On the local Artin conductor f<Subscript>Artin</Subscript> (Χ) of a character Χ of Gal(E/K) — II: Main results for the metabelian case

This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK _k. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren _G is the break in the upper ramification filtration ofG = Gal(E/K) defined by . Next, we study the basic properties of the idealf(E/K) inO _k in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]). After reviewing the Artin charactera _G : G → ℂ ofG := Gal(E/K) and Artin representationsA _g G → G →GL(V) corresponding toa _G : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5) where Χ_gr : G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then where Gal(E^ker(ρ)/E^ker(ρ)•) is any maximal abelian normal subgroup of Gal(E^ker(ρ)/K) containing Gal(E^ker(ρ) /K)′, and the break n_G/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]). We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A _G ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then Kazim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V _n denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)^• → ℂ^Χ satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂ_δ} = ω =(G/N) and where δ runs over R((G/N)^•/(G/N)), a fixed given complete system of representatives of (G/N)^•/(G/N), by declaring that ω₁ ∼ ω₂ if and only if ω₁ = ω _2,δ for some δ ∈ R((G/N)^•/(G/N)). Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.     

The 2-primary torsion on elliptic curves in the Z p -extensions of Q

Let E be an elliptic curve over Q and p a prime number. Denote by Q_p,∞ the Z_p-extension of Q. In this paper, we show that if p≠3, then where E(Q_p,∞)₍₂₎ is the 2-primary part of the group E(Q_p,∞) of Q_p,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q_2,∞)₍₂₎ in terms of E(Q)₍₂₎; in case p≥5 (or in case p=3 and E(Q)₍₂₎≠{O}), we show that E(Q_p,∞)₍₂₎=E(Q)₍₂₎.     

Analytic normal basis theorem

Let p be a prime number, ℚ _p the field of p-adic numbers, and a fixed algebraic closure of ℚ _p . We provide an analytic version of the normal basis theorem which holds for normal extensions of intermediate fields ℚ _p ⊆ K ⊆ L ⊆ .      

From the Mahler Conjecture to Gauss Linking Integrals

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product v(K)  = (Vol K)(Vol K°) of the volume of a symmetric convex body and its polar body K°. The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain is minimized when K is an ellipsoid. It implies the Mahler conjecture up to a factor of (π/4) ⁿ γ _n , where γ _n is a monotonic factor that begins at 4/π and converges to . This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of c ⁿ . The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K ^◇, with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere S ^n-1 and in hyperbolic space H ^n-1. Received: December 2006, Accepted: January 2007     

The Compositum of Wild Extensions of Local Fields of Prime Degree

In this paper we present a general view of the totally and wildly ramified extensions of degree p of a p-adic field K. Our method consists in deducing the properties of the set of all extensions of degree p of K from the study of the compositum of all its elements. We show that in fact is the maximal abelian extension of exponent p of F = F(K), where F is the compositum of all cyclic extensions of K of degree dividing p − 1. By our method, it is fairly simple to recover the distribution of the extensions of K of degree p (and also of their isomorphism classes) according to their discriminant.     

Square-free orders for CM elliptic curves modulo <Emphasis Type="Italic">p</Emphasis>

Let E be an elliptic curve defined over , of conductor N, and with complex multiplication. We prove unconditional and conditional asymptotic formulae for the number of ordinary primes , p ≤ x, for which the group of points of the reduction of E modulo p has square-free order. These results are related to the problem of finding an asymptotic formula for the number of primes p for which the group of points of E modulo p is cyclic, first studied by Serre (1977). They are also related to the stronger problem about primitive points on E modulo p, formulated by Lang and Trotter (Bull Am Math Soc 83:289–292, 1977), and the one about the primality of the order of E modulo p, formulated by Koblitz [Pacific J. Math. 131(1):157–165, 1988].     

The Poisson Kernel for Hardy Algebras

This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras H^∞(E), which we call Hardy algebras, and which are noncommutative generalizations of classical H^∞, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which “reproduces” the values, on , of the “functions” coming from H^∞(E). We present results that are natural generalizations of the Poisson integral formula. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the “model space” for the canonical model that can be attached to a point in the disc . We also connect our Poisson kernel to various “point evaluations” and to the idea of curvature. The first named author was supported in part by grants from the National Science Foundation and from the U.S.-Israel Binational Science Foundation. The second named author was supported in part by the U.S.-Israel Binational Science Foundation and by the B. and G. Greenberg Research Fund (Ottawa).     

Galois modular representation of associated Jacobians in the tamely ramified cyclic case

Let L/K be an ℓ-cyclic extension with Galois group G of algebraic function fields over an algebraically closed field k of characteristic p ≠  ℓ. In this paper, the -module structure of the ℓ-torsion of the Jacobian associated to L is explicitly determined.     

Galois representations,Mumford-Tate groups and good reduction of abelian varieties

Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent -rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of analytical nature whereas having good reduction is an arithmetical notion, linked to the ramification of Galois representations. This conjecture has been proved by Morita for particular abelian varieties with many endomorphisms (called of PEL type). Noot obtained results for abelian varieties without nontrivial endomorphisms (Mumfords example, not of PEL type). We give new results for abelian varieties not of PEL type.An erratum to this article can be found at     

A divergent Khintchine theorem in the real,complex, and <Emphasis Type="Italic">p</Emphasis>-adic fields

In this paper, we show that if the sum ∑_r=1^∞ Ψ(r) diverges, then the set of points (x, z, w) ∈ ℝ × ℂ × ℚ_p satisfying the inequalities , and for infinitely many integer polynomials P has full measure. With a special choice of parameters v _i and λ _i , i = 1, 2, 3, we can obtain all the theorems in the metric theory of transcendental numbers which were known in the real, complex, or p-adic fields separately.     

Galois actions on Q-curves and winding quotients

We prove two ``large images' results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p>13 not dividing d, provided that d is divisible by q (but d≠q) with q=2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing d D, where D= disc(K), provided that E is a semistable Q-curve. In both cases we make the (standard) assumptions that E does not have potentially good reduction at all primes p∤6 and that d is square free. The first author is supported by BFM2003-06092.     

On Galois groups of unramified pro-p extensions

Let p be an odd prime satisfying Vandiver’s conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z _p -extensions of Q(μ _p ) and the Galois group of the maximal unramified pro-p extension of Q . We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for to be abelian. The bound and the condition in the two results are given in terms of special values of a cup product pairing on cyclotomic p-units. We obtain in particular that, for p  <  1,000, Greenberg’s conjecture that X is pseudo-null holds and is in fact abelian.     

Icosahedral galois extensions and elliptic curves

Summary This paper is devoted to the last unsolved case of the Artin Conjecture in two dimensions. Given an irreducible 2-dimensional complex representation of the absolute Galois group of a number fieldF, the Artin Conjecture states that the associatedL-series is entire. The conjecture has been proved for all cases except the icosahedral one. In this paper we construct icosahedral representations of the absolute Galois group of ℚ(√5) by means of 5-torsion points of an elliptic curve defined over ℚ. We compute the L-series explicitely as an Euler product, giving algorithms for determining the factors at the difficult primes. We also prove a formula for the conductor of the elliptic representation. A feasible way of proving the Artin Conjecture in a given case is to construct a modular form whose L-series matches the one obtained from the representation. In this paper we obtain the following result: letρ be an elliptic Galois representation over ℚ(√5) of the type above, and letL(s, ρ) be the corresponding L-series. If there exists a Hilbert modular formf of weight one such thatL(s, f) ≡L(s, ρ) modulo a certain ideal above (√5), then the Artin conjecture is true forρ. This article was processed by the author using the LATEX style filecljour1m from Springer-Verlag.     

The Space of Bounded Spherical Functions on the Free 2-Step Nilpotent Lie Group

Let N be a connected and simply connected 2-step nilpotent Lie group and let K be a compact subgroup of Aut(N). We say that (K, N) is a Gelfand pair when the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In this paper we construct a one-to-one correspondence between the set Δ(K, N) of bounded spherical functions for such a Gelfand pair and a set of K-orbits in the dual of the Lie algebra for N. The construction involves an application of the Orbit Method to spherical representations of K ⋉ N. We conjecture that the correspondence is a homeomorphism. Our main result shows that this is the case for the Gelfand pair given by the action of the orthogonal group on the free 2-step nilpotent Lie group. In addition, we show how to embed the space Δ(K, N) for this example in a Euclidean space by taking eigenvalues for an explicit set of invariant differential operators. These results provide geometric models for the space of bounded spherical functions on the free 2-step group.     

The <Emphasis Type="Italic">K</Emphasis>(π,1)-conjecture for the affine braid groups

The complement of the hyperplane arrangement associated to the (complexified) action of a finite, real reflection group on ⁿ is known to be a K(,1) space for the corresponding Artin group $\Cal A$. A long-standing conjecture states that an analogous statement should hold for infinite reflection groups. In this paper we consider the case of a Euclidean reflection group of type à _n and its associated Artin group, the affine braid group $\tilde{\Cal A}$. Using the fact that $\tilde{\Cal A}$ can be embedded as a subgroup of a finite type Artin group, we prove a number of conjectures about this group. In particular, we construct a finite, $n$-dimensional K(,1)-space for $\tilde{\Cal A}$, and use it to prove the K(,1) conjecture for the associated hyperlane complement. In addition, we show that the affine braid groups are biautomatic and give an explicit biautomatic structure.     

Ranks of abelian varieties over infinite extensions of the rationals

Let S be an infinite set of rational primes and, for some p ∈ S, let be the compositum of all extensions unramified outside S of the form , for . If , let be the intersection of the fixed fields by , for i = 1, . . , n. We provide a wide family of elliptic curves such that the rank of is infinite for all n ≥ 0 and all , subject to the parity conjecture. Similarly, let be a polarized abelian variety, let K be a quadratic number field fixed by , let S be an infinite set of primes of and let be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over . We show that, under certain hypotheses, the -corank of sel _p ∞(A/F) is unbounded over finite extensions F/K contained in . As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.     

Endomorphisms of Jacobians of modular curves

Let be the modular curve associated to a congruence subgroup Γ of level N with , and let be its canonical model over . The main aim of this paper is to show that the endomorphism algebra of its Jacobian is generated by the Hecke operators T _p , with , together with the “degeneracy operators” D _M,d , D ^t _M,d , for . This uses the fundamental results of Ribet on the structure of together with a basic result on the classification of the irreducible modules of the algebra generated by these operators. Received: 18 December 2007     

On a problem of K. Mahler: Diophantine approximation and Cantor sets

Let K denote the middle third Cantor set and . Given a real, positive function ψ let denote the set of real numbers x in the unit interval for which there exist infinitely many such that |x − p/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for . One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in K—an assertion attributed to K. Mahler. Explicit examples of irrational numbers satisfying Mahler’s assertion are also given. Dedicated to Maurice Dodson on his retirement—finally!     

The limit as <Emphasis Type="Italic">p</Emphasis> → ∞ in a nonlocal <Emphasis Type="Italic">p</Emphasis>-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles

In this paper, we study the nonlocal ∞-Laplacian type diffusion equation obtained as the limit as p → ∞ to the nonlocal analogous to the p-Laplacian evolution,

Some Elements of Finite Order in K2Q

Let K2 be the Milnor functor and let Фn （x）∈ Q[X] be the n-th cyclotomic polynomial. Let Gn（Q） denote a subset consisting of elements of the form {a, Фn（a）}, where a ∈ Q^＊ and {, } denotes the Steinberg symbol in K2Q. J. Browkin proved that Gn（Q） is a subgroup of K2Q if n = 1,2, 3, 4 or 6 and conjectured that Gn（Q） is not a group for any other values of n. This conjecture was confirmed for n =2^T 3S or n = p^r, where p ≥ 5 is a prime number such that h（Q（ζp）） is not divisible by p. In this paper we confirm the conjecture for some n, where n is not of the above forms, more precisely, for n = 15, 21,33, 35, 60 or 105.