Ray class fields generated by torsion points of certain elliptic curves

We first normalize the derivative Weierstrass ???-function appearing in the Weierstrass equations which give rise to analytic parametrizations of elliptic curves, by the Dedekind ??-function. And, by making use of this normalization of ???, we associate a certain elliptic curve to a given imaginary quadratic field K and then generate an infinite family of ray class fields over K by adjoining to K torsion points of such an elliptic curve. We further construct some ray class invariants of imaginary quadratic fields by utilizing singular values of the normalization of ???, as the y-coordinate in the Weierstrass equation of this elliptic curve, which would be a partial result towards the Lang?CSchertz conjecture of constructing ray class fields over K by means of the Siegel?CRamachandra invariant.     

Torsion in Mordell-Weil groups of Fermat Jacobians

We study the torsion in the Mordell-Weil group of the Jacobian of the Fermat curve of exponent p over the cyclotomic field obtained by adjoining a primitive p-th root of 1 to Q. We show that for all (except possibly one) proper subfields of this cyclotomic field, the torsion parts of the corresponding Mordell-Weil groups are elementary abelian p-groups.     

Borne uniforme pour les homothéties dans l?image de Galois associée aux courbes elliptiques

Let K be a fixed number field and GK its absolute Galois group. We give a bound C(K), depending only on the degree, the class number and the discriminant of K, such that for any elliptic curve E defined over K and any prime number p strictly larger than C(K), the image of the representation of GK attached to the p-torsion points of E contains a subgroup of homotheties of index smaller than 12.     

Image of the group ring of the Galois representation associated to Drinfeld modules

Let φ be a Drinfeld A-module of arbitrary rank and arbitrary characteristic over a finitely generated field K, and set GK=Gal(Ksep/K). Let E=EndK(φ). We show that for almost all primes p of A the image of the group ring A[GK] in EndA(Tp(φ)) is the commutant of E. In the special case E=A it follows that the representation of GK on the p-torsion points φ[p](Ksep) of φ is absolutely irreducible for almost all p.     

Potentially good reduction of Barsotti-Tate groups

Let R be a complete discrete valuation ring of mixed characteristic (0,p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K^′ of K. We prove that there exists a constant c?2 which depends on the absolute ramification index e(K^′/Qp) and the height of G such that G has good reduction over K if and only if G[pc] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's “p-adic Néron-Ogg-Shafarevich criterion” to finite level. We use methods that can be generalized to study semi-stable p-adic Galois representations with general Hodge-Tate weights, and in particular leads to a proof of a conjecture of Fontaine and gives a constant c as above that is independent of the height of G.     

Harmonic analysis of SL(2) over a locally compact field

We carry over the pioneer work of Kunze and Stein concerning representation theory and harmonic analysis on SL(2, R) to the group G = SL(2, K), K a locally compact totally disconnected nondiscrete field. The main result is that convolution by an L^p(G) function, 1 ? p < 2, is a bounded operator on L²(G). To accomplish this result we develop the appropriate estimates (which depend upon the work of Sally et al.) that enable us to apply the Kunze and Stein interpolation theory to the Fourier-Laplace transform for the group G. Best possible estimates are obtained.     

Wild tame-by-cyclic extensions

Suppose G is a semi-direct product of the form Z/pⁿ?Z/m where p is prime and m is relatively prime to p. Suppose K is a complete discrete valuation field of characteristic p>0 with algebraically closed residue field. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration, and we calculate its dimension in terms of the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p³.     

The Galois representations associated to a Drinfeld module in special characteristic—III: Image of the group ring

Let K be a finitely generated field of transcendence degree 1 over a finite field, and set G_K?Gal(K^sep/K). Let φ be a Drinfeld A-module over K in special characteristic. Set E?End_K(φ) and let Z be its center. We show that for almost all primes p of A, the image of the group ring A_p[G_K] in End_A(T_p(φ)) is the commutant of E. Thus, for almost all p it is a full matrix ring over Z⊗_AA_p. In the special case E=A it follows that the representation of G_K on the p-torsion points φ[p] is absolutely irreducible for almost all p.     

Global construction of associated orders in complex multiplication

Let E/L be an elliptic curve defined over a number field L with complex multiplication by the ring of integers of a quadratic imaginary number field K⊆L. We fix a prime ideal p in K and assume E to have good reduction modulo p. We consider cosets of the form P+G_m, where G_m denotes the group of points of order p^m and P a point of order p^r+m, , which can be viewed as the set of all “p^m-th roots” of some point in G_r.In analogy to the Kummer theory such a coset gives rise to the definition of an order in an algebra M_P and to an associated order A defined below by its local components. These objects naturally come into play in the Galois module structure of rings of integers in ray class fields over K. It is the aim of this article to construct global generators both for and A as algebras over the ring of integers of L. For the convenience of the reader we also include from (J. Number Theory 77 (1999) 97) a simple construction of Galois generators for over A. We thereby show that these generators also fit in the setting of this article that is more general than in [Sch4]. The main results in Theorems 3, 6 and 7 are obtained assuming the base field L to contain “enough” torsion points of E.     

On local newforms for unramified U(2, 1)

Let G be the unramified unitary group in three variables defined over a p-adic field F with p ≠ 2. In this paper, we investigate local newforms for irreducible admissible representations of G. We introduce a family of open compact subgroups {K _n } _n≥0 of G to define the local newforms for representations of G as the K _n -fixed vectors. We prove the existence of local newforms for generic representations and the multiplicity one property of the local newforms for admissible representations.     

On Galois structure of the integers in elementary abelian extensions of local number fields

Let p be an odd prime number and k a finite extension of Qp. Let K/k be a totally ramified elementary abelian Kummer extension of degree p² with Galois group G. We determine the isomorphism class of the ring of integers in K as an oG-module under some assumptions. The obtained results imply there exist extensions whose rings are ZpG-isomorphic but not oG-isomorphic, where Zp is the ring of p-adic integers. Moreover we obtain conditions that the rings of integers are free over the associated orders and give extensions whose rings are not free.     

On a filtered multiplicative basis of group algebras

Let K be a field of characteristic p and G a nonabelian metacyclic finite p-group. We give an explicit list of all metacyclic p-groups G, such that the group algebra KG over a field of characteristic p has a filtered multiplicative K-basis. We also present an example of a non-metacyclic 2-group G, such that the group algebra KG over any field of characteristic 2 has a filtered multiplicative K-basis.     

Group Rings with Solvable Unit Groups of Minimal Derived Length

Let F be a field of characteristic p > 2 and G a nonabelian nilpotent group containing elements of order p. Write F G for the group ring. The conditions under which the unit group ??(F G) is solvable are known, but only a few results have been proved concerning its derived length. It has been established that if G is torsion, the minimum derived length is ?log₂(p + 1)?, and this minimum occurs if and only if |G′| = p. In the present note, we show that the same holds if G has elements of infinite order.     

The influence of minimal subgroups of focal subgroups on the structure of finite groups

A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G such that G=HK and H∩K=1. In this paper, it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of the p-focal subgroup of G is complemented in N_G(P), where P is a Sylow p-subgroup of G. As some applications, some interesting results related with complemented minimal subgroups of focal subgroups are obtained.     

Lie *-Nilpotence of Group Rings

Let KG be the group ring of a group G over a field K. Let * be an involution of a group G extended linearly to the group ring KG. Suppose that G is a torsion group without 2-elements and K is a field with characteristic different from 2. We prove that KG is Lie *-nilpotent if and only if KG is Lie nilpotent.     

A note on Ramsey multiplicity

If G₁ and G₂ are graphs and the Ramsey number r(G₁, G₂) = p, then the fewest number of G₁ in G and G₂ in ? (G complement) that occur in a graph G on p points is called the Ramsey multiplicity and denoted R(G₁, G₂). In [2, 3] the diagonal (i.e. G₁ = G₂) Ramsey multiplicities are derived for graphs on 3 and 4 points, with the exception of K₄. In this note an upper bound is established for R(K_s, K₁). Specifically, we show that R(K₄, K₄) ? 12.     

Quasi-co-local subgroups of abelian groups

Let {0}≠K be a subgroup of the abelian group G. In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Appl. Math., vol. 249, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 29-37], K was called a co-local (cl) subgroup of G if is naturally isomorphic to . We generalize this notion to the quasi-category of abelian groups and call the subgroup K≠{0} of G a quasi-co-local (qcl) subgroup of G if is naturally isomorphic to . We show that qcl subgroups behave quite differently from cl subgroups. For example, while cl subgroups K are pure in G, i.e. G/K is torsion-free if G is torsion-free, any reduced torsion group T can be the torsion subgroup t(G/K) of G/K where G is torsion-free and K is a qcl subgroup of G.     

On unitary deformations of smooth modular representations

Let G be a lo/cally ? _p -analytic group and K a finite extension of ? _p with residue field k. Adapting a strategy of B. Mazur (cf. [Maz89]) we use deformation theory to study the possible liftings of a given smooth G-representation ρ over k to unitary G-Banach space representations over K. The main result proves the existence of a universal deformation space in case ρ admits only scalar endomorphisms. As an application we let G = GL₂(? _p ) and compute the fibers of the reduction map in principal series representations.     

Deformations of locally abelian Galois representations and unramified extensions

We study the deformation theory of Galois representations whose restriction to every decomposition subgroup is abelian. As an application, we construct unramified non-solvable extensions over the field obtained by adjoining all p-power roots of unity to the field of rational numbers.