On the ideal class groups of the maximal real subfields of number fields with all roots of unity

In this paper, for a totally real number field k we show the ideal class group of k(∪_n>0μ_n)⁺ is trivial. We also study the p-component of the ideal class group of the cyclotomic Z_p-extension. Received January 15, 1998 / final version received July 31, 1998     

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.     

Integral and modular representations attached to the Selmer group of an elliptic curve with complex multiplication

Let E be an elliptic curve with complex multiplication over the ring of integers of an imaginary quadratic field K. Denote by p an odd prime that splits into in and by the unique -extension of K totally ramified above . It is well-known that the Selmer group attached to any finite extension of is analogous to the minus part of the p-class group of divisors of the cyclotomic - extensions of CM number fields. One of the most striking examples of this analogy is the existence of a translation formula à la Kida for the codimension of the Selmer group at the top of the tower. In this article we carry on the analogy with the presentations of results similar to those proven by Gold and Madan in the cyclotomic case (see [8]), which were the continuation of Kida's work. More precisely, we describe the -structure of the Selmer group when G is a cyclic group of order p or . In addition, we study the modular representation of G on the subgroup of points of order p of the Selmer group, when G is cyclic of order . Received December 3, 1997     

On congruence relations between the fundamental units of biquadratic fields

Given any distinct prime numbers p,q, and r satisfying certain simple congruence conditions, we display a congruence relation between the fundamental units for the biquadratic field , modulo a certain prime ideal of OK. This congruence in particular implies the validity of the equivariant Tamagawa number conjecture formulated by Burns and Flach for the pair (h⁰(SpecK),Z[Gal(K/Q)]).     

On the class semigroup of ZS-fields and Iwasawa invariants

Let p be a prime number. In [15], we studied the class semigroup of the ring of integers of the cyclotomic ${\mathbb{Z}}_p$-extension of the rationals. In this paper, we generalize the result to some ${\mathbb{Z}}_S$-extensions of number fields. Moreover, we investigate the relation between the class semigroup and Iwasawa invariants.     

Local units and Gauss sums

In this paper, we will determine the structure of a certain module which is related to the plus part of the ideal class groups in terms of the divisibility of Gauss sums in some local fields. This result is a generalization of a result of Iwasawa and the previous work of Ichimura and Hachimori.     

On some cubic or quartic algebraic units

Let ? be an algebraic unit such that rank of the unit group of the order Z[?] is equal to one. It is natural to ask whether ? is a fundamental unit of this order. To prove this result, we showed that it suffices to find explicit positive constants c₁, c₂ and c₃ such that for any such ? it holds that c₁c₂|?|?d??c₃|?|^2c₂, where d? denotes the absolute value of the discriminant of ?, i.e. of the discriminant of its minimal polynomial. We give a proof of this result, simpler than the original ones.     

Elliptic units and sign functions

In the first part of this paper we give a new definition of the elliptic analogue of Sinnott’s group of circular units. In this we essentially use the ideas discussed in Oukhaba (in Ann Inst Fourier, 55(33):753–772, 2005). In the second part of the paper we are interested in computing the index of this group of elliptic units. This question is closely related to the behaviour of the universal signed ordinary distributions introduced in loc. cit. Such distributions have a natural resolution discovered by Anderson. Consequently, we can apply Ouyang’s general index formula and the powerful Anderson’s theory of double complex to make the computations     

On the index of circular units in the full group of units of a compositum of quadratic fields

For a compositum of quadratic fields , where d₁,…,ds are square-free odd integers and , we study the group C of circular units of k. We construct a basis of C, compute the index of C in the full group of units of k and derive a lower bound for the divisibility of this index by a power of 2. These results give a lower bound for the divisibility of the class number of the maximal real subfield of k by a power of 2.     

On the p-adic L-function of Hilbert modular forms at supersingular primes

In this paper, I discuss the construction of the p-adic L-function attached to a Hilbert modular form f, supersingular or ordinary, which turns out to be the non-archimedean Mellin transform of an h-admissible measure. And h is explicitly given. As a special case, when the Fourier coefficient of f at p|p is zero, plus/minus p-adic L-functions are furthermore defined as bounded functions, and they interpolate special values of L(f,χ,s) for cyclotomic characters χ. This can be used to formulate Iwasawa main conjecture for supersingular elliptic curve defined over a totally real field.     

On the Birch and Swinnerton-Dyer conjecture for abelian varieties attached to Hilbert modular forms

In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character, then the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character regarded over arbitrary totally real number fields.     

Coleman maps for modular forms at supersingular primes over Lubin-Tate extensions

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Given an elliptic curve with supersingular reduction at an odd prime p, Iovita and Pollack have generalised results of Kobayashi to define even and odd Coleman maps at p over Lubin-Tate extensions given by a formal group of height 1. We generalise this construction to modular forms of higher weights.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=KQpsht0JaME.     

Enlargement of the group of circular units of a bicyclic field

We compute the index of a certain extension of Sinnott's group of circular units in the group of all units of a bicyclic field. From this index we obtain some divisibility properties for class numbers of bicyclic fields.     

On the Iwasawa invariants of certain non-cyclotomic ℤ2-extensions

Let k be an imaginary quadratic field in which the prime 2 splits. We consider the Iwasawa invariants of a certain non-cyclotomic ℤ₂-extension of k and give some sufficient conditions for the vanishing of λ- and μ-invariants.     

On elliptic units and p-adic Galois representations attached to elliptic curves

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 not ramified in K and write h_p for the class number of the ray class field of K of conductor p. Given an elliptic curve A/K with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that if then the image of a certain deformation of is “as big as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Z_p). The proof rests on the theory of Siegel functions and elliptic units as developed by Kubert, Lang and Robert.     

Exponents of the ideal class groups of CM number fields

Since class numbers of CM number fields of a given degree go to infinity with the absolute values of their discriminants, it is reasonable to ask whether the same conclusion still holds true for the exponents of their ideal class groups. We prove that under the assumption of the Generalized Riemann Hypothesis this is indeed the case. Received: 8 May 2001; in final form: 15 April 2002/Published online: 8 November 2002     

Elliptic units in K2

We present certain norm-compatible systems in K₂ of function fields of some CM elliptic curves. We demonstrate that these systems have some properties similar to elliptic units.     

Central derivatives of <Emphasis Type="Italic">L</Emphasis>-functions in Hida families

We prove a result of the following type: given a Hida family of modular forms, if there exists a weight two form in the family whose L-function vanishes to exact order one at s = 1, then all but finitely many weight two forms in the family enjoy this same property. The analogous result for order of vanishing zero is also true, and is an easy consequence of the existence of the Mazur–Kitagawa two-variable p-adic L-function. This research was supported in part by NSF grant DMS-0556174.     

The class number one problem for some totally complex quartic number fields

We prove that there are 95 non-isomorphic totally complex quartic fields whose rings of algebraic integers are generated by an algebraic unit and whose class numbers are equal to 1. Moreover, we prove Louboutin's Conjecture according to which a totally complex quartic unit εu generally generates the unit group of the quartic order Z[εu].     

On the Galois cohomology of ideal class groups

We use étale cohomology to prove some explicit results on the Galois cohomology of ideal class groups. Received: 3 May 2007