Lifting Galois representations of number fields

Let F be a number field. Given a continuous representation with insoluble image we show, under moderate assumptions at primes dividing ?∞, that for some continuous representation which is unramified outside finitely many primes. We also establish level lowering when F is totally real, is the reduction of a nearly ordinary Hilbert modular form and is distinguished at ?.     

Galois module structure of unramified covers

We use the theory of n-cubic structures to study the Galois module structure of the coherent cohomology groups of unramified Galois covers of varieties over the integers. Assuming that all the Sylow subgroups of the covering group are abelian, we show that the invariant that measures the obstruction to the existence of a “virtual normal integral basis” is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover. Partially supported by NSF grants # DMS05-01049 and # DMS01-11298 (via the Institute for Advanced Study).     

Multiplicities of modp Galois representations

We study the multiplicity with which 2-dimensional modp Galois representations occur in Jacobians of modular curves.     

On unramified Galois extensions constructed using Galois representations

 For a field k, We denote the maximal abelian extension of k by k _ab and (K _ab ^r−1 _ab by k _ab ^r . In this paper, unramified Galois extensions over k _ab ^r are constructed using Galois representations of arbitrary dimension with larger coefficient rings. Received: 31 August 2001 / Revised version: 22 March 2002 Mathematics Subject Classification (2000): 11R21     

Galois module structure for dihedral extensions of degree 8: Realizable classes over the group ring

Let k be a number field with ring of integers O_k, and let Γ be the dihedral group of order 8. For each tame Galois extension N/k with group isomorphic to Γ, the ring of integers O_N of N determines a class in the locally free class group Cl(O_k[Γ]). We show that the set of classes in Cl(O_k[Γ]) realized in this way is the kernel of the augmentation homomorphism from Cl(O_k[Γ]) to the ideal class group Cl(O_k), provided that the ray class group of O_k for the modulus 4O_k has odd order. This refines a result of the second-named author (J. Algebra 223 (2000) 367-378) on Galois module structure over a maximal order in k[Γ].     

On Galois representations for abelian varieties with complex and real multiplications

In this paper, we study the image of l-adic representations coming from Tate module of an abelian variety defined over a number field. We treat abelian varieties with complex and real multiplications. We verify the Mumford-Tate conjecture for a new class of abelian varieties with real multiplication.     

A note on integral bases of unramified cyclic extensions of prime degree. II

Let p be a prime number and K a number field containing a primitive p-th root of unity. It is known that an unramified cyclic extension L/K of degree p has a power integral basis if it has a normal integral basis. We show that for all p, the converse is not true in general. Received: 18 July 2000 / Revised version: 18 October 2000     

Multiplicities of mod p Galois representations

We study the multiplicity with which 2-dimensional mod p Galois representations occur in Jacobians of modular curves. Received: 26 February 1997 / Revised version: 2 July 1997     

Bernoulli-Hurwitz numbers, Wieferich primes and Galois representations

Let K be a quadratic imaginary number field with discriminant DK≠−3,−4 and class number one. Fix a prime p?7 which is unramified in K. Given an elliptic curve A/Q 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, for all but finitely many inert primes p, the image of a certain deformation of is “as large as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Zp). If p splits in K, then the same result holds as long as a certain Bernoulli-Hurwitz number is a p-adic unit which, in turn, is equivalent to a prime ideal not being a Wieferich place. The proof rests on the theory of elliptic units of Robert and Kubert-Lang, and on the two-variable main conjecture of Iwasawa theory for quadratic imaginary fields.     

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.     

Realizable classes of tetrahedral extensions

Let k be a number field and O_k its ring of integers. Let Γ be the alternating group A₄. Let be a maximal O_k-order in k[Γ] containing O_k[Γ] and its class group. We denote by the set of realizable classes, that is the set of classes such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Γ, for which the class of is equal to c, where O_N is the ring of integers of N. In this article we determine and we prove that it is a subgroup of provided that k and the 3rd cyclotomic field of are linearly disjoint, and the class number of k is odd.     

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.     

Normal integral bases for cyclic Kummer extensions

Let K/F be a Kummer cyclic extension of number fields. In the case when the degree is a prime number, Gómez Ayala gave an explicit criterion for the existence of a normal integral basis. More recently Ichimura proposed a generalization of that result for cyclic extensions of arbitrary degree, but we have found that Ichimura’s result is incorrect. In this paper we present a counter-example to Ichimura’s result as well as the correct generalization of Gómez Ayala’s result.     

Normal integral bases in quadratic and cyclic cubic extensions of quadratic fields

Let K be a number field and let G be a finite abelian group. We call K a Hilbert-Speiser field of type G if, and only if, every tamely ramified normal extension L/K with Galois group isomorphic to G has a normal integral basis. Now let C₂ and C₃ denote the cyclic groups of order 2 and 3, respectively. Firstly, we show that among all imaginary quadratic fields, there are exactly three Hilbert-Speiser fields of type $C_{2}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-1, -3, -7\}$. Secondly, we give some necessary and sufficient conditions for a real quadratic field $K = \mathbb{Q}(\sqrt {m})$ to be a Hilbert-Speiser field of type C₂. These conditions are in terms of the congruence class of m modulo 4 or 8, the fundamental unit of K, and the class number of K. Finally, we show that among all quadratic number fields, there are exactly eight Hilbert-Speiser fields of type $C_{3}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-11,-3, -2, 2, 5, 17, 41, 89\}$.Received: 2 April 2002     

The Galois group of the Eisenstein polynomial X 5 + aX + a

Let p be a rational prime and let a be an integer which is divisible by p exactly to the first power. Then the Galois group of the Eisenstein polynomial f = X ^p + aX + a is known to be either the full symmetric group S ^p or the affine group A(1, p), and it is conjectured that always G = S _p . In this note we settle this conjecture for p = 5 and, answering a question by J.-P. Serre, we show that this does not carry over when replacing the integer a by some rational number with 5-adic valuation equal to 1. Received: 6 June 2007     

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.