Sur les représentations modulaires de degré 2 de Gal(Q¯/Q)

Given a continuous homomorphism 

$${G_{Q,S}}G{L_2}\left( {{Z_p}} \right)$$

where Gℚ,S is the Galois group of the maximal algebraic extension of ℚ unramified outside the finite set S of primes of ℚ, the motivating problem of this paper is to study, in a systematic way, the possible liftings of ρ to p-adic representations, 

$${G_{Q,S}}\mathop \to \limits^{{\rho _o}} G{L_2}\left( {{Z_p}} \right).$$

.

Deforming Galois Representations

The inverse problem of Galois theory asks whether an arbitrary finite group G can be realized as Gal(K/Q) for some Galois extension K of Q. When such a realization has been given for a particular G then a natural sequel is to find arithmetical realizations of the irreducible representations of G. One possibility is to ask for realizations in the Mordell-Weil groups of elliptic curves over Q: Given an irreducible complex representation τ of Gal(K/Q), does there exist an elliptic curve E over Q such that τ occurs in the natural representation of Gal(K/Q) on C ⊗Z E(K)? The present paper does not attempt to investigate this question directly. Instead we adopt Greenberg’s point of view in his remarks on nonabelian Iwasawa theory [5] and consider a related question about root numbers. Let ρE denote the representation of Gal(K/Q) on C ⊗Z E(K) and 〈τ, ρE〉 the multiplicity of τ in ρE , and write L(E, τ, s) for the tensor product L-function associated to E and τ . The conjectures of Birch-Swinnerton-Dyer and Deligne-Gross imply that

/pdf/galois-theory-elliptic-curves-and-root-numbers-1bszlcf7gp.pdf

Galois theory, elliptic curves, and root numbers

Computing Galois groups over the rationals

Publisher Summary This chapter discusses the composition of functions. These problems arise in three disciplines: propositional calculus of logics, synthesis of automata, and universal algebras. They are connected with the theory of relations. The chapter focuses on the functions that can be constructed by composition from a given set of functions, or equivalently, the functions that can be realized by combinatorial circuits, built from gates of given types. The chapter discusses some results moving in one direction; the choice of material is personal and biased. It has been shown that a finite loop is complete with constants if it is simple and not an elementary abelian p -group. A finite group is complete with constants if it is a simple non-abelian group. A finite module is complete with constants if it is simple and its group is not an elementary abelian p -group.

completeness properties of multiple-valued logic algebras

Polynomials with galois group psl(2,7)

In this paper by means of simple models it is shown that the five set-theoretical axioms of Extensionalit y, Replacement, Power-Set, SumSet, and Choice are consistent and that each of the axioms of Extensionality, Replacement, and Power-Set is independent from the remaining four axioms. Although the above results are known and can be found in part in [l], it is believed that this paper has some expository merits. The abovementioned axioms are five of the six axioms of the ZermeloFraenkel Theory of Sets, the sixth being Axiom of Infinity. We accept that every element of a set is a set and (without borrowing "=" from Logic) we define two sets u and υ to be equal if and only if they possess the same elements, in which case we write u = v. With this in mind, the six axioms of the Zermelo-Fraenkel Theory of Sets can be stated as follows [2]:

Samuel LaMacchia

Papers

Polynomials with galois group psl(2,7)

On the consistency and independence of some set-theoretical axioms

Some generalized sheffer functions

Examples of generalized Sheffer functions

Some consequences of the axiom of power-set