\\,}}(K/k)$$\\end{document}. This result explains the complexity of XK∞,S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{K_{\\infty },S}$$\\end{document} as a Zp[[Gal(K∞/k)]]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_p[[{{\\,\ extrm{Gal}\\,}}(K_{\\infty }/k)]]$$\\end{document}-module when the p-rank of Gal(K/k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Gal}\\,}}(K/k)$$\\end{document} is large. Moreover, we prove an analogous theorem in the setting that K/k is non-abelian. We also study the Iwasawa adjoint of XK∞,S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{K_{\\infty },S}$$\\end{document}, and the minus part of the unramified Iwasawa module for a CM-extension. In order to prove these theorems, we systematically study the minimal resolutions of XK∞,S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{K_{\\infty },S}$$\\end{document}.

On the Brumer--Stark conjecture

Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer–Stark conjecture states that the Stickelberger element $\Theta_{S,T}^{H/F}$ annihilates the $T$-smoothed class group $\mathrm{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with ${\bf Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $\mathrm{Cl}^T(H) \otimes {\bf Z}[1/2]$ in terms of Stickelberger elements. We also show that this stronger result implies Rubin's higher rank version of the Brumer--Stark conjecture, again away from 2. Our technique is a generalization of Ribet's method, building upon on our earlier work on the Gross--Stark conjecture. Here we work with group ring valued Hilbert modular forms as introduced by Wiles. A key aspect of our approach is the construction of congruences between cusp forms and Eisenstein series that are stronger than usually expected, arising as shadows of the trivial zeroes of $p$-adic $L$-functions. These stronger congruences are essential to proving that the cohomology classes we construct are unramified at $p$.

Notes on the dual of the ideal class groups of CM-fields

In this paper, for a CM abelian extension K/k of number fields, we propose a conjecture which describes completely the Fitting ideal of the minus part of the Pontryagin dual of the T-ray class group of K for a set T of primes as a Gal(K/k)-module. Here, we emphasize that we consider the full class group, and do not throw away the ramifying primes (the object we study is not the quotient of the class group by the subgroup generated by the classes of ramifying primes). We prove that our conjecture is a consequence of the equivariant Tamagawa number conjecture, and also prove the Iwasawa theoretic version of our conjecture.

Extreme values for iterated integrals of the logarithm of the Riemann zeta-function

We give an approximate formula for the measure of extreme values of the logarithm of the Riemann zeta-function and its iterated integrals. The result recovers the unconditional best result for the minus part of the $\Omega $-result for $S_{1}(t)$ due to T

On the rational motivic homotopy category

We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of constructible objects, and Grothendieck-Verdier duality for SH_Q. Next, we prove that SH_Q is canonically SL-oriented; we compare SH_Q with the category of rational Milnor-Witt motives; and we relate the rational bivariant A^1-theory to Chow-Witt groups. These results are derived from analogous statements for the minus part of SH[1/2].

New criteria for Vandiver’s conjecture using Gauss sums – Heuristics and numerical experiments

The link between Vandiver's conjecture and Gauss sums is well known since the papers of Iwasawa (1975), Thaine (1995-1999) and Angl{\`e}s-Nuccio (2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver's conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian p-ramified pro-p-extension of the pth cyclotomic field (1984). Then we provide a specific use of Gauss sums of characters of order p of F\_ell^x and prove new criteria for Vandiver's conjecture to hold (Theorem 1.2(a) using both the sets of exponents of p-irregularity and of p-primarity of suitable twists of the Gauss sums, and Theorem 1.2(b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in \S5.2 new heuristics showing that any counterexample to the conjecture leads to excessive constraints modulo p on the above twists as ell varies and suggests analytical approaches to evidence. We perform numerical experiments to strengthen our arguments in direction of the very probable truth of Vandiver's conjecture. All the calculations are given with their PARI/GP programs.

Reconstructing rational stable motivic homotopy theory

Using a recent computation of the rational minus part of$SH(k)$by Ananyevskiy, Levine and Panin, a theorem of Cisinski and Déglise and a version of the Röndigs and Østvær theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor–Witt correspondences in the sense of Calmès and Fasel.

On the growth of the (S,{2})-refined class number

We give a new proof of the fact that the growth (with respect to n) of the p-adic valuation of hn,2− is linear, where hn,2− denotes the minus part of the (S,{2})-refined class number of the cyclotomic field Q(μpn+1), as defined by Hu and Kim in [1]. As a consequence of our proof, we obtain an explicit relation between the p-adic valuation of hn,2− and the p-adic valuation of hn−, the minus part of the class number hn of the cyclotomic field Q(μpn+1).

Capitulation in the cyclotomic ℤ2 extension of CM number fields

AbstractLet 𝕂n be the intermediate steps in the cyclotomic ℤp-extension of a CM number field 𝕂. For p ⧧ 2 the minus part of the p-class groups is given by An− = $\frac{1}{2}$(1 − j)An. We will give a new definition of the minus part for p = 2 and will prove that there is no finite submodule in lim∞←nAn−. Furthermore we will show that μ = 0 if and only if μ− = 0 in this new definition.

Abstract $\ell$-Adic 1-Motives and Tate's Canonical Class for Number Fields

In an earlier paper we constructed a new class of Iwasawa modules as \ell -adic realizations of waht we called abstract \ell -adic 1-motives in the number field setting. We proved in loc. cit. that the new Iwasawa modules satisfy an equivariant main conjecture. In this paper we link the new modules to the \ell -adified Tate canonical class, defined by Tate in 1960 [Ta1] and give an explicit construction of (the minus part og) \ell -adic Tate sequences for any Galois CM extension K/k of an arbitrary totally real number field k . These explicit constructions are significant and useful in their own right but also due to their applications (via results in [GP2]) to a proof of the minus part of the far reaching Equivariant Tamagawa Number Conjecture for the Artin motive associated to the Galois extension K/k .

Witt sheaves and the η-inverted sphere spectrum

Ananyevskiy has recently computed the stable operations and cooperations of rational Witt theory. These computations enable us to show a motivic analog of Serre's finiteness result. Theorem.Let k be a field of characteristic different from two. Then π n A 1 ( S k − ) ∗ is torsion for n > 0 . As an application, we define a category of Witt motives and show that rationally this category is equivalent to the minus part of SH ( k ) Q .

On the Stickelberger ideal of a quadratic twist of a cyclotomic field

A system of generators of the minus part of the Stickelberger ideal of a quadratic twist of a cyclotomic field is obtained.

On higher Fitting ideals of Iwasawa modules of ideal class groups over imaginary quadratic fields and Euler systems of elliptic units

Kurihara described all higher Fitting ideals of the minus part of Iwasawa modules of ideal class groups over totally real fields by using Stickelberger elements and Euler systems of “Gauss sums.” In this paper, we obtain some partial results for elliptic units which are analogues of his result. By using Kolyvagin derivative classes of Euler systems of elliptic units, we construct some ideals Ci,χell of Iwasawa algebras and prove that they give “upper bounds” of higher Fitting ideals of one- and two-variable Iwasawa modules of ideal class groups over imaginary quadratic fields.

The Reciprocity Conjecture of Khare and Wintenberger

We prove a strengthening of the “reciprocity conjecture” of Khare and Wintenberger. The input to the original conjecture is an odd prime p, a CM number field F containing the pth roots of unity, and a pair (q1, q2) of primes of the maximal totally real subfield F+ of F that are inert in the cyclotomic Zp-extension F + ∞/F +. In analogy to a statement about generalized Jacobians of curves, the conjecture asserts the equality of two procyclic subgroups of the Galois group of the maximal pro-p extension M of F+ ∞ that is unramified outside p and abelian over F+. The first is the intersection with Gal(M/F+ ∞) of the closed subgroup of Gal(M/F+) generated by the Frobenius elements of q1 and q2. The second is generated by the class of an exact sequence defining the minus part of the p-part of the ray class group of F∞ of conductor q1q2.

On the equivariant Tamagawa number conjecture in tame CM-extensions, II

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.

The ideal class groups of dihedral extensions over imaginary quadratic fields and the special values of the Artin L-function

We study the relation between the minus part of the p-class subgroup of a dihedral extension over an imaginary quadratic field and the special value of the Artin L-function at 0.

Stickelberger ideals and Fitting ideals of class groups for abelian number fields

In this paper, we determine completely the initial Fitting ideal of the minus part of the ideal class group of an abelian number field over Q up to the 2-component. This answers an open question of Mazur and Wiles (Invent Math 76:179–330, 1984) up to the 2-component, and proves Conjecture 0.1 in Kurihara (J Reine Angew Math 561:39–86, 2003). We also study Brumer’s conjecture and prove a stronger version for a CM-field, assuming certain conditions, in particular on the Galois group.

On the equivariant Tamagawa number conjecture in tame CM-extensions

Let L/K be a finite Galois CM-extension with Galois group G. The Equivariant Tamagawa Number Conjecture (ETNC) for the pair $${(h^0({\rm Spec} (L))(0), {\mathbb Z}G)}$$ naturally decomposes into p-parts, where p runs over all rational primes. If p is odd, these p-parts in turn decompose into a plus and a minus part. Let L/K be tame above p. We show that a certain ray class group of L defines an element in $${K_0({\mathbb Z}{_p}G_-, \mathbb Q_p)}$$ which is determined by a corresponding Stickelberger element if and only if the minus part of the ETNC at p holds. For this we use the Lifted Root Number Conjecture for small sets of places which is equivalent to the ETNC in the number field case. For abelian G, we show that the minus part of the ETNC at p implies the Strong Brumer–Stark Conjecture at p. We prove the minus part of the ETNC at p for almost all primes p.

On Jacobi sums in Q(ζp)

We study the arithmetic of Jacobi Sums Q(\zeta_p) and we show that they play a somehow parallel role in the minus part as to Units in the plus part of the Class Field Theory of Cyclotomic fields.

On a conjecture concerning minus parts in the style of Gross

This paper is devoted to Gross's conjecture on tori over the base field Q. We call it the Minus Conjecture, since it involves a regulator built from units in the minus part. We recall and develop its relation to a conjecture of Burns, which is now known to hold generally in the absolutely abelian setting; however in many situations it is not clear at all how one should deduce the Minus Conjecture from it. We prove a somewhat weaker statement (order of vanishing) rather generally, and we give a proof of the Minus Conjecture for some specific classes of absolutely abelian extensions K/Q, for which K^+/Q is l-elementary and ramified in at most two primes. The field K is assumed to be of the form FK^+ where F is an arbitrary imaginary quadratic field. Our methods involve a good deal of explicit calculation; among other things, we use p-adic Gamma-functions and the Gross-Koblitz formula.